Abstract
We construct special cycles on the moduli stack of hermitian shtukas. We prove an identity between (1) the \(r^{\mathrm{th}}\) central derivative of nonsingular Fourier coefficients of a normalized Siegel–Eisenstein series, and (2) the degree of special cycles of “virtual dimension 0” on the moduli stack of hermitian shtukas with \(r\) legs. This may be viewed as a functionfield analogue of the KudlaRapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series.
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1 Introduction
The classical Siegel–Weil formula ([22, 26]) relates the special values of Siegel–Eisenstein series on the symplectic group (resp. the unitary group) to theta functions, which are generating series of representation numbers of quadratic (resp. Hermitian) forms over number fields. In particular, by exploiting the factorization of the nonsingular Fourier coefficients into a product of local terms, one arrives at Siegel’s formula for representation numbers of global quadratic or Hermitian forms in terms of local representation densities.
In [11] Kudla began to study an arithmetic version of the Siegel–Weil formula and he discovered a relation between an “arithmetic theta function” — a generating series of arithmetic cycles on an integral model of a Shimura curve—and the first central derivative of a Siegel–Eisenstein series on \(\operatorname{Sp}_{4}\). In a series of papers, Kudla and Rapoport developed this paradigm by defining the nonsingular terms of a generating series of special cycles on suitable integral models of Shimura varieties for \(\operatorname{SO}(n1,2)\) with \(n\leq 4\) and for all \(\mathrm {U}(n1,1)\). Of particular relevance to our paper, in [12, 13] Kudla and Rapoport defined the soughtafter special cycles on integral models of unitary Shimura varieties, now known as Kudla–Rapoport cycles, and conjectured a relationship to the nonsingular Fourier coefficients of the central derivative of the Siegel–Eisenstein series. Their conjecture has been recently proved by Li and one of us [18]; we also refer to the introduction of [18] for a more detailed account of recent advances in some other related directions (see also [19] for the orthogonal analog). With the Kudla–Rapoport conjecture proved in [18] and its archimedean counterpart proved by Liu [20] and independently by Garcia and Sankaran [6] as some of the key ingredients, Li and Liu [16, 17] have recently proved an arithmetic Rallis inner product formula relating the height pairing of the generating series to the first derivative of Lfunctions for unitary groups, from which they deduced cases of Beilinson–Bloch conjecture for certain high rank motives.
In this paper we study a function field analogue of the arithmetic Siegel–Weil formula, for unitary groups. In particular, we will construct special cycles on the moduli space of hermitian shtukas with arbitrary number of legs. Then we formulate and prove the analogue of the KudlaRapoport conjecture for derivatives of arbitrary order at the center of the Siegel–Eisenstein series, relating the nonsingular Fourier coefficients of such higher derivatives to the degrees of special cycles. We remark that the proofs here follow a completely different strategy than in [18].
In the sequel [4], we will construct the complete generating series of special cycles (including singular terms) and give evidence for their modularity.
1.1 Statement of main result
To formulate the result, let \(X\) be a smooth, proper and geometrically connected curve over \(k=\mathbf {F}_{q}\) of characteristic \(p\ne 2\), and \(\nu \colon X' \rightarrow X\) be an étale double cover, with the nontrivial automorphism denoted \(\sigma \in \operatorname{Aut}(X'/X)\). Let \(F\) be the function field of \(X\) and let \(F'\) be the ring of rational functions on \(X'\) (we allow \(X'=X\coprod X\)). In §6 we recall the definition of the moduli stack \(\operatorname{Sht}_{U(n)}^{r}\) parametrizing rank \(n\) “Hermitian shtukas” with \(r\) legs. Roughly speaking it classifies chains of vector bundles with Hermitian structures
related by elementary modifications. It admits a fibration \(\operatorname{Sht}_{U(n)}^{r} \rightarrow (X')^{r}\), and will play the role of Shimura varieties in the function field context.
1.1.1 Special cycles
Drawing inspiration from the construction of KudlaRapoport cycles on unitary Shimura varieties [13], we introduce in §7 certain stacks \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) over \(\operatorname{Sht}_{U(n)}^{r}\) indexed by ℰ, a vector bundle of rank \(m\) with \(1\leq m\leq n\) on \(X'\), and a Hermitian map^{Footnote 1}\(a \colon \mathcal {E}\rightarrow \sigma ^{*} \mathcal {E}^{\vee}\) where \(\mathcal {E}^{\vee}:= \underline{\operatorname{Hom}}(\mathcal {E}, \omega _{X'})\) is the Serre dual of ℰ. They classify Hermitian shtukas as in (1.1) together with compatible maps \(\mathcal {E}\to \mathcal {F}_{i}\) such that \(a\) is induced from the Hermitian form on \(\mathcal {F}_{0}\).
If ℰ is a line bundle on \(X'\), \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) is an analogue of KudlaRapoport divisors although they have dimension \(r\) less than \(\operatorname{Sht}_{U(n)}^{r}\). In general, \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) are analogs of special cycles for function fields.
We will be particularly interested in the case \(m=n\) and \(a \colon \mathcal {E}\rightarrow \sigma ^{*} \mathcal {E}^{\vee}\) is injective (by this we shall always mean as a map of coherent sheaves). In this case, the “virtual dimension” of \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) is 0. However, as is already seen in the number field context [13], the literal dimension of \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) is often significantly larger; this problem is exacerbated as \(r\) increases. Nevertheless, under the assumption that \(a \colon \mathcal {E}\rightarrow \sigma ^{*} \mathcal {E}^{\vee}\) is injective (as a map of coherent sheaves), we are able to construct an appropriate “virtual fundamental cycle” \([\mathcal {Z}_{\mathcal {E}}^{r}(a) ] \in \operatorname{Ch}_{0}(\mathcal {Z}_{\mathcal {E}}^{r}(a))_{\mathbf {Q}}\). Interestingly, it turns out that there are some new difficulties present in this construction that do not appear in the Shimura variety setting. For \(a\) injective, it turns out that \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) is proper over \(\mathbf {F}_{q}\), so that \([\mathcal {Z}_{\mathcal {E}}^{r}(a) ] \) has a welldefined degree \(\deg [\mathcal {Z}_{\mathcal {E}}^{r}(a) ] \in \mathbf {Q}\).
1.1.2 The main result
Let \(E(g,s, \Phi )\) be the Siegel–Eisenstein series for the standard split \(F'/F\)skewHermitian space of dimension \(2n\), with respect to the unramified standard section \(\Phi \). For a rank \(n\) vector bundle ℰ on \(X'\) as above, \(E(g,s, \Phi )\) admits a Fourier expansion with respect to ℰ indexed by Hermitian maps \(a \colon \mathcal {E}\rightarrow \sigma ^{*} \mathcal {E}^{\vee}\). We let \(\widetilde {E}_{a}(m(\mathcal {E}),s,\Phi )\) be the \(a^{\mathrm {th}}\) Fourier coefficient multiplied by certain normalization factors, explained precisely in (12.1).
In our normalization, \(s=0\) is the center of the functional equation for \(\widetilde {E}_{a}(m(\mathcal {E}), s, \Phi )\). Our main theorem relates the Taylor expansion at this central point to the degrees of special cycle classes.
Theorem 1.1
Let \(n \geq 1\) and \(r \geq 0\). Let ℰ be a rank \(n\) vector bundle on \(X'\) and \(a \colon \mathcal {E}\rightarrow \sigma ^{*} \mathcal {E}^{\vee}\) be an injective Hermitian map. Then we have
where \(d=\deg (\mathcal {E})+n\deg \omega _{X} =\chi (X',\mathcal {E})\).
1.1.3 Initial comments on the proof
We stress that (1.2) holds for all \(r\), regardless of the order of vanishing of \(\widetilde {E}_{a}(m(\mathcal {E}),s,\Phi )\) at \(s=0\). This is a distinguishing novelty of Theorem 1.1 compared to all other works on the SeigelWeil or arithmetic SiegelWeil formula. The first results of this nature, giving motivic interpretations of Taylor coefficients of automorphic \(L\)functions even “beyond the leading term”, were proved in [28, 29] for \(\operatorname{PGL}_{2}\). Our results here are the first higher derivative formulas to be proved for groups of arbitrary rank. Our proof shares some common ingredients with these earlier works, but also has a number of interesting new ones. For example, a key discovery for us was a connection between the Fourier coefficients of Siegel–Eisenstein series and certain perverse sheaves arising from Springer theory. Another key realization was that the special cycles are governed by certain variants of the Hitchin fibration, whose geometry can also be described in terms of Springer theory. In particular, the geometry behind Theorem 1.1 is much more complicated than that in [28, 29] as soon as \(n>2\). An overview of the proof will be given in §1.2.
Another feature of the proof of Theorem 1.1 is that it is completely uniform in \(r\), and in particular unites the “Siegel–Weil formula” and “arithmetic Siegel–Weil formula” in the same framework. For this reason, we propose to call (1.2) a higher Siegel–Weil formula. This formula will serve as the first step to establish a higher order derivative version of the aforementioned recent work of Li and Liu [16, 17] over function fields, which would give a geometric interpretation of higher derivatives of Langlands \(L\)functions.
Remark 1.2
When \(r=0\), the coarse moduli space of \(\operatorname{Sht}_{U(n)}^{r}\) is just the discrete set of points which form the domain of everywhere unramified automorphic forms for \(U(n)\). In that case, Theorem 1.1 specializes to (the nonsingular Fourier coefficients of) the classical Siegel–Weil formula, which can be found in [26].
One should imagine that when \(r=1\), \(\operatorname{Sht}_{U(n)}^{r} \rightarrow X'\) is analogous to (the integral model of) a unitary Shimura variety. Now, under the technical assumptions of the present paper (namely the everywhere unramifiedness assumptions) this space is always empty, corresponding to the fact that the sign of the functional equation for the Siegel–Eisenstein series is \(+1\) (so that all odd order derivatives vanish). However, with a mild modification of the setup, the same methods may be used to prove variants of Theorem 1.1 in which the sign of the functional equation is −1. More precisely, in this paper we consider rank \(n\) vector bundles on \(X'\) with a Hermitian pairing valued in the canonical bundle \(\omega _{X'}\cong \nu ^{*} \omega _{X}\); if we replace \(\omega _{X} \) here by a line bundle on \(X\) which is not a norm from \(X'\), then the sign of the functional equation is −1 when \(n\) is odd. The precise formulation is in [4, §9]. We mention also the work [25] over function fields, which should be thought of as being similar to the special case of Theorem 1.1 for \(r=1\) and \(n=1\).
When \(r>1\), no analogue of the spaces \(\operatorname{Sht}_{U(n)}^{r}\) is presently known in the number field setting. Consequently, we do not know how to formulate an analogue of the main result for number fields.
1.1.4 Construction of virtual fundamental cycles
For a vector bundle ℰ of rank \(m\) on \(X'\) and a Hermitian map \(a\colon \mathcal {E}\to \sigma ^{*}\mathcal {E}^{\vee}\), the dimension of \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) differs from its “virtual dimension”, which is \(r(nm)\). The situation gets worse if \(a\) is singular (i.e., not injective, in analogy to the terminology of [13]). For example, when \(a=0\), \(\mathcal {Z}^{r}_{\mathcal {E}}(0)\) contains \(\operatorname{Sht}^{r}_{U(n)}\) as a substack. It is a nontrivial task to define a cycle class \([\mathcal {Z}_{\mathcal {E}}^{r}(a)]\) in the expected dimension \(r(nm)\).
Our companion paper [4] proposes two solutions to this problem, one using classical intersection theory and the other using derived algebraic geometry. There, we construct cycle classes \([\mathcal {Z}^{r}_{\mathcal {E}}(a)]\) for all ℰ of rank \(\le n\) and possibly singular \(a \colon \mathcal {E}\rightarrow \sigma ^{*} \mathcal {E}^{\vee}\). Moreover, we assemble them into generating series valued in the Chow groups of \(\operatorname{Sht}_{U(n)}^{r}\) and conjecture it to be automorphic, in analogy to known results over number fields [2], which fall under the umbrella of the Kudla program.
In this paper, we use a more elementary method to define the 0cycle \([\mathcal {Z}_{\mathcal {E}}^{r}(a)]\) in the case \(m=n\) and \(a\) injective. First, we prove that when ℒ is a line bundle and \(a \colon \mathcal {L}\rightarrow \sigma ^{*} \mathcal {L}^{\vee}\) is an injective Hermitian map, \(\mathcal {Z}_{\mathcal {L}}^{r}(a)\) has the expected dimension (cf. Proposition 7.9 and Remark 7.10). Next, when \(\mathcal {E}=\oplus _{i=1}^{n} \mathcal {L}_{i}\) is a direct sum of line bundles, the class \([\mathcal {Z}_{\mathcal {E}}^{r}(a) ] \in \operatorname{Ch}_{0}(\mathcal {Z}_{\mathcal {E}}^{r}(a))_{\mathbf {Q}}\) can be defined as (the restriction to \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) of) the intersection product of \(\mathcal {Z}_{\mathcal {L}_{i}}^{r}(a_{ii})\) for the diagonal entries \(a_{ii}\) of \(a\); this is similar to the number field case. However, compared to the number field case, a new difficulty arises since ℰ is not necessarily a direct sum of line bundles. We overcome this difficulty in §7.9 by introducing the notion of a good framing for ℰ to reduce to the case of a sum of line bundles. A nontrivial task is to verify that the cycle class \([\mathcal {Z}_{\mathcal {E}}^{r}(a) ]\) is independent of the choice of the good framing, which occupies much of the sections §8–§10.
1.2 Method of proof
To summarize, we prove Theorem 1.1 by constructing two perverse sheaves that encode the two sides of (1.2) in the sense of sheaffunction correspondence, and then identifying these two perverse sheaves using a Hermitian variant of Springer theory, which labels these perverse sheaves by representations of the appropriate Weyl group. In this way, Theorem 1.1 is eventually unraveled into an elementary identity between representations of the Weyl group for type B/C.
On the geometric side, the connection between special cycles and Springer theory comes via the geometry of a moduli stack that resembles the Hitchin moduli space. On the other side, the connection between the Fourier coefficients of SiegelEisenstein series and Springer theory goes through local density formulas of ChoYamauchi.
Let us briefly explain the connection between the higher SiegelWeil formula and the Hitchin moduli stack and Hermitian Springer theory, and defer details to the later paragraphs. The degree of the special cycle that appear on the right side of (1.2) is essentially an intersection number of cycles on \(\operatorname{Sht}^{r}_{U(n)}\). The ambient space \(\operatorname{Sht}^{r}_{U(n)}\) can itself be realized an intersection of a Hecke correspondence with the graph of a Frobenius endomorphism. We use this to “unfold” all the intersections, and then redo them in a different order, performing the linear intersections (i.e., those not involving the Frobenius map) first, and leaving the Frobenius semilinear intersection till the last step (cf. (10.15) – (10.19)). In this process, a Hitchintype moduli stack \(\mathcal {M}_{d}\) appears naturally as we perform linear intersections (cf. (10.18)). The degree of the special cycle \([\mathcal {Z}^{r}_{\mathcal {E}}(a)]\) can be expressed as a weighted counting of \(k\)points on the fiber of a map \(f_{d}: \mathcal {M}_{d}\to \mathcal {A}_{d}\) (analogue of Hitchin fibration) over the point \((\mathcal {E},a)\in \mathcal {A}_{d}(k)\), where \((\mathcal {E},a)\) are as in the statement of Theorem 1.1.
The cokernel \(\mathcal {Q}=\operatorname{coker}(a)\) is a torsion sheaf on \(X'\) with a Hermitian structure inherited from \(a\). This motivates the introduction of the moduli stack \(\operatorname{Herm}_{2d}(X'/X)\) that parametrizes torsion coherent sheaves on \(X'\) of length \(2d\) together with a Hermitian structure, so that \(\mathcal {Q}\) is a \(k\)point of \(\operatorname{Herm}_{2d}(X'/X)\) (where \(2d=\dim _{k}\Gamma (X',\mathcal {Q})\)). We show that the fiber of \(f_{d}:\mathcal {M}_{d}\to \mathcal {A}_{d}\) over \((\mathcal {E},a)\) depends only on \(\mathcal {Q}=\operatorname{coker}(a)\), therefore the degree of \([\mathcal {Z}^{r}_{\mathcal {E}}(a)]\) depends only on the \(k\)point \(\mathcal {Q}\) of \(\operatorname{Herm}_{2d}(X'/X)\).
On the other hand, the Eisenstein series side of (1.2) can be written as a product of local terms – representation density functions for Hermitian lattices. These density functions again only depend on the torsion sheaf \(\mathcal {Q}\) together with its Hermitian structure, i.e., a \(k\)point in \(\operatorname{Herm}_{2d}(X'/X)\).
Therefore we reduce to proving that two quantities attached to a \(k\)point in \(\operatorname{Herm}_{2d}(X'/X)\) are equal. A key realization is that both quantities are of motivic nature: they come by the sheaftofunction correspondence from two (graded, virtual) perverse sheaves on \(\operatorname{Herm}_{2d}(X'/X)\). This is where Hermitian Springer theory enters. Classically, starting with a reductive Lie algebra \(\mathfrak {g}\), Springer theory outputs a perverse sheaf \(\operatorname{Spr}_{\mathfrak {g}}\) on \(\mathfrak {g}\), defined as the direct image complex of the GrothendieckSpringer resolution \(\pi _{\mathfrak {g}}: \widetilde {\mathfrak {g}}\to \mathfrak {g}\), together with an action of the Weyl group \(W\). In our setting, \(\operatorname{Herm}_{2d}(X'/X)\) will play the role of \(\mathfrak {g}\). In §4, we construct a perverse sheaf \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}\) on \(\operatorname{Herm}_{2d}(X'/X)\) together with an action of \(W_{d}=(\mathbf {Z}/2\mathbf {Z})^{d}\rtimes S_{d}\) analogous to the Springer sheaf. If \(\operatorname{Herm}_{2d}(X'/X)\) is replaced by \(\operatorname{Coh}_{d}(X)\), the moduli of torsion coherent sheaves on \(X\) of length \(d\), such a Springer sheaf was constructed by Laumon [15]. The Springer sheaf on \(\operatorname{Coh}_{d}(X)\) (resp. \(\operatorname{Herm}_{2d}(X'/X)\)) can be viewed as a global version of the Springer sheaf for \(\mathfrak {gl}_{d}\) (resp. \(\mathfrak{o}_{2d}\)). The perverse sheaves on \(\operatorname{Herm}_{2d}(X'/X)\) that govern both sides of (1.2) will be constructed from direct summands of the Hermitian Springer sheaf \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}\).
Thus, the proof of Theorem 1.1 is completed in three steps:

(1)
Construct a graded perverse sheaf on \(\operatorname{Herm}_{2d}(X'/X)\)
$$ \mathcal {K}^{\operatorname{Eis}}_{d}=\bigoplus _{i=0}^{d}\mathcal {K}^{\operatorname{Eis}}_{d,i} $$whose Frobenius trace at \(\mathcal {Q}\) is related to the LHS of (1.2). More precisely,
$$ \widetilde {E}_{a}(m(\mathcal {E}),s, \Phi )= \sum _{i=0}^{d}\operatorname{Tr}(\operatorname{Fr}_{\mathcal {Q}}, (\mathcal {K}^{ \operatorname{Eis}}_{d,i})_{\mathcal {Q}})q^{2is}. $$ 
(2)
Construct a graded perverse sheaf on \(\operatorname{Herm}_{2d}(X'/X)\)
$$ \mathcal {K}^{\operatorname{Int}}_{d}=\bigoplus _{i=0}^{d}\mathcal {K}^{\operatorname{Int}}_{d,i} $$whose Frobenius trace at \(\mathcal {Q}\) is relate to the RHS of (1.2). More precisely,
$$ \deg [\mathcal {Z}_{\mathcal {E}}^{r}(a)]=\sum _{i=0}^{d}\operatorname{Tr}(\operatorname{Fr}_{\mathcal {Q}}, (\mathcal {K}^{ \operatorname{Int}}_{d,i})_{\mathcal {Q}})\cdot (d2i)^{r}. $$(1.3) 
(3)
Prove that
$$ \mathcal {K}^{\operatorname{Eis}}_{d}\cong \mathcal {K}^{\operatorname{Int}}_{d} $$(1.4)as graded perverse sheaves on \(\operatorname{Herm}_{2d}(X'/X)\).
These three steps correspond to the three parts of the paper. We elaborate on the main ideas involved in each step.
1.2.1 Step (1)
After a standard procedure expressing the nonsingular Fourier coefficients of Eisenstein series in terms of local density of Hermitian lattices, we use the formula of Cho and Yamauchi [3] for these densities (more precisely, the unitary variant developed in [18]). We also need an extension of their formula in the split case (Theorem 2.3). The formula of Cho and Yamauchi depends only on the Hermitian torsion sheaf \(\mathcal {Q}=\operatorname{coker}(a)\), which gives the hope that the local density, as a function on the set of Hermitian torsion sheaves, comes from a sheaf on \(\operatorname{Herm}_{2d}(X'/X)\) via Grothendieck’s sheaftofunction dictionary. We do this by developing an analog of Springer theory over \(\operatorname{Herm}_{2d}(X'/X)\) (§3§4).
The key observation here is that the term in the Cho–Yamauchi formula resembles the Frobenius trace function for a certain linear combination of Springer sheaves for \(\mathfrak {gl}_{d}\) or \(\operatorname{Coh}_{d}(X)\), except for some signs. To match the signs exactly we consider an analogous linear combination of Springer sheaves on \(\operatorname{Herm}_{2d}(X'/X)\), and we compare the Frobenius actions on the cohomology of Springer fibers over \(\operatorname{Coh}_{d}(X)\) and over \(\operatorname{Herm}_{2d}(X'/X)\), see §4.5 and §4.6.
1.2.2 Step (2)
This step consists of three substeps.

First, we define special cycles for nonsingular \(a\) (§6§7). When ℰ is a direct sum of line bundles \(\mathcal {L}_{i}\), we define, following Kudla and Rapoport, \([\mathcal {Z}_{\mathcal {E}}^{r}(a)]\) as the intersection of cycle classes \([\mathcal {Z}_{\mathcal {L}_{i}}^{r}(a_{ii})]\), which, despite not being divisors in our setting, always have the “expected” dimension (more precisely, codimension \(r\) in \(\operatorname{Sht}^{r}_{U(n)}\)). The definition of \([\mathcal {Z}^{r}_{\mathcal {E}}(a)]\) for general vector bundles ℰ requires choosing a “good framing” on ℰ, i.e., an injective map from a direct sum of line bundles \(\mathcal {E}'=\oplus _{i=1}^{n}\mathcal {L}_{i}\hookrightarrow \mathcal {E}\) satisfying certain conditions. In any case, the RHS of (1.2) is an intersection number of cycles on \(\operatorname{Sht}^{r}_{U(n)}\).

The welldefinedness of \([\mathcal {Z}^{r}_{\mathcal {E}}(a)]\) is proved in the second substep (§8§10), which also gives a different definition of these cycle classes without any choices. The idea is similar to the one used in [28], namely by exchanging the order of intersection, we perform “linear intersections” first to form Hitchintype moduli stacks (denoted \(\mathcal {M}_{d}\), making sense over any base field), and in the last step we perform a shtukatype construction by intersecting with the graph of Frobenius.

In the last substep (§11) we use the Lefschetz trace formula to express the degree of \([\mathcal {Z}^{r}_{\mathcal {E}}(a)]\), formulated using the Hitchintype moduli stack \(\mathcal {M}_{d}\), as the trace of Frobenius composed with the \(r^{\mathrm {th}}\) power of an endomorphism \(C\) on the direct image complex \(Rf_{*}\overline {\mathbf {Q}}_{\ell}\) of the Hitchin map \(f:\mathcal {M}_{d}\to \mathcal {A}_{d}\). Now, the “Hitchin base” \(\mathcal {A}_{d}\) has a canonical smooth map to \(\operatorname{Herm}_{2d}(X'/X)\), and it turns out that \(Rf_{*}\overline {\mathbf {Q}}_{\ell}\) together with its endomorphism \(C\) descends through this map to a perverse sheaf \(\mathcal {K}^{\operatorname{Int}}_{d}\) on \(\operatorname{Herm}_{2d}(X'/X)\) with an endomorphism \(\overline{C}\). The decomposition of \(\mathcal {K}^{\operatorname{Int}}_{d}\) into graded pieces \(\mathcal {K}^{\operatorname{Int}}_{d,i}\) is according to the eigenvalues of the \(\overline{C}\)action, which are of the form \((d2i)\). Combining these facts we get (1.3).
1.2.3 Step (3)
Both \(\mathcal {K}^{\operatorname{Eis}}_{d}\) and \(\mathcal {K}^{\operatorname{Int}}_{d}\) are linear combinations of isotypical summands of \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}\) under the action of \(W_{d}\). The isomorphism (1.4) then comes from an isomorphism of two graded virtual representations of \(W_{d}\), which we verify directly.
1.3 Notation
Throughout this paper, \(k=\mathbf {F}_{q}\) is a finite field of odd characteristic \(p\). Let \(\ell \ne p\) be a prime. Let \(\psi _{0}: k\to \overline {\mathbf {Q}}_{\ell}^{\times}\) be a nontrivial character. For a stack \(\mathcal {Y}\) over \(k\), we write \(\operatorname{Fr}\) or \(\operatorname{Fr}_{\mathcal {Y}}\) for its \(q\)power Frobenius endomorphism. We will use \(\operatorname{Frob}\) or \(\operatorname{Frob}_{y}\) for geometric Frobenius at an \(\mathbf {F}_{q}\)point \(y\).
Let \(X\) denote a smooth curve over \(k\). With the exception of §3 and §4, \(X\) is assumed to be projective and geometrically connected. Let \(\omega _{X}\) be the line bundle of 1forms on \(X\).
Let \(F=k(X)\) denote the function field of \(X\). Let \(X\) be the set of closed points of \(X\). For \(v\in X\), let \(\mathcal {O}_{v}\) be the completed local ring of \(X\) at \(v\) with fraction field \(F_{v}\) and residue field \(k_{v}\). Let \(\mathbb{A} = \mathbb{A} _{F}\) denote the ring of adèles of \(F\), and \(\widehat {\mathcal {O}}=\prod _{v\in X}\mathcal {O}_{v}\). Let \(\deg (v)=[k_{v}:k]\), and \(q_{v}=q^{\deg (v)}=\#k_{v}\). A uniformizer of \(\mathcal {O}_{v}\) is typically denoted \(\varpi _{v}\). Let \(\cdot _{v}: F^{\times}_{v}\to q^{\mathbf {Z}}_{v}\) be the absolute value such that \(\varpi _{v}_{v}=q^{1}_{v}\). Let \(\cdot _{F}: \mathbb{A} _{F}^{\times}\to q^{\mathbf {Z}}\) be the absolute value that is \(\cdot _{v}\) on \(F_{v}^{\times}\).
Let \(X'\) be another smooth curve over \(k\) and \(\nu :X'\to X\) be a finite map of degree 2 that is generically étale. We denote by \(\sigma \) the nontrivial automorphism of \(X'\) over \(X\). With the exception of §4.1 and §4.2, \(\nu \) is assumed to be étale. We emphasize that the case \(X'=X\coprod X\) is allowed. Let \(F'\) be the ring of rational functions on \(X'\), which is either a quadratic extension of \(F\) or \(F\times F\). We let \(k'\) be the ring of constants in \(F'\). The notations \(\omega _{X'}, X', F'_{v'}, \mathcal {O}_{v'}, k_{v'}, \mathbb{A} _{F'}, \cdot _{v'}, \cdot _{F'}, q_{v'}\) and \(\deg (v')\) (for \(v'\in X'\)) are defined similarly as their counterparts for \(X\). Additionally, for \(v\in X\), we use \(\mathcal {O}'_{v}\) to denote the completion of \(\mathcal {O}_{X'}\) along \(\nu ^{1}(v)\), and define \(F'_{v}\) to be its total ring of fractions.
For a vector bundle ℰ on \(X'\), let \(\mathcal {E}^{\vee}=\underline{\operatorname{Hom}}(\mathcal {E}, \omega _{X'})\) be its Serre dual. For a torsion sheaf \(\mathcal {T}\) on \(X'\), let \(\mathcal {T}^{\vee}=\underline{\operatorname{Ext}}^{1}(\mathcal {T}, \omega _{X'})\).
When \(X\) (hence \(X'\)) is projective, let \(\operatorname{Bun}_{\operatorname{GL}_{n}}\) (resp. \(\operatorname{Bun}_{\operatorname{GL}_{n}'}\)) be the moduli stack of rank \(n\) vector bundles over \(X\) (resp. \(X'\)). Let \(g\) be the genus of \(X\) and \(g'\) be the arithmetic genus of \(X'\). Note that whenever \(\nu \) is étale, we have \(g' = 2g1\).
For an algebraic stack \(\mathcal {Y}\), \(\operatorname{Ch}(\mathcal {Y})\) denotes its rationalized Chow group and \(D^{b}(\mathcal {Y},\overline {\mathbf {Q}}_{\ell})\) its bounded derived category of constructible \(\overline {\mathbf {Q}}_{\ell}\)sheaves.
Part I. The analytic side
2 Fourier coefficients of Eisenstein series
In this section we will define the SiegelEisenstein series featuring into our main theorem, and explain how to express their nonsingular Fourier coefficients in terms of local density polynomials, which will be geometrized in later sections.
2.1 Siegel–Eisenstein series
For any onedimensional \(F\)vector space \(L\), let \(\operatorname{Herm}_{n}(F,L)\) be the \(F\)vector space of \(F'/F\)Hermitian forms \(h: F^{\prime \,n}\times F^{\prime \,n}\to L\otimes _{F}F'\) (with respect to the involution \(1\otimes \sigma \) on \(L\otimes _{F}F'\)). For any \(F\)algebra \(R\), \(\operatorname{Herm}_{n}(R, L):=\operatorname{Herm}_{n}(F,L)\otimes _{F}R\) is the set of \(L\otimes _{F}R'\)valued \(R'/R\)Hermitian forms on \(R^{\prime \,n}\), where \(R'=R\otimes _{F}F'\). When \(L=F\) we write \(\operatorname{Herm}_{n}(F)=\operatorname{Herm}_{n}(F,F)\) and \(\operatorname{Herm}_{n}(R)=\operatorname{Herm}_{n}(F)\otimes _{F} R\) for any \(F\)algebra \(R\).
Let \(W\) be the standard split \(F'/F\)skewHermitian space of dimension \(2n\). Let \(H_{n}=U(W)\). Write \(\mathbb{A} := \mathbb{A} _{F}\) for the ring of adeles of \(F\). Let \(P_{n}( \mathbb{A} )=M_{n}( \mathbb{A} )N_{n}( \mathbb{A} )\) be the standard Siegel parabolic subgroup of \(H_{n}( \mathbb{A} )\), where
Let \(\eta : \mathbb{A} _{F}^{\times}/F^{\times}\rightarrow \mathbb{C}^{\times}\) be the quadratic character associated to \(F'/F\) by class field theory. Fix \(\chi : \mathbb{A} _{F'}^{\times}/F^{\prime \,\times }\rightarrow \mathbb{C}^{\times}\) a character such that \(\chi _{ \mathbb{A} _{F}^{\times}}=\eta ^{n}\). We may view \(\chi \) as a character on \(M_{n}( \mathbb{A} )\) by \(\chi (m(\alpha ))=\chi (\det (\alpha ))\) and extend it to \(P_{n}( \mathbb{A} )\) trivially on \(N_{n}( \mathbb{A} )\). Define the degenerate principal series to be the unnormalized smooth induction
For a standard section \(\Phi (, s)\in I_{n}(s,\chi )\), define the associated Siegel–Eisenstein series
which converges for \(\Re (s)\gg 0\) and admits meromorphic continuation to \(s\in \mathbb{C}\). Notice that \(E(g,s,\Phi )\) depends on the choice of \(\chi \).
Remark 2.1
In this paper, we will choose \(\chi \) to be unramified everywhere. To see that such \(\chi \) exists, observe that since \(\mathbf {C}^{\times}\) is injective (in the category of abelian groups), it suffices to check that \(\eta ^{n}\) is trivial on \(\ker (\operatorname{Pic}(X) \rightarrow \operatorname{Pic}(X') )\). If \(X'/X\) is the trivial double cover or the double cover corresponding to \(\mathbf {F}_{q^{2}}/\mathbf {F}_{q}\), then this kernel is trivial so the result is immediate. Otherwise, the cover is geometrically nontrivial. Since \(\mathrm {char}(k) \neq 2\), the kernel consists of the 2torsion line bundle whose class in \(\mathrm {H}^{1}({X, \mu _{2}})\) agrees with \(\eta \in H^{1}(X, \mathbf {Z}/2\mathbf {Z})\) under the isomorphism \(\mu _{2} \cong \mathbf {Z}/2\mathbf {Z}\). If \(n\) is even then there is nothing to check; if \(n\) is odd then the desired vanishing property amounts (when \(\mathrm {char}(k) \neq 2\)) to the alternating property of the cup product pairing \(H^{1}(X_{\overline {\mathbf {F}}_{q}},\mathbf {Z}/2\mathbf {Z})\times \mathrm {H}^{1}({X_{\overline {\mathbf {F}}_{q}},\mathbf {Z}/2\mathbf {Z}})\to \mathbf {Z}/2\), which follows from the graded commutativity of the cup product and the fact that the geometric \(\mathbf {Z}_{2}\)cohomology of curves in characteristic \(\neq 2\) is torsionfree.
As justified by Remark 2.1, we may choose \(\chi \) to be everywhere unramified. Then \(I_{n}(s,\chi )\) is unramified and we fix \(\Phi (, s)\in I_{n}(s,\chi )\) as the unique \(K=H_{n}(\widehat {\mathcal {O}})\)invariant section normalized by
Similarly we normalize \(\Phi _{v} \in I_{n}(s,\chi _{v})\) for every \(v\in X\) and we then have a factorization \(\Phi =\bigotimes _{v\in X}\Phi _{v}\).
2.2 Fourier expansion
Let \(\omega _{F}\) be the generic fiber of the canonical bundle of \(X\), and \(\mathbb{A} _{\omega _{F}}= \mathbb{A} \otimes _{F}\omega _{F}\). The sum of the residues induces a pairing \(\mathbb{A} _{\omega _{F}}\times \mathbb{A} _{F}\to k\) induces a pairing
given by \(\langle {T,b}\rangle =\operatorname{Res}(\operatorname{Tr}(Tb))\). Composing this pairing with the fixed nontrivial additive character \(\psi _{0}: k\to \mathbb{C} ^{\times}\) exhibits \(\operatorname{Herm}_{n}( \mathbb{A} , \omega _{F})\) as the Pontryagin dual of \(\operatorname{Herm}_{n}( \mathbb{A} )\). Moreover, it exhibits \(\operatorname{Herm}_{n}(F,\omega _{F})\) as the Pontryagin dual of \(\operatorname{Herm}_{n}(F)\backslash \operatorname{Herm}_{n}( \mathbb{A} )=N_{n}(F)\backslash N_{n}( \mathbb{A} )\). The global residue pairing is the sum of local residue pairings \(\langle {\cdot ,\cdot}\rangle _{v}: \operatorname{Herm}_{n}(F_{v}, \omega _{F_{v}})\times \operatorname{Herm}_{n}(F_{v}) \to k\) defined by \(\langle {T,b}\rangle _{v}=\operatorname {tr}_{k_{v}/k}\operatorname{Res}_{v}(\operatorname{Tr}(Tb))\).
We have a Fourier expansion
where
and the Haar measure \(\mathrm {d}n(b)\) is normalized such that \(N_{n}(F)\backslash N_{n}( \mathbb{A} )\) has volume 1. For any \(\alpha \in M_{n}( \mathbb{A} )\) we have
Suppose \(T\) is nonsingular, meaning that for one (equivalently, any) choice of trivialization of \(\omega _{F}\) it has nonvanishing determinant, for a factorizable \(\Phi =\bigotimes _{v\in X}\Phi _{v}\) we have a factorization of the Fourier coefficient into a product (cf. [11, §4])
where the local (generalized) Whittaker function is defined by
and has analytic continuation to \(s\in \mathbb{C}\). Here the local Haar measure \(\mathrm {d}_{v} n(b)\) is the one such that the volume of \(N_{n}(\mathcal {O}_{v})\) is 1. The factor \(\omega _{X}_{F}^{n^{2}/2}\) is the ratio between the global measure \(\mathrm {d}n\) and the product of the local measures \(\prod _{v}\mathrm {d}_{v}n\).
Note that for \(\alpha \in M_{n}(F_{v})\),
We define the regular part of the Eisenstein series to be
2.3 Local densities for Hermitian lattices
The local density for Hermitian lattices in the nonsplit case has been studied in [18, §3] following the strategy of Cho–Yamauchi [3]. Here we recall the result of [18] and extend the results to the split case.
From now on until §2.5, let \(F\) be a nonarchimedean local field of characteristic not equal to 2 (but possibly with residue characteristic 2). Let \(F'\) be either an unramified quadratic field extension or the split quadratic \(F\)algebra \(F'=F\times F\). Denote by \(\mathcal {O}_{F}\) (resp. \(\mathcal {O}_{F'}\)) the ring of integers in \(F\) (resp. \(F'\)). In the split case we have \(\mathcal {O}_{F'}=\mathcal {O}_{F}\times \mathcal {O}_{F}\). Let \(\eta =\eta _{F'/F}: F^{\times}\to \{\pm 1\}\) be the quadratic character attached to \(F'/F\) by class field theory. Let \(\varpi \) be a uniformizer of \(F\), \(k\) the residue field, \(q=\#k\).
Let \(L, M\) be two Hermitian \(\mathcal {O}_{F'}\)lattices. In the split case, the datum of a Hermitian \(\mathcal {O}_{F'}\)lattice \(L\) is a pair \((L_{1},L_{2})\) of \(\mathcal {O}_{F}\)lattices together with an \(\mathcal {O}_{F}\)bilinear pairing
that is perfect after base change to \(F\). We will define \(L^{\vee}=(L_{1}^{\vee}, L_{2}^{\vee})\) where \(L_{1}^{\vee}= \{x\in L_{1}\otimes _{\mathcal {O}_{F}} F: (x,L_{2})\subset \mathcal {O}_{F} \}\) and similarly for \(L_{2}^{\vee}\).
Let \(\operatorname{Rep}_{M,L}\) be the scheme of integral representations of \(M\) by \(L\), an \(\mathcal {O}_{F}\)scheme such that for any \(\mathcal {O}_{F}\)algebra \(R\),
where \(\operatorname{Herm}\) denotes the set of Hermitian \(R\)module homomorphisms. In the split case, if we write \(L\) and \(M\) in terms of pairs \((L_{1},L_{2})\) and \((M_{1},M_{2})\) with their \(\mathcal {O}_{F}\)bilinear pairings, then a Hermitian module homomorphism consists of a pair of \(R\)linear maps \(\phi _{i}:L_{i} \otimes _{\mathcal {O}_{F}}R\to M_{i} \otimes _{\mathcal {O}_{F}}R \) preserving the base change to \(R\) of the \(\mathcal {O}_{F}\)bilinear pairings.
The local density of integral representations of \(M\) by \(L\) is defined to be
Note that if \(L, M\) have \(\mathcal {O}_{F'}\)rank \(n, m\) respectively and the generic fiber \((\operatorname{Rep}_{M,L})_{F}\ne \varnothing \), then \(n\le m\) and
2.4 Cho–Yamauchi formula for local density
Definition 2.2
For \(a\in \mathbf {Z}_{\ge 0}\) we define a polynomial of degree \(a\)
Note that \(\mathfrak {m}(a; T)\) depends on \(F'/F\).
In both the nonsplit and the split cases, for a finite torsion \(\mathcal {O}_{F}\)module \(\mathcal {T}\) we define
For an \(\mathcal {O}_{F'}\)Hermitian lattice \(L\), we define its type
where we view the finite torsion \(\mathcal {O}_{F'}\)module \(L^{\vee}/L\) as an \(\mathcal {O}_{F}\)module.
When \(F'/F\) is nonsplit, for a finite torsion \(\mathcal {O}_{F'}\)module \(\mathcal {T}\) we define
Then we have
When \(F'=F\times F\) is split, for a finite torsion \(\mathcal {O}_{F'}\)module \(\mathcal {T}\) we may define \(\ell '(\mathcal {T})\) and \(t'(\mathcal {T})\) by (2.6). Moreover, for \(\mathcal {O}_{F'}\)Hermitian lattices \(L=(L_{1},L_{2})\) and \(L'=(L'_{1},L'_{2})\) such that \(L\subset L'\) (meaning that \(L_{1} \subset L_{1}'\) and \(L_{2} \subset L_{2}'\)), we have
and
In both the split and nonsplit case, we define
We have the following analog of Cho–Yamauchi formula [3].
Theorem 2.3
Let \(j\ge 0\) be an integer. Let \(\langle 1\rangle ^{j}\) be the selfdual Hermitian \(\mathcal {O}_{F'}\)lattice of rank \(j\) with Hermitian form given the identity matrix \(\mathbf{1}_{j}\). Let \(L\) be a Hermitian \(\mathcal {O}_{F'}\)lattice of rank \(n\).

(1)
We have
$$ \operatorname{Den}(\langle 1\rangle ^{n+j}, \langle 1\rangle ^{n})=\prod _{i=1}^{n}(1(\eta (\varpi )q)^{i}T) \bigg_{T= (\eta (\varpi )q)^{j}}. $$ 
(2)
There is a (unique) polynomial \(\operatorname{Den}(T,L)\in \mathbb{Z}[T]\), called (normalized) local Siegel series of \(L\), such that for all \(j\geq 0\),
$$ \operatorname{Den}((\eta (\varpi ) q)^{j},L)= \frac{\operatorname{Den}(\langle 1\rangle ^{n+j}, L)}{\operatorname{Den}(\langle 1\rangle ^{n+j}, \langle 1\rangle ^{n})}. $$ 
(3)
We have
$$ \operatorname{Den}(T,L) = \sum _{L \subset L '\subset L^{\prime \,\vee} \subset L^{\vee}} T^{2 \ell '(L'/L)}\mathfrak {m}( t'(L');T). $$(2.7)Here the sum is over \(\mathcal {O}_{F'}\)lattices \(L'\) (in the \(F'\)Hermitian space spanned by \(L\)) containing \(L\) on which the Hermitian form is integral.
Proof
The nonsplit case is proved in [18, Thm. 3.5.1] and here we indicate the necessary change in the split case. Now suppose \(F'=F\times F\) and hence \(k'=k\times k\). Let \(L_{k}=L\otimes _{\mathcal {O}_{F}} k\) and \(\langle 1\rangle ^{m}_{k}=\langle 1\rangle ^{m}\otimes _{\mathcal {O}_{F}} k\), which are free \(k'\)modules with the induced \(k'/k\)Hermitian forms. In particular, \(\langle 1\rangle ^{m}_{k}\) is nondegenerate and the radical of \(L_{k}=L\otimes _{\mathcal {O}_{F}} k\) has \(k'\)rank equal to \(t'(L)=t(L_{1}^{\vee}/L_{1})=t(L_{2}^{\vee}/L_{2})\). Let \(\operatorname{Isom}_{\langle 1\rangle ^{m}_{k}, L_{k}}\) be the \(k\)scheme of “isometric embeddings” from \(L_{k}\) to \(\langle 1\rangle ^{m}_{k}\), i.e., injective \(k'\)linear maps from \(L_{k}\) to \(\langle 1\rangle ^{m}_{k}\) preserving the Hermitian forms.
Similar to the orthogonal case [3, §3.3], we have
where \(\dim \operatorname{Rep}(\langle 1\rangle ^{m},L)_{F}=m^{2}(mn)^{2}=2mnn^{2}\).
It remains to show that
where \(a=t'(L)\) is the \(k'\)rank of the radical of \(L_{k}\). Note that uptoisomorphism, \(L_{k}\) is determined by its rank and the rank of its radical. Let \(U_{na,a}\) be a \(k'/k\)Hermitian space of rank \(n\) with radical of rank \(a\). Let \(V_{m}=U_{m,0}\) be a (nondegenerate) \(k'/k\)Hermitian space of dimension \(m\ge n\). Then it is easy to see that \(U_{na,a}\simeq U_{na,0}\oplus U_{0,a}\) and
By (2.9) (note that \(\#\operatorname{Isom}_{\langle 1\rangle ^{m}_{k}, L_{k}}(k)= \# \operatorname{Isom}_{V_{m}, U_{na,a}}(k)\)), it suffices to show (2.8) in the two extreme cases: \(a=0\) and \(a=n\).
First we consider the case \(a=n\). Then, to give an isometric embedding from \(U=U_{0,n}=k^{\prime \,n}\) to \(V=U_{m,0}=k^{\prime \,m}\) is equivalent to give an injective \(k\)linear map \(\phi :k^{n}\to k^{m}\) and then an injective \(k\)linear map \(\varphi :k^{n}\to {\mathrm{Im}}(\phi )^{\perp}\subset k^{m}\). Therefore, denoting by \(\operatorname{Hom}^{*}_{k}(k^{n},k^{m})\) the set of injective \(k\)linear maps \(\phi :k^{n}\to k^{m}\), we have
It remains to consider the case \(a=0\). Then a similar argument shows
This completes the proof. □
Remark 2.4
By Theorem 2.3, the polynomial \(\operatorname{Den}(T,L)\) depends only on the induced Hermitian form on the torsion module \(L^{\vee}/L\). Indeed, for a Hermitian torsion module^{Footnote 2}\(Q\) we define \(\operatorname{Den}(T,Q)\) by the formula
Then by (2.7), we have \(\operatorname{Den}(T, L) = \operatorname{Den}(T, L^{\vee}/L)\).
Remark 2.5
In the split case, write \(L=(L_{1},L_{2})\) and \(L'=(L'_{1},L'_{2})\). Then the formula reads
Remark 2.6
The local Siegel series satisfies a functional equation
A proof in the inert case can be found in [8, Theorem 5.3]. By Theorem, 2.3 the constant term of \(\operatorname{Den}(T,L)\) is 1. It follows that the degree of the polynomial \(\operatorname{Den}(T,L)\) is equal to \(\ell '(L^{\vee}/L)\). We will not use this fact in this paper. See Corollary 11.14 for the (global) geometric analog.
2.5 Relation with local Whittaker functions
We continue to let \(F\) be a local field. Define the local Lfunction
Lemma 2.7
Let \(L\) be a Hermitian \(\mathcal {O}_{F'}\)lattice of rank \(n\). Let \(T=((x_{i}, x_{j}))_{1\le i,j\le n}\) be the fundamental matrix of an \(\mathcal {O}_{F'}\)basis \(\{x_{1},\ldots , x_{n}\}\) of \(L\), an \(n\times n\) Hermitian matrix over \(F\). Let \(\theta \) be a generator of \(\omega _{\mathcal {O}_{F}}\) so that \(T\theta \in \operatorname{Herm}_{n}(F,\omega _{F})\). Then
Here \(\Phi \) is the local unramified section normalized by \(\Phi (1_{2n},s)=1\).
Proof
Note that by Theorem 2.3
It is known that \(W_{T\theta }(1, s, \Phi )\) is a rational function in \(q^{s}\). Therefore the formula is equivalent to
for all integer \(j\geq 0\). In the nonsplit case this is essentially [13, Prop. 10.1] (cf. [18, §3.3]), which can be easily modified to the split case. We note that \(W_{T\theta }\) is the same as \(W_{T}\) in loc. cit.. □
2.6 Fourier coefficients revisited
Now we return to the global situation. We need the following global Lfunction to normalize the Eisenstein series
We now consider the restriction of the regular part \(E^{\mathrm{reg}}(\cdot ,s,\Phi )\) (as a function in \(g\in H_{n}( \mathbb{A} )\), cf. (2.4)) to the Levi subgroup \(M_{n}( \mathbb{A} )\). Since the restriction is left \(M_{n}(F)\)invariant and right \(K\)invariant, it descends to a function on
via the canonical identifications. From now on we will freely switch between \(g=m(\alpha )\in M_{n}( \mathbb{A} )\) and the corresponding element \(\mathcal {E}\in \operatorname{Bun}_{\operatorname{GL}_{n}'}(k)\) and we will write
Note that the absolute value on \(\mathbb{A} _{F'}^{\times}\) is normalized such that \(\det (\alpha )_{F'}=q^{\deg (\mathcal {E})}\). By abuse of notation we also view \(\chi \) as a function on \(\operatorname{Bun}_{\operatorname{GL}_{1}'}(k)\).
Recall that \(\mathcal {E}^{\vee}=\underline {\operatorname{Hom}}_{\mathcal {O}_{X'}}(\mathcal {E},\omega _{X'})\) denotes the Serre dual of ℰ. Consider a rational Hermitian map \(a:\mathcal {E}\dashrightarrow \sigma ^{*}\mathcal {E}^{\vee}\) (i.e., \(a\) is a map defined over the generic point of \(X'\), such that \(\sigma ^{*} a = a\)). Given a pair \((\mathcal {E},a)\) as above, we shall define the Fourier coefficient
as follows. For any generic trivialization \(\tau : \mathcal {E}_{F'}\stackrel {\sim }{\to }(F')^{n}\), the pair \((\mathcal {E},\tau )\) gives a point \(\alpha =\alpha (\mathcal {E},\tau )\in M_{n}( \mathbb{A} )/M_{n}(\widehat {\mathcal {O}})\) such that ℰ is glued from \((F')^{n}\) and the lattices \(\alpha _{v}\mathcal {O}^{n}_{F'_{v}}\). Under \(\tau \), the restriction of \(a\) at the generic point gives an \(\omega _{F}\)valued Hermitian form on \((F')^{n}\) which we denote by \(T=T(a,\tau )\). Then we define
If we change \(\tau \) to \(\gamma \tau \) for some \(\gamma \in M_{n}(F)=\operatorname{GL}_{n}(F')\), then \(\alpha (\mathcal {E},\gamma \tau )=\gamma \alpha (\mathcal {E},\tau )\) and \(T(a,\gamma \tau )={}^{t}\overline{\gamma }^{1}T(a,\tau )\gamma ^{1}\). By (2.1), we have
for all \(\gamma \in M_{n}(F)\). Therefore \(E_{a}(m(\mathcal {E}),s,\Phi )\) is welldefined.
Now suppose \(a: \mathcal {E}\hookrightarrow \sigma ^{*}\mathcal {E}^{\vee}\) is an injective Hermitian map. Let \((\mathcal {E}_{v},a)\) denote the Hermitian \(\mathcal {O}_{v}'\)lattice (valued in \(\omega _{\mathcal {O}_{F_{v}}}:=\omega _{X}\otimes _{\mathcal {O}_{X}} \mathcal {O}_{F_{v}}\)) induced by \(a\) at \(v\in X\). Choosing a generator of the free \(\mathcal {O}_{F_{v}}\)module \(\omega _{\mathcal {O}_{F_{v}}}\) of rank one, we obtain a Hermitian lattice \(\mathcal {E}_{v}\) (valued in \(\mathcal {O}_{F_{v}}\)) and hence the density polynomial \(\operatorname{Den}(T, \mathcal {E}_{v})\) defined by (2.7) relative to \(F'_{v}/F_{v}\). We define the density polynomial for \((\mathcal {E}_{v},a)\) as
It is easy to see that the result is independent of the choice of the generator of \(\omega _{\mathcal {O}_{F_{v}}}\). We then define
Note that the degree of \(\operatorname{Den}(q^{2s}, (\mathcal {E},a))\) (as a polynomial of \(q^{s}\)) is
Theorem 2.8
Let ℰ be a vector bundle over \(X'\) of rank \(n\). Then
where the sum runs over all injective Hermitian maps \(a: \mathcal {E}\to \sigma ^{*}\mathcal {E}^{\vee }\). Moreover, we have
Proof
From the definitions it is clear that
where \(a\) runs over rational Hermitian maps \(\mathcal {E}\dashrightarrow \sigma ^{*} \mathcal {E}^{\vee}\) that are generically nonsingular.
Now let \(a:\mathcal {E}\dashrightarrow \sigma ^{*}\mathcal {E}^{\vee}\) be such a rational nonsingular Hermitian map. We continue with the convention defining (2.10)
By (2.2) and (2.3), and noting that the character \(\chi \) is trivial on the norm of \(\mathbb{A} _{F'}^{\times}\), we have
If the (\(\omega _{F_{v}}\)valued) Hermitian form \(T_{v}\) does not have integral entries, then \(W_{T_{v}}(1,s, \Phi _{v})=0\) (since \(\Phi \) is invariant under \(N_{n}(\mathcal {O}_{v})\)). Therefore \(E_{T}(m(\alpha ),s,\Phi )\) is nonzero only when \(T_{v}\) is integral for all \(v\), i.e., \(a\) is an everywhere regular Hermitian map \(\mathcal {E}\hookrightarrow \sigma ^{*}\mathcal {E}^{\vee}\). This proves (2.12).
For such \(a: \mathcal {E}\hookrightarrow \sigma ^{*}\mathcal {E}^{\vee}\), by Lemma 2.7, the right side of (2.15) is
where, by choosing a generator of \(\omega _{\mathcal {O}_{F_{v}}}\), the \(\mathcal {O}_{F_{v}'}\)module \(\mathcal {E}_{v}\) is endowed with the Hermitian form (valued in \(\mathcal {O}_{F_{v}}\)) induced by the \(\omega _{X}\)valued Hermitian form \(a\).
By \(\chi (\det (\alpha ))=\chi (\det (\mathcal {E}))\), \(\det (\alpha )_{F'}=q^{\deg (\mathcal {E})}\), and (2.11), we obtain
Combining these facts with (2.14), (2.15) and (2.16), we get (2.13). □
3 Springer theory for torsion coherent sheaves
In this section we review the construction of the Springer sheaf on the moduli stack of torsion coherent sheaves on a curve following Laumon [15]. We also compute the Frobenius trace function of a particular summand of the Springer sheaf called the Steinberg sheaf.
In this section let \(X\) be any smooth (not necessarily projective or connected) curve over \(k=\mathbf {F}_{q}\). For \(d\in \operatorname{\mathbf{N}}\), let \(X_{d}\) be the \(d^{\mathrm {th}}\) symmetric power of \(X\).
3.1 Local geometry of \(\operatorname{Coh}_{d}\)
Let \(\operatorname{Coh}_{d}=\operatorname{Coh}_{d}(X)\) be the moduli stack of torsion coherent sheaves on \(X\) of length \(d\). For any \(k\)scheme \(S\), \(\operatorname{Coh}_{d}(S)\) is the groupoid of coherent sheaves on \(X \times S\) whose pushforward to \(S\) is locally free of rank \(d\).
Let \(s^{\operatorname{Coh}}_{d}: \operatorname{Coh}_{d}\to X_{d}\) be the support map. Recall that for any \(k\)scheme \(S\), \([\mathfrak {gl}_{d}/\operatorname{GL}_{d}](S)\) is the groupoid of \((V, T)\) where \(V\) is a vector bundle of rank \(d\) on \(S\) and \(T\) is an endomorphism of \(V\). When \(X=\mathbf {A}^{1}\), we have a canonical isomorphism
given as follows. For \(\mathcal {Q}\in \operatorname{Coh}_{d}(\mathbf {A}^{1})(S)\), \(\Gamma (\mathbf {A}^{1}_{S},\mathcal {Q})\) is a locally free rank \(d\) \(\mathcal {O}_{S}\)module equipped with an endomorphism given by the affine coordinate \(t\) for \(\mathbf {A}^{1}\), giving an \(S\)point of \([\mathfrak {gl}_{d}/\operatorname{GL}_{d}]\); conversely, given an object \((V,T) \in [\mathfrak {gl}_{d}/\operatorname{GL}_{d}]\) we may view \(V\) as an \(\mathcal {O}_{S}[t]\)module \(\mathcal {Q}\) (viewed as a coherent sheaf on \(\mathbf {A}^{1}_{S}\)) with \(t\) acting as \(T\).
Let \(U\subset X_{\overline{k}}\) be open and \(f:U\to \mathbf {A}^{1}_{\overline{k}}\) be an étale map. Such a pair \((U,f)\) is called an étale chart for \(X_{\overline{k}}\). It induces a map \(f^{\operatorname{Coh}}_{d}: \operatorname{Coh}_{d}(U)\to \operatorname{Coh}_{d}(\mathbf {A}^{1}_{\overline{k}})\) sending \(\mathcal {Q}\) to \(f_{*}\mathcal {Q}\) which is compatible with the symmetric power \(f_{d}: U_{d}\to (\mathbf {A}^{1}_{\overline{k}})_{d}\) under \(s^{\operatorname{Coh}}_{d}\). Let \(\mathfrak {D}_{d,\mathbf {A}^{1}}\subset (\mathbf {A}^{1}_{\overline{k}})_{d}\) and \(\mathfrak {D}_{d,U}\subset U_{d}\) be the discriminant divisors, i.e., they parametrize divisors with multiplicities. Clearly \(\mathfrak {D}_{d,U}\subset f^{1}_{d}(\mathfrak {D}_{d,\mathbf {A}^{1}})\), therefore we may write \(f^{1}_{d}(\mathfrak {D}_{d,\mathbf {A}^{1}})=\mathfrak {D}_{d,U}+\mathfrak {R}_{d,f}\) as Cartier divisors on \(U_{d}\).
Lemma 3.1
Let \(D\) be an effective divisor of degree \(d\) on \(U\). Then \(D\in U_{d}\backslash \mathfrak {R}_{d,f}\) if and only if for all pairs of distinct points \(x,y\) in the support of \(D\), \(f(x)\ne f(y)\).
Proof
Let \(\pi _{d}: U^{d}\to U_{d}\) be the quotient map by the symmetric group \(S_{d}\). We compute the divisor \(\pi _{d}^{1}(\mathfrak {R}_{d,f})\). Consider \(U\times _{\mathbf {A}^{1}_{\overline{k}}} U\). Since \(U\) is étale over \(\mathbf {A}^{1}_{\overline{k}}\), \(\Delta (U)\subset U\times _{\mathbf {A}^{1}} U\) is open and closed, hence we can write \(U\times _{\mathbf {A}^{1}_{\overline{k}}}U=\Delta (U)\coprod \mathfrak {R}\). For geometric points \(x,y\in U\), \((x,y)\in \mathfrak {R}\) if and only if \(x\ne y\) and \(f(x)=f(y)\).
For \(1\le i< j\le d\), let \(p_{ij}: U^{d}\to U\times U\) be the projection to the \(i\)th and \(j\)th coordinates. Let
By definition, \(\pi _{d}^{1}(\mathfrak {D}_{d,\mathbf {A}^{1}})=\sum _{1\le i< j\le d}\widetilde {\Delta }_{ij}\), \(\pi _{d}^{1}(\mathfrak {D}_{d,U})=\sum _{1\le i< j\le d}\Delta _{ij}\) as divisors on \(U^{d}\). Therefore
From this we see, if \(D=x_{1}+x_{2}+\cdots +x_{d}\), where \(x_{i}\in U(\overline{k})\), then \(D\notin \mathfrak {R}_{d,f}\) if and only if \((x_{1},\ldots , x_{d})\notin \pi _{d}^{1}(\mathfrak {R}_{d,f})\). By (3.1), the latter happens if and only if for all \(1\le i< j\le d\), \((x_{i},x_{j})\notin \mathfrak {R}\), i.e., either \(x_{i}=x_{j}\) or \(f(x_{i})\ne f(x_{j})\). □
Let \(\operatorname{Coh}_{d}(U)^{f}\subset \operatorname{Coh}_{d}(U)\) be the preimage of \(U_{d}\backslash \mathfrak {R}_{d,f}\). Then \(\operatorname{Coh}_{d}(U)^{f}\) is an open substack of \(\operatorname{Coh}_{d}(X)_{\overline{k}}=\operatorname{Coh}_{d}(X_{\overline{k}})\).
The following lemma shows that \(\operatorname{Coh}_{d}(X)\) is étale locally isomorphic to \(\operatorname{Coh}_{d}(\mathbf {A}^{1})\cong [\mathfrak {gl}_{d}/\operatorname{GL}_{d}]\).
Lemma 3.2

(1)
For any étale chart \((U,f)\) of \(X_{\overline{k}}\), the map \(f^{\operatorname{Coh}}_{d}:\operatorname{Coh}_{d}(U)\to \operatorname{Coh}_{d}(\mathbf {A}^{1})_{\overline{k}}\) is étale when restricted to \(\operatorname{Coh}_{d}(U)^{f}\).

(2)
The stack \(\operatorname{Coh}_{d}(X)_{\overline{k}}\) is covered by the substacks \(\operatorname{Coh}_{d}(U)^{f}\) for various étale charts \((U,f)\) of \(X_{\overline{k}}\).
Proof
(1) For any \(\mathcal {Q}\in \operatorname{Coh}_{d}(U)^{f}(\overline{k})\), the tangent map of \(f_{d}^{\operatorname{Coh}}\) at \(\mathcal {Q}\) is \(\operatorname{Ext}^{*}_{U}(\mathcal {Q},\mathcal {Q})\to \operatorname{Ext}^{*}_{\mathbf {A}^{1}}(f_{*}\mathcal {Q}, f_{*}\mathcal {Q})\). By Lemma 3.1, different points in the support of \(\mathcal {Q}\) map to different points in \(\mathbf {A}^{1}\), the above map is the direct sum of \(\tau _{z}: \operatorname{Ext}^{*}_{\mathcal {O}_{U,z}}(\mathcal {Q}_{z},\mathcal {Q}_{z})\to \operatorname{Ext}^{*}_{ \mathcal {O}_{\mathbf {A}^{1},f(z)}}(\mathcal {Q}_{z}, \mathcal {Q}_{z})\) over \(z\in \operatorname{supp}(\mathcal {Q})\). Since \(f\) is étale at each such \(z\), \(\tau _{z}\) are isomorphisms, and hence \(f_{d}^{\operatorname{Coh}}\) is étale at \(\mathcal {Q}\) by the Jacobian criterion.
(2) For every point \(\mathcal {Q}\in \operatorname{Coh}_{d}(X)(\overline{k})\) we will construct an étale chart \((U,f)\) such that \(\mathcal {Q}\in \operatorname{Coh}_{d}(U)^{f}(\overline{k})\). Let \(Z\subset X(\overline{k})\) be the support of \(\mathcal {Q}\). For \(z\in Z\), let \(\mathcal {O}_{z}\) be the completed local ring of \(X_{\overline{k}}\) at \(z\) with a uniformizer \(\varpi _{z}\). The map of sheaves \(r: \mathcal {O}_{X_{\overline{k}}}\to \oplus _{x\in Z}\mathcal {O}_{z}/\varpi _{z}^{2}\) is surjective. Let \(c:Z\to \overline{k}\) be any injective map of sets. Then there exists an open neighborhood \(U_{1}\) of \(Z\) and \(f\in \mathcal {O}(U_{1})\) such that \(r(f)=(c(z)+\varpi _{z})_{z\in Z}\). Viewing \(f\) as a map \(f: U_{1}\to \mathbf {A}^{1}_{\overline{k}}\), it is then étale at \(Z\), hence étale in an open neighborhood \(U\subset U_{1}\) of \(Z\), i.e., \((U,f)\) is an étale chart. Since \(\{f(z)=c(z)\}_{z\in Z}\) are distinct points in \(\mathbf {A}^{1}_{\overline{k}}\), we see that \(\mathcal {Q}\in \operatorname{Coh}_{d}(U)^{f}_{\overline{k}}\) by Lemma 3.1. □
3.2 Springer theory for \(\operatorname{Coh}_{d}\)
Let \(\widetilde {\operatorname{Coh}}_{d}(X)\) be the moduli stack classifying a full flag of torsion sheaves on \(X\)
where \(\mathcal {Q}_{j}\) has length \(j\). Let
be the forgetful map recording only \(\mathcal {Q}=\mathcal {Q}_{d}\).
Lemma 3.3
Laumon [15, Theorem 3.3.1]
The stacks \(\widetilde {\operatorname{Coh}}_{d}(X)\) and \(\operatorname{Coh}_{d}(X)\) are smooth of dimension zero, and the map \(\pi ^{\operatorname{Coh}}_{d,X}\) is proper and small.
Proof
It is enough to check the same statements after base change to \(\overline{k}\). We give a quick alternative proof using Lemma 3.2: for an étale chart \((U,f)\) (over \(\overline{k}\)), we have a diagram in which both squares are Cartesian:
Here \(\widetilde {\operatorname{Coh}}_{d}(U)^{f}\) is the preimage of \(\operatorname{Coh}_{d}(U)^{f}\) in \(\widetilde {\operatorname{Coh}}_{d}(U)\). Since the horizontal maps are étale and the \(\operatorname{Coh}_{d}(U)^{f}\) cover \(\operatorname{Coh}_{d}(X)_{\overline{k}}\) by Lemma 3.2, the desired properties of \(\pi ^{\operatorname{Coh}}_{d,X}\) follow from the same properties of \(\pi ^{\operatorname{Coh}}_{d,\mathbf {A}^{1}}\), which is the Grothendieck alteration \(\pi _{\mathfrak {gl}_{d}}: [\widetilde {\mathfrak {gl}}_{d}/\operatorname{GL}_{d}]\to [\mathfrak {gl}_{d}/\operatorname{GL}_{d}]\). □
Let \(X_{d}^{\circ }\subset X_{d}\) be the open subset of multiplicityfree divisors (i.e., the complement of \(\mathfrak {D}_{d,X}\)), and let \(\operatorname{Coh}_{d}(X)^{\circ }\) (resp. \(\widetilde {\operatorname{Coh}}_{d}(X)^{\circ }\)) be its preimage under \(s^{\operatorname{Coh}}_{d}\) (resp. under \(s^{\operatorname{Coh}}_{d}\circ \pi _{d}^{\operatorname{Coh}}\)). Then \(\widetilde {\operatorname{Coh}}_{d}(X)^{\circ }\to \operatorname{Coh}_{d}(X)^{\circ }\) is an \(S_{d}\)torsor.
Corollary 3.4
Laumon [15, p.320]
The complex
is a perverse sheaf on \(\operatorname{Coh}_{d}(X)\) that is the middle extension from its restriction to \(\operatorname{Coh}_{d}(X)^{\circ }\). In particular, the natural \(S_{d}\)action on \(\operatorname{Spr}_{d}_{\operatorname{Coh}_{d}(X)^{\circ }}\) extends to the whole \(\operatorname{Spr}_{d}\).
3.3 Springer fibers
Let \(\mathcal {Q}\in \operatorname{Coh}_{d}(X)(\overline{k})\) with image \(D\) in \(X_{d}(\overline{k})\), an effective divisor of degree \(d\). Let \(Z=(\operatorname{supp}D)(\overline{k})\). Let \(\Sigma (Z)\) be the set of maps \(y:\{1,2,\ldots , d\}\to Z\) such that \(\sum _{i=1}^{d}y(i)=D\). Let \(\mathcal {B}_{\mathcal {Q}}\) be the fiber of \(\pi ^{\operatorname{Coh}}_{d}\) over \(\mathcal {Q}\). Then \(\mathcal {B}_{\mathcal {Q}}\) classifies complete flags of subsheaves \(0\subset \mathcal {Q}_{1}\subset \mathcal {Q}_{2}\subset \cdots \subset \mathcal {Q}_{d1} \subset \mathcal {Q}\). We write \(\mathrm {H}^{*}({}) := \mathrm {H}^{*}({; \mathbf {Q}_{\ell}})\) for \(\ell \)adic cohomology (regarded as a graded \(\mathbf {Q}_{\ell}\)vector space). By Corollary 3.4, \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}})=(\operatorname{Spr}_{d})_{\mathcal {Q}}\) carries an action of \(S_{d}\).
For \(y\in \Sigma (Z)\), let \(\mathcal {B}_{\mathcal {Q}}(y)\) be the open and closed subscheme of \(\mathcal {B}_{\mathcal {Q}}\) defined by the condition \(\operatorname{supp} \mathcal {Q}_{i}/\mathcal {Q}_{i1}=y(i)\). Then \(\mathcal {B}_{\mathcal {Q}}\) is the disjoint union of \(\mathcal {B}_{\mathcal {Q}}(y)\) for \(y\in \Sigma (Z)\). Hence
There is an action of \(S_{d}\) on \(\Sigma (Z)\) by precomposing.
Lemma 3.5
The action of \(w\in S_{d}\) on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}})\) sends \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y)})\) to \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y\circ w^{1})})\), for all \(y\in \Sigma (Z)\).
Proof
It suffices to check the statement for each simple reflection \(s_{i}\) switching \(i\) and \(i+1\) (\(1\le i\le d1\)). Let \(\widetilde {\operatorname{Coh}}^{i}_{d}(X)\) be the moduli stack classifying chains of torsion coherent sheaves \(0\subset \mathcal {Q}_{1}\subset \cdots \subset \mathcal {Q}_{i1}\subset \mathcal {Q}_{i+1} \subset \cdots \subset \mathcal {Q}_{d}\) with \(\mathcal {Q}_{i}\) missing. Then we have a factorization
The map \(\rho _{i}\) is an étale double cover over the open dense locus \(\widetilde {\operatorname{Coh}}^{i,\heartsuit }_{d}(X)\) where \(\mathcal {Q}_{i+1}/\mathcal {Q}_{i1}\) (which has length 2) is supported at two distinct points. The map \(\rho _{i}\) is small by Lemma 3.3, and \(R\rho _{i*}\overline {\mathbf {Q}}_{\ell}\) carries an involution \(\widetilde {s}_{i}\), which induces an involution \(\widetilde {s}_{i}\) on \(R\pi _{i*}R\rho _{i*}\overline {\mathbf {Q}}_{\ell}\cong \operatorname{Spr}_{d}\). This action coincides with the action of \(s_{i}\) over \(\operatorname{Coh}_{d}(X)^{\circ }\), hence coincides with \(s_{i}\) everywhere.
Let \(\mathcal {B}_{\mathcal {Q}}^{i}=\pi _{i}^{1}(\mathcal {Q})\). By considering the support of the successive quotients, we have a decomposition of \(\mathcal {B}_{\mathcal {Q}}^{i}\) by the orbit set \(\Sigma (Z)/\langle {s_{i}}\rangle \). When \(y\in \Sigma (Z)\) satisfies \(y\ne y\circ s_{i}\), the \(s_{i}\)orbit \(\eta =\{y,y\circ s_{i}\}\) gives an open and closed substack \(\mathcal {B}^{i}_{\mathcal {Q}}(\eta )\subset \mathcal {B}_{\mathcal {Q}}^{i}\), such that \(\rho ^{1}_{i}(\mathcal {B}^{i}_{\mathcal {Q}}(\eta ))=\mathcal {B}_{\mathcal {Q}}(y)\coprod \mathcal {B}_{\mathcal {Q}}(y \circ s_{i})\), and \(\mathcal {B}^{i}_{\mathcal {Q}}(\eta )\subset \widetilde {\operatorname{Coh}}^{i,\heartsuit }_{2d}\). Therefore in this case the action of \(\widetilde {s}_{i}\) on \(\mathrm {H}^{*}({\rho ^{1}_{i}(\mathcal {B}^{i}_{\mathcal {Q}}(\eta ))})\) comes from the involution on \(\mathcal {B}_{\mathcal {Q}}(y)\coprod \mathcal {B}_{\mathcal {Q}}(y\circ s_{i})\) that interchanges the two components. Since \(\widetilde {s}_{i}=s_{i}\), this proves the statement for \(s_{i}\) and \(y\) such that \(y\ne y\circ s_{i}\). For \(y=y\circ s_{i}\) the statement is vacuous. This finishes the proof. □
Let \(\mathcal {Q}_{x}\) be the direct summand of \(\mathcal {Q}\) supported at \(x\in Z\). Let \(d_{x}=\dim _{\overline{k}}\mathcal {Q}_{x}\). Then for any \(y\in \Sigma (Z)\), there is a canonical isomorphism over \(\overline{k}\)
sending \((\mathcal {Q}_{i})\in \mathcal {B}_{\mathcal {Q}}(y)\) to the full flag of \(\mathcal {Q}_{x}\) given by taking the summands of \(\mathcal {Q}_{i}\) supported at \(x\).
The proof above implies the following statement that we record for future reference.
Lemma 3.6
Let \(y,y'\in \Sigma (Z)\) and let \(w\in S_{d}\) be such that \(y\circ w^{1}=y'\). Assume that \(w\) has minimal length (in terms of the simple reflections \(s_{1},\ldots , s_{d1}\)) among such elements (such \(w\) is unique). Then the Springer action \(w: \mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y)})\to \mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y')})\) is induced by the composition of the canonical isomorphisms \(\beta _{y,y'}:=\beta _{y'}^{1}\circ \beta _{y}: \mathcal {B}_{\mathcal {Q}}(y)\stackrel {\sim }{\to }\mathcal {B}_{\mathcal {Q}}(y')\). In particular, \(w\) sends the fundamental class of \(\mathcal {B}_{\mathcal {Q}}(y)\) to the fundamental class of \(\mathcal {B}_{\mathcal {Q}}(y')\).
Proof
Let \(w^{1}=s_{i_{1}}\cdots s_{i_{N}}\) be a reduced word for \(w^{1}\). Let \(y_{j}=ys_{i_{1}}\cdots s_{i_{j}}\), \(1\le j\le N\). Let \(y_{0}=y\), and \(y'=y_{N}\). Since \(w\) has minimal length among \(w'\in S_{d}\) such that \(y\circ w^{\prime \,1}=y'\), for each \(1\le j\le N\), \(y_{j1}\ne y_{j}\) for otherwise one could delete \(s_{i_{j}}\) to shorten \(w\). Since \(y_{j}=y_{j1}\circ s_{i_{j}}\ne y_{j1}\), the proof of Lemma 3.5 shows that the Springer action of \(s_{i_{j}}: \mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y_{j1})})\to \mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y_{j})})\) is induced by the canonical isomorphism \(\sigma _{j}=\beta ^{1}_{y_{j}}\circ \beta _{y_{j1}}: \mathcal {B}_{\mathcal {Q}}(y_{j1})\stackrel {\sim }{\to }\mathcal {B}_{\mathcal {Q}}(y_{j})\). The action \(w: \mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y)})\to \mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y')})\), being the composition \(\sigma _{N}\circ \cdots \circ \sigma _{1}\), is then equal to \(\beta _{y'}^{1}\circ \beta _{y}:\mathcal {B}_{\mathcal {Q}}(y)\stackrel {\sim }{\to }\mathcal {B}_{\mathcal {Q}}(y')\). □
Corollary 3.7
Let \(y\in \Sigma (Z)\) and \(S_{y}\cong \prod _{x\in Z}S_{d_{x}}\) be the stabilizer of \(y\) under \(S_{d}\). There is an isomorphism of graded \(S_{d}\)representations
Here on the right side, each factor \(S_{d_{x}}\) of \(S_{y}\) acts on the tensor factor indexed by \(x\) (for \(x\in Z\)) via the Springer action in Corollary 3.4on \((\operatorname{Spr}_{d_{x}})_{\mathcal {Q}_{x}}\).
Proof
By Lemma 3.5, \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y\circ w^{1})})=w\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y)})\) for \(w\in S_{d}\). In particular, \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y)})\) is stable under \(S_{y}\), and \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}})\cong \operatorname{Ind}^{S_{d}}_{S_{y}} \mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y)})\). By (3.2) and the Künneth formula, we have \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y)})\cong \otimes _{x\in Z} \mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}_{x}}})\).
It remains to check that the action of \(S_{y}\) on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y)})\) (as the restriction of the \(S_{d}\)action on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}})\)) is the same as the tensor product of the Springer action of \(S_{d_{x}}\) on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}_{x}}})\). Since the action of \(S_{d}\) on \(\Sigma (Z)\) is transitive, it suffices to check this statement for a particular \(y\in \Sigma (Z)\).
Order points in \(Z\) as \(x_{1},\ldots , x_{r}\). Let \(y_{0}\in \Sigma (Z)\) be the unique increasing function, i.e. such that if \(i< j\) then the index of \(y_{0}(i)\) is less than or equal to the index of \(y_{0}(j)\). Let \(d_{i}=d_{x_{i}}\). Let \(\delta =(\delta _{i})_{1\le i\le r}\) be the increasing sequence \(\delta _{i}=d_{1}+\cdots +d_{i}\). Let \(\operatorname{Coh}_{\delta }(X)\) be the moduli stack of partial chains of torsion coherent sheaves \(0\subset \mathcal {Q}_{\delta _{1}}\subset \cdots \subset \mathcal {Q}_{\delta _{r1}} \subset \mathcal {Q}_{\delta _{r}}=\mathcal {Q}\) such that \(\mathcal {Q}_{\delta _{i}}\) has length \(\delta _{i}\). The map \(\pi ^{\operatorname{Coh}}_{d}\) then factorizes as
We have a Cartesian diagram
where \(c\) sends \((\mathcal {Q}_{\delta _{i}})\) to \((\mathcal {Q}_{\delta _{i}}/\mathcal {Q}_{\delta _{i1}})\). By proper base change we have \(R\pi _{\delta *}\overline {\mathbf {Q}}_{\ell}\cong c^{*}(\boxtimes _{i=1}^{r}\operatorname{Spr}_{d_{i}})\), and the latter carries the Springer action of \(S_{d_{1}}\times \cdots \times S_{d_{r}}=S_{y_{0}}\) (pulled back along \(c\)). Pushing forward along \(\nu _{\delta }\), this induces an action of \(S_{y_{0}}\) on \(R\nu _{\delta *}R\pi _{\delta *}\overline {\mathbf {Q}}_{\ell}=\operatorname{Spr}_{d}\). This action coincides with the restriction of the action of \(S_{d}\) because both actions come from deck transformations over \(\operatorname{Coh}_{d}(X)^{\circ }\).
Now \(\nu _{\delta }^{1}(\mathcal {Q})\) contains the point \(\mathcal {Q}^{\dagger }\in \operatorname{Coh}_{\delta }(X)\) where \(\operatorname{supp} \mathcal {Q}_{\delta _{i}}/\mathcal {Q}_{\delta _{i1}}=\{x_{i}\}\) for \(1\le i\le r\). This is an isolated point in \(\nu ^{1}_{\delta }(\mathcal {Q})\), and \(\mathcal {B}_{\mathcal {Q}}(y_{0})=\pi _{\delta }^{1}(\mathcal {Q}^{\dagger })\). Moreover, the isomorphism (3.2) is the one given by taking the Cartesian diagram (3.3) and restricting to \(\mathcal {Q}^{\dagger }\in \operatorname{Coh}_{\delta }(X)\). The above discussion shows that the action of \(S_{y_{0}}\subset S_{d}\) on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}(y_{0})})\subset \mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}})\) is the same as the Springer action of \(\prod _{i}S_{d_{i}}\) on \(\otimes _{i}\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}_{x_{i}}}})\) via the isomorphism (3.2). □
3.4 The Steinberg sheaf
Let \(\operatorname{St}_{d}\in D^{b}(\operatorname{Coh}_{d}(X),\overline {\mathbf {Q}}_{\ell})\) be the direct summand of \(\operatorname{Spr}_{d}\) where \(S_{d}\) acts through the sign representation. We will describe its Frobenius trace function below. The result is wellknown but we include a selfcontained proof.
We call \(\mathcal {Q}\in \operatorname{Coh}_{d}(X)(\overline{k})\) semisimple if it is a direct sum of skyscraper sheaves at closed points.
Proposition 3.8

(1)
If \(\mathcal {Q}\in \operatorname{Coh}_{d}(X)(\overline{k})\) is not semisimple, then the stalk of \(\operatorname{St}_{d}\) at \(\mathcal {Q}\) is zero.

(2)
Let \(\mathcal {Q}=\oplus _{v\in X}k_{v}^{\oplus d_{v}}\in \operatorname{Coh}_{d}(X)(k)\) be semisimple. Then the stalk of \(\operatorname{St}_{d}\) at \(\mathcal {Q}\) is 1dimensional, and the geometric Frobenius \(\operatorname{Frob}\) acts on the stalk \(\operatorname{St}_{d,\mathcal {Q}}\) by the scalar
$$ \varepsilon (\mathcal {Q})\prod _{v\in \operatorname{supp} \mathcal {Q}}q_{v}^{d_{v}(d_{v}1)/2} $$where \(\varepsilon (\mathcal {Q})\in \{\pm 1\}\) is the sign of Frobenius permuting the geometric points in the support of \(\mathcal {Q}\) counted with multiplicities (as a multiset of cardinality \(d\)).
Proof
Let \(\mathcal {Q}\in \operatorname{Coh}_{d}(X)(\overline{k})\). Let \(Z\subset X(\overline{k})\) be the geometric points in the support of \(\mathcal {Q}\) and \(y\in \Sigma (Z)\). By Corollary 3.7 and Frobenius reciprocity,
(1) By the above factorization of \(\operatorname{St}_{d,\mathcal {Q}}\), it suffices to show that if \(\mathcal {Q}_{x}\) is not semisimple, then \(\operatorname{St}_{d_{x}, \mathcal {Q}_{x}}=0\). By Lemma 3.2 we may reduce to the case \(X=\mathbf {A}^{1}\) and \(\mathcal {Q}\) is concentrated at \(x=0\). In this case \(\operatorname{Spr}\) is the usual Springer sheaf on \([\mathfrak {gl}_{d}/\operatorname{GL}_{d}]\), and \(\mathcal {Q}\) corresponds to a nilpotent element \(e\in \mathscr{N}_{d}\subset \mathfrak {gl}_{d}\) (here \(\mathscr{N}_{d}\) is the nilpotent cone in \(\mathfrak {gl}_{d}\)). It is wellknown that \(\operatorname{St}_{d}_{\mathscr{N}_{d}}\cong \delta _{0}[d(d1)]\) where \(\delta _{0}\) is the skyscraper sheaf at \(0\in \mathscr{N}_{d}\). Indeed, by [1, §3.4, Corollary (b)], for any nonzero nilpotent element \(e\in \mathscr{N}_{d}\), the sign representation of \(S_{d}\) does not appear in \(\mathrm {H}^{*}({\mathcal {B}_{e}})\) (\(\mathcal {B}_{e}\) is the Springer fiber for \(e\)). For \(e=0\), the sign representation of \(S_{d}\) only appears in the top degree \(\mathrm {H}^{d(d1)}({\mathcal {B}_{e}})\), which is onedimensional. This implies that \(\operatorname{St}_{d}_{\mathscr{N}}\cong \delta _{0}[d(d1)]\). In particular, \(\operatorname{St}_{d,e}=0\) for all nilpotent \(e\ne 0\).
(2) Let \(\mathcal {Q}\in \operatorname{Coh}_{d}(X)(k)\) be semisimple. Let \(Z\) be the set of closed points in the support of \(\mathcal {Q}\). The above discussion shows that \(\operatorname{St}_{d_{x},\mathcal {Q}_{x}}\cong \mathrm {H}^{top}({\mathcal {B}_{\mathcal {Q}_{x}}}) = \mathrm {H}^{d_{x}(d_{x}1)}({\operatorname{Fl}_{d_{x}}})\) where \(\operatorname{Fl}_{d_{x}}\) is the flag variety for \(\operatorname{GL}_{d_{x}}\). By (3.4), \(\operatorname{St}_{d,\mathcal {Q}}\) is 1dimensional and is in the top degree cohomology of \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}})\). Let
(here \(d_{v}=d_{x}\) for any \(xv\)). Let \(0\ne \xi \in \operatorname{St}_{d,\mathcal {Q}}\subset \oplus _{y\in \Sigma (Z)} \mathrm {H}^{2N}({\mathcal {B}_{\mathcal {Q}}(y)})\). Let \(\operatorname{Fr}: \mathcal {B}_{\mathcal {Q}}\to \mathcal {B}_{\mathcal {Q}}\) be the Frobenius morphism. We need to show that \(\operatorname{Fr}^{*}\xi =\varepsilon (\mathcal {Q})q^{N}\xi \).
For \(y\in \Sigma (Z)\), let \(\eta _{y}\in \mathrm {H}^{2N}({\mathcal {B}_{\mathcal {Q}}(y)})\) be the fundamental class of \(\mathcal {B}_{\mathcal {Q}}(y)\). Then \(\operatorname{Fr}\) sends \(\mathcal {B}_{\mathcal {Q}}(y)\) onto \(\mathcal {B}_{\mathcal {Q}}(\operatorname{Fr}(y))\) (here \(\operatorname{Fr}(y)\) means postcomposing \(y\) with the Frobenius permutation on \(Z\)), and hence \(\operatorname{Fr}^{*}\eta _{\operatorname{Fr}(y)}=q^{N}\eta _{y}\). On the other hand, let \(w\in S_{d}\) be the minimal length element such that \(\operatorname{Fr}(y)=y\circ w^{1}\). By Lemma 3.6, the Springer action of \(w\) satisfies \(w\eta _{y}=\eta _{\operatorname{Fr}(y)}\). Write \(\xi =(\xi _{y})_{y\in \Sigma (Z)}\) where \(\xi _{y}=c_{y}\eta _{y}\) for some \(c_{y}\in \overline {\mathbf {Q}}_{\ell}^{\times}\). Since \(w\xi =\operatorname {sgn}(w)\xi \), we see that \(w\xi _{y}=\operatorname {sgn}(w)\xi _{\operatorname{Fr}(y)}\). Since \(w\eta _{y}=\eta _{\operatorname{Fr}(y)}\), we have \(c_{y}=\operatorname {sgn}(w)c_{\operatorname{Fr}(y)}\). Therefore
Note that, for any choice of \(y\) and \(w\) above, \(\operatorname {sgn}(w)\) is equal to the sign of the Frobenius permutation of the multiset \(\{y(i)\}_{1\le i\le d}\), which is \(\varepsilon (\mathcal {Q})\). This implies \(\operatorname{Fr}^{*}\xi =\varepsilon (\mathcal {Q})q^{N}\xi \) as desired. □
4 Springer theory for Hermitian torsion sheaves
In this section we extend the construction in §3 to the case of Hermitian torsion sheaves. The main output is a perverse sheaf \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}\) on the moduli stack of Hermitian torsion sheaves with an action of \(W_{d}:=(\mathbf {Z}/2\mathbf {Z})^{d}\rtimes S_{d}\). We will compare the stalks and Frobenius trace functions of \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}\) with those of \(\operatorname{Spr}_{d}\).
As in §3, \(X\) is a smooth curve over \(k\) (not necessarily projective or connected). Recall from §1.3 that \(\nu : X'\to X\) is a finite map of degree 2 that is assumed to be generically étale (and \(X'\) is smooth over \(k\)). We develop the Hermitian Springer theory in this generality. Starting from §4.3 we will assume \(\nu \) to be étale, which is the case needed for proving the main theorem. Let \(\sigma \in \operatorname {Gal}(X'/X)\) be the nontrivial involution.
4.1 Local geometry of \(\operatorname{Herm}_{d}\)
We say that a map of torsion coherent sheaves \(a \colon \mathcal {Q}\rightarrow \sigma ^{*} \mathcal {Q}^{\vee}\) is Hermitian if \(\sigma ^{*} a^{\vee}= a\). Let \(d\in \operatorname{\mathbf{N}}\). Let
be the moduli stack of pairs \((\mathcal {Q},h)\) where \(\mathcal {Q}\) is a torsion coherent sheaf on \(X'\) of length \(d\), and \(h\) is a Hermitian isomorphism \(\mathcal {Q}\stackrel {\sim }{\to }\sigma ^{*}\mathcal {Q}^{\vee}:=\sigma ^{*}\underline{\operatorname{Ext}}^{1}(\mathcal {Q},\omega _{X'})\).
We offer two other ways to think about a Hermitian torsion sheaf \((\mathcal {Q},h)\). For a torsion sheaf \(\mathcal {Q}\) on \(X'\) of length \(d\), the datum of a Hermitian isomorphism \(h \colon \mathcal {Q}\xrightarrow{\sim} \sigma ^{*} \mathcal {Q}^{\vee}\) is equivalent to either:

(1)
a symmetric \(k\)bilinear nondegenerate pairing
$$ (\cdot ,\cdot ): V\times V\to k $$on \(V=\Gamma (X', \mathcal {Q})\) satisfying \((fv_{1},v_{2})=(v_{1},\sigma ^{*}(f)v_{2})\) for any function \(f\) on \(X'\) regular near the support of \(\mathcal {Q}\), or

(2)
an \(\mathcal {O}_{X'}\)sesquilinear nondegenerate pairing
$$ \langle {\cdot ,\cdot}\rangle : \mathcal {Q}\times \mathcal {Q}\to \omega _{F'}/\omega _{X'} $$satisfying \(\langle {v_{1},v_{2}}\rangle =\sigma ^{*}\langle {v_{2},v_{1}}\rangle \). Here \(\omega _{F'}\) is the constant (and quasicoherent) sheaf on \(X'\) whose local sections are the rational 1forms on \(X'\).
For example, to pass from (2) to (1), form cohomology and apply the trace map \(\mathrm {H}^{0}({\mathcal {Q}}) \rightarrow k\). To pass from \(h\) to the pairing in (1), observe that \(\underline {\operatorname{Ext}}^{1}(Q, \omega _{X'}) \cong \underline {\operatorname{Hom}}(Q, \omega _{F'}/ \omega _{X'})\) by the long exact sequence associated to \(\omega _{X'} \rightarrow \omega _{F'} \rightarrow \omega _{F'}/ \omega _{X'}\). Therefore, \(h\) is equivalent to a sesquilinear pairing \(\mathcal {Q}\times \mathcal {Q}\rightarrow \omega _{F'}/\omega _{X'}\), which upon taking global sections and applying the residue map \(\mathrm {H}^{0}({\omega _{F'}/\omega _{X'}}) \rightarrow k\) gives the pairing in (1).
We refer to \(h\), or any of the above equivalent data, as a Hermitian structure on \(\mathcal {Q}\).
We have the support map
Note that we have an isomorphism \((X'_{2d})^{\sigma } \cong X_{d}\), sending a \(\sigma \)invariant divisor on \(X'\) to its descent on \(X\). so we will also allow ourselves to view the support map as \(s^{\operatorname{Herm}}_{2d}: \operatorname{Herm}_{2d}(X'/X)\to X_{d}\).
Remark 4.1
When \(\nu \) is étale and \(d\) is odd, \((X'_{d})^{\sigma }=\varnothing \) hence \(\operatorname{Herm}_{d}(X'/X)=\varnothing \).
In general, when \(\nu \) is ramified over the points \(R\subset X(\overline{k})\), \((X'_{d})^{\sigma }\) has a decomposition into open and closed subschemes according to the parity of the multiplicities of the divisor at each point \(x\in R\).
Let \(\mathbf {A}^{1}_{\sqrt{t}}\to \mathbf {A}^{1}_{t}\) be the square map of affine lines.
Lemma 4.2
There is a canonical isomorphism
Here \(\mathrm {O}_{d}\) denotes the orthogonal group on a \(d\)dimensional nondegenerate quadratic space over \(k\) and \(\mathfrak {o}_{d}\) is its Lie algebra (the stack \([\mathfrak {o}_{d}/\mathrm {O}_{d}]\) is independent of the quadratic form).
Proof
We give the map \(\operatorname{Herm}_{d}(\mathbf {A}^{1}_{\sqrt{t}}/\mathbf {A}^{1}_{t})\to [\mathfrak {o}_{d}/\mathrm {O}_{d}]\) on \(S\)points. For an \(S\)point \((\mathcal {Q},h)\) of \(\operatorname{Herm}_{d}(\mathbf {A}^{1}_{\sqrt{t}}/\mathbf {A}^{1}_{t})\), \(V=\Gamma (\mathbf {A}^{1}_{S}, \mathcal {Q})\) is a locally free \(\mathcal {O}_{S}\)module of rank \(d\) with a nondegenerate symmetric selfduality \((\cdot ,\cdot )\), i.e., an \(\mathrm {O}_{d}\)torsor over \(S\). Moreover the action of \(\sqrt{t}\) on \(V\) satisfies \((\sqrt{t}v_{1},v_{2})=(v_{1},\sqrt{t}v_{2})\) since \(\sigma ^{*}\sqrt{t}=\sqrt{t}\). Therefore \(\sqrt{t}\) gives a section of the adjoint bundle of \(V\). It is easy to check this map is an equivalence of groupoids \(\operatorname{Herm}_{d}(\mathbf {A}^{1}_{\sqrt{t}}/\mathbf {A}^{1}_{t})(S)\stackrel {\sim }{\to }[\mathfrak {o}_{d}/\mathrm {O}_{d}](S)\). □
An \(\sigma \)equivariant étale chart of \(X'_{\overline{k}}\) is a pair \((U,f)\), where \(U\subset X_{\overline{k}}\) is an open subset (with preimage \(U'\subset X'_{\overline{k}}\)) and a regular function \(f: U'\to \mathbf {A}^{1}_{\sqrt{t}, \overline{k}}\) that is an étale map satisfying \(\sigma ^{*}f=f\). Note that if \(\nu \) is étale, the image of \(f\) has to lie in \(\mathbf {A}^{1}_{\sqrt{t}, \overline{k}}\backslash \{0\}\).
A \(\sigma \)equivariant étale chart \((U,f)\) of \(X'_{\overline{k}}\) induces a map
by sending \(\mathcal {Q}\) to \(f_{*}\mathcal {Q}\). Let \(\operatorname{Herm}_{d}(U'/U)^{f}\) be the preimages of \((U'_{d})^{\sigma }\backslash \mathfrak {R}^{\sigma }_{d,f}\) under the support maps (here \(\mathfrak {R}_{d,f}\subset U'_{d}\) is defined using the map \(f:U'\to \mathbf {A}^{1}_{\sqrt{t}, \overline{k}}\); see §3.1).
We have an analog of Lemma 3.2 in the Hermitian setting.
Lemma 4.3

(1)
Let \((U,f)\) be a \(\sigma \)equivariant étale chart for \(X'_{\overline{k}}\). Then the map \(f^{\operatorname{Herm}}_{d}\) is étale when restricted to \(\operatorname{Herm}_{d}(U'/U)^{f}\).

(2)
Assume \(\nu \) is ramified at most one point (over \(\overline{k}\)). Then the stack \(\operatorname{Herm}_{d}(X'/X)_{\overline{k}}\) is covered by \(\operatorname{Herm}_{d}(U'/U)^{f}\) for various \(\sigma \)equivariant étale charts \((U,f)\) of \(X'_{\overline{k}}\). In particular, \(\operatorname{Herm}_{d}(X'/X)\) is étale locally isomorphic to \([\mathfrak {o}_{d}/\mathrm {O}_{d}]\).

(3)
In general, \(\operatorname{Herm}_{d}(X'/X)\) is smooth of dimension 0.
Proof
(1) is similar to that of Lemma 3.2(1).
(2) We only need to construct for \((\mathcal {Q},h)\in \operatorname{Herm}_{d}(X'/X)(\overline{k})\), a \(\sigma \)equivariant étale chart \((U,f)\) such that \((\mathcal {Q},h)\in \operatorname{Herm}_{d}(U'/U)^{f}\). Let \(Z'\) be its support in \(X'\); since the Hermitian property of \((\mathcal {Q}, h)\) implies that \(Z'\) is stable under \(\sigma \), hence is the preimage of some \(Z\subset X(\overline{k})\) under \(\nu \). Let \(\mathcal {L}=(\nu _{*}\mathcal {O}_{X'_{\overline{k}}})^{\sigma =1}\), a line bundle over \(X\). Then the map \(r=(r_{z})_{z\in Z}: \mathcal {L}\to \oplus _{z\in Z}\mathcal {L}_{z}/\varpi _{z}^{2}\) is surjective. Let \(Z_{0}\subset Z\) be the points over which \(\nu \) is étale (so \(ZZ_{0}\) is empty or has one point). For each \(z\in Z_{0}\), upon choosing \(z'\in Z'\) over \(z\), we may identify \(\mathcal {L}_{z}\) with \(\mathcal {O}_{z}=\overline{k}[[\varpi _{z}]]\); changing \(z'\) to \(\sigma (z')\) changes the identification by a sign. If \(z\in ZZ_{0}\), then \(\mathcal {L}_{z}\cong \sqrt{\varpi _{z}}\overline{k}[[\varpi _{z}]]\). Choose a map \(c:Z_{0}\to \overline{k}^{\times}\) such that \(c(z)^{2}\) are distinct for \(z\in Z\). Let \(f\) be a section of ℒ over some open neighborhood \(U_{1}\subset X_{\overline{k}}\) of \(Z\) such that \(r_{z}(f)=c(z)+\varpi _{z}\) for \(z\in Z_{0}\) under one of the two identifications \(\mathcal {L}_{z}\cong \mathcal {O}_{z}\), and \(r_{z}(f)\equiv \sqrt{\varpi _{z}}\ \ \mathrm{mod}\ {\varpi }_{z}\) for \(z\in ZZ_{0}\). Then \(f\) restricts to an étale map \(U'=\nu ^{1}(U)\to \mathbf {A}^{1}_{\sqrt{t}, \overline{k}}\) for some open neighborhood \(U\) of \(Z\) in \(U_{1}\). The definition of ℒ implies \(\sigma ^{*}(f)=f\). Now \(\{f(z')z'\in Z'\}\) is the union of \(\{c(z),c(z)z\in Z_{0}\}\) and possibly \(\{0\}\) if \(ZZ_{0}\) is nonempty, which are all distinct points in \(\mathbf {A}^{1}_{\sqrt{t}, \overline{k}}\) by construction. We conclude that \((\mathcal {Q},h)\in \operatorname{Herm}_{d}(U'/U)^{f}\).
(3) Let \(R\subset X_{\overline{k}}\) be the ramification locus of \(\nu \). The case \(R\le 1\) is treated in (2), so we may assume \(R\ge 2\). For \(x\in R\), let \(Y_{x}=X\backslash (R\backslash \{x\})\) and let \(Y'_{x}=\nu ^{1}(Y_{x})\). For any function \(\delta :R\to \mathbf {Z}_{\ge 0}\) such that \(\sum _{x\in R} \delta (x)=d\) we have a map \(\mathfrak {Y}_{\delta }:=\prod _{x\in R}(Y'_{x,d(x)})^{\sigma }\to (X'_{d})^{\sigma }\) by adding divisors. Let \(\mathfrak {Y}_{\delta }^{\heartsuit }\subset \mathfrak {Y}_{\delta }\) be the open locus where the divisors indexed by different \(x\in R\) are disjoint. It is clear that \(\mathfrak {Y}^{\heartsuit }_{\delta }\to (X'_{d})^{\sigma }\) is étale and for varying \(\delta \) their images cover \((X'_{d})^{\sigma }\). To prove the statement it suffices to show that the base change \(\operatorname{Herm}_{d}(X'/X)_{\mathfrak {Y}_{\delta }^{\heartsuit }}\) is smooth of dimension 0 for each \(\delta \). Observe that \(\operatorname{Herm}_{d}(X'/X)_{\mathfrak {Y}_{\delta }^{\heartsuit }}\) is isomorphic to the restriction of the product \(\prod _{x\in R}\operatorname{Herm}_{\delta (x)}(Y'_{x}/Y_{x})\) to \(\mathfrak {Y}_{\delta }^{\heartsuit }\). Since \(\nu _{Y_{x}}: Y'_{x}\to Y_{x}\) is ramified at one point, by (2) \(\operatorname{Herm}_{\delta (x)}(Y'_{x}/Y_{x})\) is smooth of dimension 0. Therefore \(\operatorname{Herm}_{d}(X'/X)_{\mathfrak {Y}_{\delta }^{\heartsuit }}\cong \prod _{x\in R}\operatorname{Herm}_{\delta (x)}(Y'_{x}/Y_{x})_{ \mathfrak {Y}_{\delta }^{\heartsuit }}\) is smooth of dimension 0. □
Remark 4.4
There is an obvious notion of skewHermitian torsion sheaves. Let \(\mathrm {SkHm}_{d}(X'/X)\) be the moduli stack of skewHermitian torsion sheaves on \((X',\sigma )\) of length \(d\). Then \(d\) is even if \(\mathrm {SkHm}_{d}(X'/X)\ne \varnothing \). The skewHermitian analog of Lemma 4.3 says that \(\mathrm {SkHm}_{d}(X'/X)\) is étale locally isomorphic to \([\mathfrak {sp}_{d}/\operatorname{Sp}_{d}]\), at least when \(\nu \) is étale.
4.2 The Hermitian Springer sheaf
Let \(\widetilde {\operatorname{Herm}}_{d}(X'/X)\) be the moduli stack classifying \((\mathcal {Q},h)\in \operatorname{Herm}_{d}(X'/X)\) together with a full flag
where \(\mathcal {Q}_{i}\) has length \(i\) and \(\mathcal {Q}_{di}=\mathcal {Q}_{i}^{\bot}\) (the orthogonal of \(\mathcal {Q}_{i}\) under the Hermitian pairing \(\mathcal {Q}\times \mathcal {Q}\to \omega _{F'}/\omega _{X'}\)). Let
be the forgetful map. Let \(\operatorname{Herm}_{d}(X'/X)^{\circ }\subset \operatorname{Herm}_{d}(X'/X)\) be the preimage of the multiplicityfree part \(X^{\prime \,\circ }_{d}\) under the support map \(s^{\operatorname{Herm}}_{d}\).
We recall the Grothendieck alteration for the full orthogonal group \(\mathrm {O}(V,Q)\) for some vector space \(V\) of dimension \(d\) over \(k\) and a nondegenerate quadratic form \(Q\) on \(V\). Let \(\operatorname{Fl}(V,Q)\) be the flag variety that parametrizes full isotropic flags \(V_{\bullet }=(V_{1}\subset \cdots \subset V_{d}=V)\) in \(V\). Note that when \(d\) is even, this is different from the flag variety of \(\operatorname{SO}(V,Q)\) but rather a double cover of it because there are two choices for \(V_{d/2}\) given the rest of members of a flag. Let \(\mathfrak {o}(V,Q)\) be the Lie algebra of \(\mathrm {O}(V,Q)\). Let \(\widetilde {\mathfrak {o}}(V,Q)\) be the moduli space of pairs \((A, V_{\bullet })\in \mathfrak {o}(V,Q)\times \operatorname{Fl}(V,Q)\) such that \(AV_{i}\subset V_{i}\) for all \(i\). The Grothendieck alteration for \(\mathrm {O}(V,Q)\) is the \(\mathrm {O}(V,Q)\)equivariant map \(\widetilde {\mathfrak {o}}(V,Q)\to \mathfrak {o}(V,Q)\) forgetting the flag. The quotient stacks \([\mathfrak {o}(V,Q)/\mathrm {O}(V,Q)]\) and \([\widetilde {\mathfrak {o}}(V,Q)/\mathrm {O}(V,Q)]\) are canonical; they are independent of the quadratic form \(Q\) and only depend on \(d=\dim V\). Therefore we also write the Grothendieck alteration as \(\pi _{\mathrm {O}_{d}}: [\widetilde {\mathfrak {o}}_{d}/\mathrm {O}_{d}]\to [\mathfrak {o}_{d}/\mathrm {O}_{d}]\).
Proposition 4.5

(1)
If \(\nu \) is ramified at most one point, then the map \(\pi ^{\operatorname{Herm}}_{d}\) is étale locally isomorphic to the Grothendieck alteration \(\pi _{\mathrm {O}_{d}}:[\widetilde {\mathfrak {o}}_{d}/\mathrm {O}_{d}]\to [\mathfrak {o}_{d}/\mathrm {O}_{d}]\).

(2)
In general, \(\widetilde {\operatorname{Herm}}_{d}(X'/X)\) is smooth of dimension 0 and \(\pi ^{\operatorname{Herm}}_{d}\) is a small map. In particular, the complex
$$ \operatorname{Spr}^{\operatorname{Herm}}_{d}:=R\pi ^{\operatorname{Herm}}_{d,*}\overline {\mathbf {Q}}_{\ell}$$is the middle extension perverse sheaf of its restriction to \(\operatorname{Herm}_{d}(X'/X)^{\circ }\).
Proof
(1) The proof is similar to that of Corollary 3.4. For a \(\sigma \)equivariant étale chart \((U,f)\) for \(X'_{\overline{k}}\) we have a diagram with Cartesian squares and étale horizontal maps by Lemma 4.3
Using the isomorphism in Lemma 4.2, we identify \(\pi ^{\operatorname{Herm}}_{d, \mathbf {A}^{1}_{\sqrt{t}}/\mathbf {A}^{1}_{t}}\) with the Grothendieck alteration \(\pi _{\mathrm {O}_{d}}\). Since \(\operatorname{Herm}_{d}(U'/U)^{f}\) cover \(\operatorname{Herm}_{d}(X'/X)\) by Lemma 4.3(2), \(\pi ^{\operatorname{Herm}}_{d,X'/X}\) is étale locally isomorphic to \(\pi ^{\operatorname{Herm}}_{d, \mathbf {A}^{1}_{\sqrt{t}}/\mathbf {A}^{1}_{t}}=\pi _{\mathrm {O}_{d}}\).
(2) We use the notation from the proof of Lemma 4.3(3). We may assume \(R\ge 2\). For each function \(\delta : R\to \mathbf {Z}_{\ge 0}\) satisfying \(\sum _{x\in R}\delta (x)=d\), the base change of \(\pi ^{\operatorname{Herm}}_{d}\) along \(\mathfrak {Y}_{\delta }^{\heartsuit }\to (X'_{d})^{\sigma }\) is a disjoint union of the restriction of
to \(\mathfrak {Y}_{\delta }^{\heartsuit }\). The disjoint union comes from different ways to distribute \(\operatorname{supp}(\mathcal {Q}_{i}/\mathcal {Q}_{i1})\) among various factors in the product \(\prod _{x}(Y'_{x,\delta (x)})^{\sigma }\). By (1), \(\pi ^{\operatorname{Herm}}_{\delta (x)}: \widetilde {\operatorname{Herm}}_{\delta (x)}(Y'_{x}/ Y_{x})\to \operatorname{Herm}_{ \delta (x)}(Y'_{x}/Y_{x})\) has smooth 0dimensional source and is small for each \(x\in R\), the same holds true for the base change of \(\pi ^{\operatorname{Herm}}_{d}\) to \(\mathfrak {Y}_{\delta }^{\heartsuit }\). Since \(\{\mathfrak {Y}_{\delta }^{\heartsuit }\}_{\delta }\) form an étale covering of \((X'_{d})^{\sigma }\), the same is true for \(\pi ^{\operatorname{Herm}}_{d}\). □
4.3 The action of \(W_{d}\)
From now on we assume that \(\nu :X'\to X\) is an étale double cover. In this case, \((X'_{2d})^{\sigma }\) can be identified with \(X_{d}\) via \(\nu ^{1}(D)\leftrightarrow D\). Let
be the Weyl group for \(\mathrm {O}_{2d}\). Then \(\pi ^{\operatorname{Herm}}_{2d}\) is a \(W_{d}\)torsor over \(\operatorname{Herm}_{2d}(X'/X)^{\circ }\).
Corollary 4.6
of Proposition 4.5(1)
If \(\nu :X'\to X\) is an étale double cover, then there is a canonical action of \(W_{d}\) on \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}\) extending the geometric action on its restriction to \(\operatorname{Herm}_{2d}(X'/X)^{\circ }\).
Definition 4.7

(1)
For any representation \(\rho \) of \(W_{d}\), we define \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}[\rho ]\) to be the perverse sheaf on \(\operatorname{Herm}_{2d}(X'/X)\):
$$ \operatorname{Spr}^{\operatorname{Herm}}_{2d}[\rho ]=(\rho ^{\vee} \otimes \operatorname{Spr}^{\operatorname{Herm}}_{2d})^{W_{d}} \in D^{b}(\operatorname{Herm}_{2d}(X'/X),\overline {\mathbf {Q}}_{\ell}). $$ 
(2)
We define the Hermitian analog of the Springer sheaf \(\operatorname{Spr}\) as
$$\begin{aligned} \operatorname{HSpr}_{d}:=(\operatorname{Spr}^{\operatorname{Herm}}_{2d})^{(\mathbf {Z}/2\mathbf {Z})^{d}}\in D^{b}(\operatorname{Herm}_{2d}(X'/X), \overline {\mathbf {Q}}_{\ell}). \end{aligned}$$
Note that the notation shifts from the subscript \(2d\) to \(d\). By Corollary 4.6, \(\operatorname{HSpr}_{d}\) carries a canonical \(S_{d}\)action.
Remark 4.8
In the case \(\nu \) is ramified with ramification locus \(R\subset X(\overline{k})\), the stack \(\operatorname{Herm}_{2d}(X'/X)_{\overline{k}}\) decomposes into the disjoint union of open and closed substacks \(\operatorname{Herm}^{\varepsilon }_{2d}(X'/X)_{\overline{k}}\) indexed by \(\varepsilon : R\to \{0,1\}\) where the length of \(\mathcal {Q}_{x}\) has parity \(\varepsilon (x)\) for all \(x\in R\). Then \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}_{\operatorname{Herm}^{\varepsilon }_{2d}(X'/X)_{\overline{k}}}\) carries a canonical action of \(W_{d'}\) where \(d'=(d\sum _{x\in R}\varepsilon (x))/2\).
4.4 The Springer fibers over \(\operatorname{Herm}_{2d}\)
Let \((\mathcal {Q},h)\in \operatorname{Herm}_{2d}(X'/X)(\overline{k})\) and consider its Hermitian Springer fiber
This is a proper scheme over \(\overline{k}\). In this subsection we prove the Hermitian analogs of results in §3.3.
Let \(Z'=\operatorname{supp} \mathcal {Q}\subset X'(\overline{k})\). Let \(D'=s^{\operatorname{Herm}}_{2d}(\mathcal {Q})\in (X'_{2d})^{\sigma }(\overline{k})\), which is of the form \(D'=\nu ^{1}(D)\) for some \(D\in X_{d}(\overline{k})\). Let \(Z=\nu (Z')\), the support of \(D\). Write \(D'=\sum _{z\in Z'}d_{z}z\).
Let \(\Sigma (Z')\) be the set of maps \(y': \{1,2,\ldots ,2d\}\to Z'\) satisfying \(y'(2d+1i)=\sigma (y'(i))\) for all \(i\) and \(\sum _{i=1}^{2d}y'(i)=D'\). Identifying \(W_{d}\) with permutations of \(\{1,2,\ldots ,2d\}\) commuting with the involution \(i\mapsto 2d+1i\), we get an action of \(W_{d}\) on \(\Sigma (Z')\) by \(w: y'\mapsto y'\circ w^{1}\).
Similarly let \(\Sigma (Z)\) be the set of maps \(y:\{1,\ldots , d\}\to Z\) such that \(\sum _{i=1}^{d}y(i)=D\). Then the natural map \(\Sigma (Z')\to \Sigma (Z)\) (sending \(y'\) to \(y\) defined by \(y(i)=\nu (y'(i))\)) is a \((\mathbf {Z}/2\mathbf {Z})^{d}\)torsor.
For \(y'\in \Sigma (Z')\), let \(\mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}(y')\) be the subscheme of \(\mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}\) consisting of isotropic flags \(\mathcal {Q}_{\bullet }\) such that \(\operatorname{supp}(\mathcal {Q}_{i}/\mathcal {Q}_{i1})=y'(i)\) for all \(1\le i\le 2d\). Then we have a decomposition into open and closed subschemes
Accordingly we get a decomposition of cohomology
Lemma 4.9
The action of \(w\in W_{d}\) on \(\mathrm {H}^{*}({\mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}})\) sends the direct summand \(\mathrm {H}^{*}({\mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}(y')})\) to the direct summand \(\mathrm {H}^{*}({\mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}(y'\circ w^{1})})\).
Proof
It suffices to check the statement for each simple reflection \(s_{i}\), \(i=1,\ldots , d\). Here, for \(1\le i\le d1\), \(s_{i}=(i,i+1)(2di,2d+1i)\); for \(i=d\), \(s_{d}=(d,d+1)\). For \(1\le i\le d\), let \(\widetilde {\operatorname{Herm}}^{i}_{2d}\) be the moduli stack classifying isotropic flags that only misses the terms \(\mathcal {Q}_{i}\) and \(\mathcal {Q}_{2di}\) (for \(i=d\) only misses \(\mathcal {Q}_{d}\)). Then we have a factorization
The map \(\rho _{i}\) is an étale double cover over the open dense locus \(\widetilde {\operatorname{Herm}}^{i,\heartsuit }_{2d}\) where \(\mathcal {Q}_{i+1}/\mathcal {Q}_{i1}\) (which has length 2) is supported at two distinct points. The map \(\rho _{i}\) is small, and \(R\rho _{i*}\overline {\mathbf {Q}}_{\ell}\) carries an involution \(\widetilde {s}_{i}\), which induces an involution \(\widetilde {s}_{i}\) on \(R\pi _{i*}R\rho _{i*}\overline {\mathbf {Q}}_{\ell}\cong R\pi ^{\operatorname{Herm}}_{2d*}\overline {\mathbf {Q}}_{\ell}\). This action coincides with the action of \(s_{i}\) over \(\operatorname{Herm}_{2d}^{\circ }\), hence coincides with \(s_{i}\) everywhere.
Let \((\mathcal {Q},h)\in \operatorname{Herm}_{2d}(\overline{k})\), and \(\mathcal {B}_{\mathcal {Q}}^{i}=\pi _{i}^{1}(\mathcal {Q},h)\). We have a decomposition of \(\mathcal {B}_{\mathcal {Q}}^{i}\) by the orbit set \(\Sigma (Z')/\langle {s_{i}}\rangle \). When \(y'\in \Sigma (Z')\) satisfies \(y'\ne y'\circ s_{i}\), the \(s_{i}\)orbit of \(\eta '=\{y',y'\circ s_{i}\}\) gives an open closed substack \(\mathcal {B}^{i}_{\mathcal {Q}}(\eta ')\subset \mathcal {B}_{\mathcal {Q}}^{i}\), such that \(\rho ^{1}_{i}(\mathcal {B}^{i}_{\mathcal {Q}}(\eta '))=\mathcal {B}_{\mathcal {Q}}(y')\coprod \mathcal {B}_{\mathcal {Q}}(y' \circ s_{i})\), and \(\mathcal {B}^{i}_{\mathcal {Q}}(\eta ')\subset \widetilde {\operatorname{Herm}}^{i,\heartsuit }_{2d}\). Therefore in this case the action of \(\widetilde {s}_{i}\) on \(\mathrm {H}^{*}({\rho ^{1}_{i}(\mathcal {B}^{i}_{\mathcal {Q}}(\eta '))})\) comes from the involution on \(\mathcal {B}_{\mathcal {Q}}(y')\coprod \mathcal {B}_{\mathcal {Q}}(y'\circ s_{i})\) that interchanges the two components. Since \(\widetilde {s}_{i}=s_{i}\), this proves the statement for \(s_{i}\) and \(y'\) such that \(y'\ne y'\circ s_{i}\). For \(y'=y'\circ s_{i}\) the statement is vacuous. This finishes the proof. □
Choose \(Z^{\sharp }\subset Z'\) such that \(Z^{\sharp }\coprod \sigma (Z^{\sharp })=Z'\). Then for each \(x\in Z\) there is a unique \(x^{\sharp }\in Z^{\sharp }\) above \(x\). For \(y'\in \Sigma (Z')\) with image \(y\in \Sigma (Z)\), we have an isomorphism
mapping \((\mathcal {Q}_{i})_{1\le i\le 2d}\) to the (nonstrictly increasing) flag \((\mathcal {Q}_{i,x^{\sharp }})\) of \(\mathcal {Q}_{x^{\sharp }}\).
If \(y',y''\in \Sigma (Z')\), the composition
is independent of the choice of \(Z^{\sharp }\).
Lemma 4.10
Let \(y',y''\in \Sigma (Z')\) and let \(w\in W_{d}\) be a minimal length element such that \(y''=y'\circ w^{1}\). Then the Springer action \(w: \mathrm {H}^{*}({\mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}(y')})\to \mathrm {H}^{*}({\mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}(y'')})\) is induced by the isomorphism \(\gamma _{y', y''}\).
Proof
Similar to the proof of Lemma 3.6. □
4.5 Comparing stalks of \(\operatorname{HSpr}_{d}\) and \(\operatorname{Spr}_{d}\)
In this subsection we abbreviate \(\operatorname{Herm}_{2d}(X'/X)\) by \(\operatorname{Herm}_{2d}\). Consider the stack \(\operatorname{Lagr}_{2d}\) classifying pairs \((\mathcal {L}\subset \mathcal {Q})\) where \(\mathcal {Q}\in \operatorname{Herm}_{2d}\) and \(\mathcal {L}\subset \mathcal {Q}\) is a Lagrangian subsheaf, i.e., ℒ has length \(d\) and the composition \(\mathcal {L}\hookrightarrow \mathcal {Q}\xrightarrow {h}\sigma ^{*}\mathcal {Q}^{\vee}\to \sigma ^{*}\mathcal {L}^{\vee}\) is zero. We have natural maps
where \(\upsilon _{2d}(\mathcal {L}\subset \mathcal {Q})=\mathcal {Q}\) and \(\varepsilon '_{d}(\mathcal {L}\subset \mathcal {Q})=\mathcal {L}\). Let \(\varepsilon _{d}=\nu _{*}\circ \varepsilon '_{d}: \operatorname{Lagr}_{2d}\to \operatorname{Coh}_{d}(X)\).
Let \((X'_{d})^{\diamondsuit }\subset X'_{d}\) be the open subscheme parametrizing \(D\in X'_{d}\) such that \(D\cap \sigma (D)=\varnothing \). Let \(\operatorname{Lagr}^{\diamondsuit }_{2d}\subset \operatorname{Lagr}_{2d}\) be the preimage of \((X'_{d})^{\diamondsuit }\) under the map \(\operatorname{Lagr}_{2d}\xrightarrow {\varepsilon '_{d}}\operatorname{Coh}_{d}(X')\xrightarrow {s^{\operatorname{Coh}}_{d,X'}} X'_{d}\). It is easy to see that \(\varepsilon '_{d}\) restricts to an isomorphism
whose inverse is given by \(\mathcal {L}\mapsto (\mathcal {L}\subset \mathcal {Q}=\mathcal {L}\oplus \sigma ^{*}\mathcal {L}^{\vee})\).
Let \(\upsilon ^{\diamondsuit }_{2d}\) and \(\varepsilon ^{\diamondsuit }_{d}\) be the restrictions of \(\upsilon _{2d}\) and \(\varepsilon _{d}\) to \(\operatorname{Lagr}^{\diamondsuit }_{2d}\). Thus we view \(\operatorname{Lagr}^{\diamondsuit }_{2d}\cong \operatorname{Coh}_{d}(X')^{\diamondsuit }\) as a correspondence between \(\operatorname{Herm}_{2d}\) and \(\operatorname{Coh}_{d}(X)\)
Note that both \(\upsilon ^{\diamondsuit }_{2d}\) and \(\varepsilon ^{\diamondsuit }_{d}\) are surjective. Now both \(\operatorname{Herm}_{2d}\) and \(\operatorname{Coh}_{d}(X)\) carry Springer sheaves \(\operatorname{HSpr}_{d}\) and \(\operatorname{Spr}_{d}\) with \(S_{d}\)actions. The following proposition says that they become isomorphic after pullback to \(\operatorname{Lagr}^{\diamondsuit }_{2d}\).
Proposition 4.11
There is a canonical \(S_{d}\)equivariant isomorphism of perverse sheaves on \(\operatorname{Lagr}^{\diamondsuit }_{2d}\)
Proof
The map \(\pi ^{\operatorname{Herm}}_{2d}\) factors as
Let \(\lambda ^{\diamondsuit }_{2d}: \widetilde {\operatorname{Herm}}^{\diamondsuit }_{2d}\to \operatorname{Lagr}^{\diamondsuit }_{2d}\) be the restriction of \(\lambda _{2d}\) to \(\operatorname{Lagr}^{\diamondsuit }_{2d}\). We have a commutative diagram
Here \(\operatorname{Coh}_{d}(X')^{\diamondsuit }\) and \(\widetilde {\operatorname{Coh}}_{d}(X')^{\diamondsuit }\) are the preimages of \((X'_{d})^{\diamondsuit }\) under the support map. We have:

The middle square is Cartesian. This is true even before restricting to the ♢ locus.

Since \(\varepsilon ^{\prime \,\diamondsuit }_{d}\) is an isomorphism, so is \(\widetilde {\varepsilon }^{\prime \,\diamondsuit }_{d}\).

The rightmost square is Cartesian.
From these properties we get maps
To check \(\alpha \) is an isomorphism, it suffices to check on geometric stalks. Let \(\mathcal {L}\in \operatorname{Coh}_{d}(X')^{\diamondsuit }(\overline{k})\) with support \(Z^{\sharp }\subset X'(\overline{k})\). Let \(\mathcal {Q}=\mathcal {L}\oplus \sigma ^{*}\mathcal {L}^{\vee}\in \operatorname{Herm}_{2d}(\overline{k})\). The support of \(\mathcal {Q}\) is \(Z'=Z^{\sharp }\coprod \sigma (Z^{\sharp })\), with image \(Z\subset X(\overline{k})\). We have \((\mathcal {L}\subset \mathcal {Q})\in \operatorname{Lagr}^{\diamondsuit }_{2d}(\overline{k})\), with image \(\nu _{*}\mathcal {L}\in \operatorname{Coh}_{d}(X)(\overline{k})\). The stalk of \(\alpha \) at \((\mathcal {L}\subset \mathcal {Q})\) is
Recall \(\mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}=\coprod _{y'\in \Sigma (Z')}\mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}(y')\). Let \(\Sigma (Z^{\sharp })\subset \Sigma (Z')\) be the set of \(y'\) such that \(y'(i)\in Z^{\sharp }\) for \(1\le i\le d\). Then we have a natural bijection \(\Sigma (Z)\leftrightarrow \Sigma (Z^{\sharp }), y\mapsto y^{\sharp }\). The fiber \(\lambda ^{1}_{2d}(\mathcal {L}\subset \mathcal {Q})\) is the disjoint union \(\coprod _{y\in \Sigma (Z)}\mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}(y^{\sharp })\). Recall the isomorphism \(\gamma _{Z^{\sharp }, y^{\sharp }}: \mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}(y^{\sharp })\stackrel {\sim }{\to }\mathcal {B}_{\nu _{*} \mathcal {L}}(y)\) from (4.1). Using these descriptions we may rewrite (4.3) as
It remains to show that the first map above is an isomorphism. But this follows from the fact that \((\mathbf {Z}/2\mathbf {Z})^{d}\) acts freely on \(\Sigma (Z')\) with orbit representatives \(\Sigma (Z^{\sharp })\), and Lemma 4.9. This shows that \(\alpha \) is an isomorphism.
Finally we show that \(\alpha \) is \(S_{d}\)equivariant. By Proposition 4.5, \(\pi ^{\operatorname{Herm}}_{2d}\) and hence \(\lambda _{2d}\) is small, \(R\lambda _{2d*}\overline {\mathbf {Q}}_{\ell}\) is the middle extension from a dense open substack of \(\operatorname{Lagr}_{2d}\). Therefore the same is true for \(R\lambda ^{\diamondsuit }_{2d*}\overline {\mathbf {Q}}_{\ell}\). Since \(\alpha \) is an isomorphism, both \(\upsilon ^{\diamondsuit ,*}_{2d}\operatorname{HSpr}_{d}\) and \(\varepsilon ^{\diamondsuit *}_{d}\operatorname{Spr}_{d}\) are middle extension perverse sheaves from a dense open substack of \(\operatorname{Lagr}^{\diamondsuit }_{2d}\). To check that \(\alpha \) is \(S_{d}\)equivariant it suffices to check it over the dense open substack which is the preimage of the multiplicityfree locus \(X_{d}^{\circ }\). Over \(X_{d}^{\circ }\), all squares in (4.2) are Cartesian, and all vertical maps are \(S_{d}\)torsors. The \(S_{d}\)actions on \(\operatorname{HSpr}_{d}_{\operatorname{Herm}_{2d}^{\circ }}\) and \(\operatorname{Spr}_{d}_{\operatorname{Coh}_{d}(X)^{\circ }}\) come from the vertical \(S_{d}\)torsors in the diagram, so \(\alpha \) is \(S_{d}\)equivariant when restricted over \(X_{d}^{\circ }\). This finishes the proof. □
4.6 Comparing Frobenius traces of \(\operatorname{HSpr}_{d}\) and \(\operatorname{Spr}_{d}\)
In this subsection we will prove a relationship between Frobenius trace functions for \(\operatorname{HSpr}_{d}\) and for \(\operatorname{Spr}_{d}\). Since these sheaves live on different stacks, to make sense of the comparison we first need to identify the isomorphism classes of the \(k\)points of these stacks.
For a groupoid \(\mathcal {G}\), let \(\mathcal {G}\) denote its set of isomorphism classes.
Lemma 4.12
There is a canonical bijection of sets
respecting the support maps to \(X_{d}(k)\).
Proof
Let \(\mathcal {P}(d)\) be the set of partitions of \(d\in \mathbf {Z}_{\ge 0}\), and \(\mathcal {P}=\coprod _{d\ge 0}\mathcal {P}(d)\). Let \(\mathcal {P}_{X}\) be the set of functions \(\lambda : X\to \mathcal {P}\) such that \(\lambda (v)\) is the zero partition for almost all \(v\in X\). For \(\lambda \in \mathcal {P}_{X}\), let \(\lambda =\sum _{v}\lambda (v)\deg (v)\). Let \(\mathcal {P}_{X}(d)\) be the subset of those \(\lambda \in \mathcal {P}_{X}\) with \(\lambda =d\). Let \(s_{d}: \mathcal {P}_{X}(d)\to X_{d}(k)\) be the map sending \(\lambda \) to the divisor \(\sum _{v}\lambda (v)v\).
By taking the Jordan type of a torsion sheaf at each closed point, we get a canonical bijection \(\Lambda ^{\operatorname{Coh}}_{d}: \operatorname{Coh}_{d}(X)(k)\stackrel {\sim }{\to }\mathcal {P}_{X}(d)\). The map \(s_{d}^{\operatorname{Coh}}:\operatorname{Coh}_{d}(X)(k)\to X_{d}(k)\) corresponds to \(s_{d}\) under this bijection.
We define a map \(\Lambda ^{\operatorname{Herm}}_{d}: \operatorname{Herm}_{2d}(X'/X)(k)\to \mathcal {P}_{X}(d)\) as follows. For \((\mathcal {Q},h)\in \operatorname{Herm}_{2d}(X'/X)(k)\) and \(v\in X\), let \(\lambda (v)\) be the Jordan type of \(\mathcal {Q}_{v'}\) (the summand \(\mathcal {Q}_{v'}\) supported at \(v'\)) for any \(v'\in X'\) above \(v\). When \(v\) is split in \(X'\), the two choices of \(v'\) give the same Jordan type. The support map \(s_{2d}^{\operatorname{Herm}}: \operatorname{Herm}_{2d}(X'/X)(k)\to X_{d}(k)\) is the composition \(s_{d}\circ \Lambda ^{\operatorname{Herm}}_{d}\).
We claim that \(\Lambda ^{\operatorname{Herm}}_{d}\) is a bijection. Then \(\Lambda ^{\operatorname{Coh},1}_{d}\circ \Lambda ^{\operatorname{Herm}}_{d}: \operatorname{Herm}_{2d}(X'/ X)(k)\stackrel {\sim }{\to } \operatorname{Coh}_{d}(X)(k)\) is the desired bijection.
To prove the claim, since any torsion coherent sheaf splits as a direct sum over the finitely many points in its support, it suffices to fix a closed point \(v\in X\) and show that the set of isomorphism classes \(\operatorname{Herm}_{v}\) of Hermitian torsion sheaves supported above \(v\) maps bijectively (by the restriction of \(\Lambda ^{\operatorname{Herm}}_{d}\)) to \(\mathcal {P}\) which are supported at \(v\).
If \(v\) splits into \(v'\) and \(v''\) in \(X'\), then any \((\mathcal {Q},h)\in \operatorname{Herm}_{v}\) has the form \(\mathcal {Q}_{v'}\oplus \sigma ^{*}\mathcal {Q}_{v'}\) equipped with the canonical Hermitian structure. In this case we see that the isomorphism class of \((\mathcal {Q},h)\) is determined by the Jordan type of \(\mathcal {Q}_{v'}\), and conversely each Jordan type arises from some \((\mathcal {Q}, h)\).
If \(v\) is inert with preimage \(v'\in X'\), let \((\mathcal {Q},h)\in \operatorname{Herm}_{v}\) be of length \(d\) over \(k_{v'}\). Then \(V=\Gamma (X',\mathcal {Q})\) is a \(d\)dimensional Hermitian \(k_{v'}\)vector space with a selfadjoint nilpotent endomorphism \(e\) given by the action of a uniformizer \(\varpi \in \mathcal {O}_{v}\). Fix a \(d\)dimensional Hermitian space \((V_{d},h)\) over \(k_{v'}\) (unique up to isomorphism), then the isomorphism classes of \((\mathcal {Q},h)\in \operatorname{Herm}_{v}\) with length \(d\) over \(k_{v'}\) is in bijection with the adjoint orbits of the unitary group \(\mathrm {U}(V_{d},h)(k_{v})\) acting on the nilpotent cone \(\mathcal {N}(V_{d},h)(k_{v})\) of selfadjoint nilpotent endomorphisms of \(V_{d}\). Being a Galois twisted version of the usual nilpotent orbits under \(\operatorname{GL}_{d}\), the orbits \(\mathcal {N}(V_{d},h)(k_{v})/\mathrm {U}(V_{d},h)(k_{v})\) are again classified by partitions of \(d\) according the Jordan types of \(e\in \mathcal {N}(V_{d},h)(k_{v})\) (here we use that the centralizer \(C_{\operatorname{GL}_{d}}(e)\) is connected, and Lang’s theorem implies \(\mathrm {H}^{1}({k_{v}, C_{\operatorname{GL}_{d}}(e)})=\{1\}\)). Therefore the isomorphism class of \((\mathcal {Q},h)\in \operatorname{Herm}_{v}\) is determined by the Jordan type of \(\mathcal {Q}\), and conversely each Jordan type arises from a \((\mathcal {Q}, h)\). This shows that \(\Lambda ^{\operatorname{Herm}}_{d}\) is a bijection. □
4.6.1 Further notations
Now let \((\mathcal {Q},h)\in \operatorname{Herm}_{2d}(k)\). We write \(\mathcal {Q}_{\overline{k}}\) for the base change of \(\mathcal {Q}\) over \(X'_{\overline{k}}\), and adapt the notations \(Z'\subset X'(\overline{k}),Z\subset X(\overline{k}),\Sigma (Z'),\Sigma (Z)\) from §4.4. Let \(Z'\) and \(Z\) be the set of closed points contained in \(Z'\) and \(Z\). We have a decomposition
into split and inert places. For each closed point \(v\in Z\) we choose a geometric point \(x'_{v}\in Z'\) above \(v\) and denote its image in \(Z\) by \(x_{v}\).
Let \(\operatorname{Fr}:X'\to X'\) be the Frobenius morphism. Let \(Z^{\sharp }\) be the following subset of \(Z'\)
When \(v\) splits into \(v',v''\) in \(X'\), with \(x'_{v}v'\), then \(Z^{\sharp }\) contains all geometric points above \(v'\) and not any above \(v''\). When \(v\) is inert with preimage \(v'\in X'\), \(Z^{\sharp }\) contains half of the geometric points above \(v'\) which form a chain under the Frobenius, starting with \(x'_{v}\). Therefore \(Z'=Z^{\sharp }\coprod \sigma (Z^{\sharp })\). For \(x\in Z\) let \(x^{\sharp }\in Z^{\sharp }\) be the unique element above \(x\). This induces a section \(\Sigma (Z)\stackrel {\sim }{\to }\Sigma (Z^{\sharp })\subset \Sigma (Z')\) which we denote \(y\mapsto y^{\sharp }\).
Let \(\mathcal {Q}^{\flat}\in \operatorname{Coh}_{d}(X)(k)\) be the point corresponding to the isomorphism class of \((\mathcal {Q},h)\) under the bijection in Lemma 4.12. Then \(\mathcal {Q}^{\flat}_{\overline{k}}\cong \nu _{*}(\mathcal {Q}_{Z^{\sharp }})\). Recall the isomorphism (3.2) for each \(y\in \Sigma (Z)\)
From this and the Künneth formula we get an identification
By [9, Corollary 2.3(1)], \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}_{x}}})\) is concentrated in even degrees. Let \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}}(y)})^{+}\) be the direct sum of all \(\otimes _{x\in Z} \mathrm {H}^{2i_{x}}({\mathcal {B}_{\mathcal {Q}^{\flat}_{x}}})\) (for varying \((i_{x})_{x\in Z}\)) such that
Similarly, let \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}}(y)})^{}\) be the direct sum of all \(\otimes _{x\in Z} \mathrm {H}^{2i_{x}}({\mathcal {B}_{\mathcal {Q}^{\flat}_{x}}})\) such that the quantity in (4.4) is odd. We have
Taking direct sum over all \(y\in \Sigma (Z)\) we get a decomposition
Note that this decomposition depends on the choice of a geometric point \(x_{v}\) over each inert \(v\). By Corollary 3.7, the action of \(S_{y}\subset S_{d}\) on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}}(y)})\) preserves the decomposition (4.5) since it is the same as the tensor product of the Springer actions on each factor \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}_{x}}})\). Therefore the decomposition (4.6) is stable under the \(S_{d}\)action.
Now \((\mathcal {Q}_{Z^{\sharp }}\subset \mathcal {Q})\) gives a geometric point of \(\operatorname{Lagr}^{\diamondsuit }_{2d}\), which is not defined over \(k\) if \(Z_{i}\ne \varnothing \). Using this geometric point in \(\operatorname{Lagr}^{\diamondsuit }_{2d}\), Proposition 4.11 gives an isomorphism \(\alpha ^{\sharp }:=\alpha _{(\mathcal {Q}_{Z^{\sharp }}\subset \mathcal {Q})}\) on the level of stalks (see (4.3)):
This isomorphism is \(S_{d}\)equivariant. Both sides now carry geometric Frobenius actions which we denote by \(\operatorname{Frob}_{\mathcal {Q}}\) and \(\operatorname{Frob}_{\mathcal {Q}^{\flat}}\), which are not necessarily intertwined under \(\alpha ^{\sharp }\) because the point \((\mathcal {Q}_{Z^{\sharp }}\subset \mathcal {Q})\) is not necessarily defined over \(k\). The next result gives the relation between the two Frobenius actions.
Proposition 4.13
Let \(\theta \) be the involution on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}}})\) which is 1 on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}}})^{+}\) and −1 on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}}})^{}\). Then under the isomorphism \(\alpha ^{\sharp }\), \(\operatorname{Frob}_{\mathcal {Q}}\) corresponds to \(\operatorname{Frob}_{\mathcal {Q}^{\flat}} \circ \theta \).
Proof
Recall from the proof of Proposition 4.11 that \(\alpha ^{\sharp }\) is the composition
Here \(\gamma _{Z^{\sharp },y^{\sharp }}\) is defined in (4.1).
Then \(\operatorname{Fr}^{*}\) on \(\mathrm {H}^{*}({\mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}})\) maps \(\mathrm {H}^{*}({\mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}(\operatorname{Fr}(y^{\sharp }))})\) to \(\mathrm {H}^{*}({\mathcal {B}^{\operatorname{Herm}}_{\mathcal {Q}}(y^{\sharp })})\). Note that \(\operatorname{Fr}(y^{\sharp })\) and \((\operatorname{Fr}y)^{\sharp }\) are in general different: if \(y(i)=\operatorname{Fr}^{1}(x_{v})\) for some inert \(v\), then \(\operatorname{Fr}(y^{\sharp })(i)=\sigma (x'_{v})\) while \((\operatorname{Fr}y)^{\sharp }(i)=x'_{v}\). In other words, the only difference between \(\operatorname{Fr}(y^{\sharp })\) and \((\operatorname{Fr}y)^{\sharp }\) is the switch of all \(x'_{v}\) and \(\sigma (x'_{v})\) for all inert \(v\). Therefore there is a unique element \(\tau _{y}\in (\mathbf {Z}/2\mathbf {Z})^{d}\) such that \((\operatorname{Fr}y)^{\sharp }=\operatorname{Fr}(y^{\sharp })\circ \tau _{y}\).
Identifying \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}^{\operatorname{Herm}}})^{(\mathbf {Z}/2\mathbf {Z})^{d}}\) with \(\oplus _{y\in \Sigma (Z)}\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}^{\operatorname{Herm}}(y^{\sharp })})\), the geometric Frobenius endomorphism \(\operatorname{Frob}_{\mathcal {Q}}\) on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}}^{\operatorname{Herm}}})^{(\mathbf {Z}/2\mathbf {Z})^{d}}\) is the direct sum of the following compositions
where the first map is the Springer action of \(\tau _{y}\in W_{d}\) on \((\operatorname{Spr}^{\operatorname{Herm}}_{2d})_{\mathcal {Q}}\).
On the other hand, let \(w_{y}\in W_{d}\) be the minimal length element such that \((\operatorname{Fr}y)^{\sharp }= \operatorname{Fr}(y^{\sharp })\circ w_{y}\). Write
Note that \(\operatorname{Stab}_{S_{d}}(Fy)=\prod _{x\in Z}S_{I_{x}}\) where \(I_{x}\subset \{1,2,\ldots , d\}\) is the preimage of \(x\) under \(y\). An easy calculation shows that \(u_{y}=(u_{y,x})_{x\in Z}\) where \(u_{y,x}\in S_{I_{x}}\) is
Here \(w_{I_{x}}\in S_{I_{x}}\) is the involution that reverses the order of \(I_{x}\).
We use abbreviated notation
For each \(y\in \Sigma (Z)\), consider the following diagram
The left square is commutative by the \(S_{d}\)equivariance of \(\alpha ^{\sharp }\) proved in Proposition 4.11 (here \(u_{y}\in \operatorname{Stab}_{W_{d}}((\operatorname{Fr}y)^{\sharp })\subset S_{d}\)). The middle upper triangle is commutative by Lemma 4.10. The map \(\delta ^{*}\) is defined to make the lower middle triangle commutative. The right square is clearly commutative. The composite of the upper row is the restriction of \(\operatorname{Fr}_{\mathcal {Q}}\) to \(H((\operatorname{Fr}y)^{\sharp })\). Let us compute the composite of the lower row.
The map \(\delta ^{*}\) is the pullback along the automorphism \(\delta \) of \(\mathcal {B}_{\mathcal {Q}^{\flat}}(\operatorname{Fr}y)\) that makes the following diagram commutative
Under the isomorphism \(\beta _{\operatorname{Fr}y}: \mathcal {B}_{\mathcal {Q}^{\flat}}(\operatorname{Fr}y)\stackrel {\sim }{\to }\prod _{x\in Z}\mathcal {B}_{ \mathcal {Q}^{\flat}_{x}}\), \(\delta \) is the product of automorphisms \(\delta _{x}\) for each \(\mathcal {B}_{\mathcal {Q}^{\flat}_{x}}\). If \(x\) is not of the form \(x=x_{v}\) for \(v\in Z_{i}\), \(\delta _{x}\) is the identity. If \(x=x_{v}\) for some \(v\in Z_{i}\), then \(\mathcal {Q}^{\flat}_{x}=\mathcal {Q}_{x'_{v}}\) and the Hermitian structure on \(\mathcal {Q}\) gives an isomorphism \(\iota : \mathcal {Q}_{\sigma (x'_{v})}\cong \mathcal {Q}_{x'_{v}}^{\vee}\). On the other hand, \(\operatorname{Fr}^{\deg (v)}\) gives an isomorphism \(\phi : \mathcal {Q}_{x'_{v}}\cong \mathcal {Q}_{\sigma (x'_{v})}\) since \(\sigma (x'_{v})=\operatorname{Fr}^{\deg (v)}(x'_{v})\). Combining \(\iota \) and \(\phi \) we get a perfect symmetric pairing \((\cdot ,\cdot )_{x'_{v}}\) on \(\mathcal {Q}_{x'_{v}}\) itself. Then \(\delta _{x}\) sends a full flag \(\mathcal {R}_{\bullet }\) of \(\mathcal {Q}^{\flat}_{x_{v}}=\mathcal {Q}_{x'_{v}}\) to \(\mathcal {R}^{\bot}_{\bullet }\) under the pairing \((\cdot ,\cdot )_{x'_{v}}\).
By the description of \(u_{y}\) and \(\delta \) above, under the isomorphism \(\beta _{Fy}\), the composition \(\delta ^{*}\circ u_{y}\) takes the form
The automorphisms \(\delta _{x}^{*}\circ u_{y,x}\) are the identity maps except when \(x=x_{v}\) for some \(v\in Z_{i}\), in which case \(u_{y,x}=w_{I_{x}}\). Let us compute \(\delta _{x}^{*}\circ w_{I_{x}}\) on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}_{x}}})\) for \(x=x_{v}\) and \(v\in Z_{i}\). For this we switch to the following notation. Let \(V=\mathcal {Q}_{x'_{v}}=\mathcal {Q}^{\flat}_{x_{v}}\), a vector space of dimension \(m\) over \(\overline{k}\). We have argued that \(V\) carries a symmetric selfduality \((\cdot ,\cdot )\); the action of a uniformizer at \(x_{v}\) gives a nilpotent element \(e\in \operatorname{End}_{\overline{k}}(V)\), which is selfadjoint under \((\cdot ,\cdot )\). Let ℬ be the flag variety of \(\operatorname{GL}(V)\) and \(\mathcal {B}_{e}\) be the Springer fiber of \(e\). Then \(S_{m}\) acts on \(\mathrm {H}^{*}({\mathcal {B}_{e}})\). Let \(w_{0}\) be the longest element in \(S_{m}\). Let \(\delta : \mathcal {B}_{e}\stackrel {\sim }{\to }\mathcal {B}_{e}\) be the map sending a flag \(V_{\bullet }\) to \(V_{\bullet }^{\bot}\). We claim that \(\delta ^{*}\circ w_{0}\) acts on \(\mathrm {H}^{2i}({\mathcal {B}_{e}})\) by \((1)^{i}\). Indeed, by [9, Corollary 2.3(2)], the restriction map \(\mathrm {H}^{*}({\mathcal {B}})\to \mathrm {H}^{*}({\mathcal {B}_{e}})\) is surjective and is clearly equivariant under \(\delta ^{*}\circ w_{0}\), so it suffices to show that \(\delta ^{*}\circ w_{0}\) acts by \((1)^{i}\) on \(\mathrm {H}^{2i}({\mathcal {B}})\). Since \(\delta ^{*}\circ w_{0}\) preserves the cup product on \(\mathrm {H}^{*}({\mathcal {B}})\), it suffices to show it acts by −1 on \(\mathrm {H}^{2}({\mathcal {B}})\) (which generates all of \(\mathrm {H}^{2*}({\mathcal {B}})\) under the cup product). For \(1\le j\le m\), let \(\xi _{j}\) be the Chern class of the tautological line bundle on ℬ whose fiber at \(V_{\bullet }\) is \(V_{j}/V_{j1}\). Then \(\mathrm {H}^{2}({\mathcal {B}})\) is spanned by \(\xi _{j}\) for \(1\le j\le m\). Now we have \(w_{0}(\xi _{j})=\xi _{m+1j}\) since \(\mathrm {H}^{2}({\mathcal {B}})\) is the reflection representation of \(S_{m}\), and \(\delta ^{*}\xi _{j}=\xi _{m+1j}\) by the definition of \(\delta \). Therefore \(\delta ^{*}\circ w_{0}(\xi _{j})=\xi _{j}\) for all \(1\le j\le m\), which proves that \(\delta ^{*}\circ w_{0}\) acts by −1 on \(\mathrm {H}^{2}({\mathcal {B}})\), and hence acts by \((1)^{i}\) on \(\mathrm {H}^{2i}({\mathcal {B}})\) and on \(\mathrm {H}^{2i}({\mathcal {B}_{e}})\).
The above argument shows that \(\delta ^{*}\circ u_{y}\) acts by 1 on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}}(\operatorname{Fr}y)})^{+}\) and acts by −1 on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}}(\operatorname{Fr}y)})^{}\). Therefore the bottom row of (4.7) is \(\operatorname{Fr}_{\mathcal {Q}^{\flat}} \circ \theta \). By the commutativity of (4.7), \(\operatorname{Fr}_{\mathcal {Q}^{\flat}} \circ \theta \) corresponds under \(\alpha ^{\sharp }\) to the composite of the top row, which is \(\operatorname{Fr}_{\mathcal {Q}}\). This finishes the proof. □
5 Geometrization of local densities
The goal of this section is to give a sheaftheoretic interpretation of the formula of ChoYamauchi on the representation density of Hermitian lattices, see Theorem 5.3. This will complete the geometrization of the analytic side of our proposed SiegelWeil formula, at least for nonsingular Fourier coefficients. The technical part of the proof of the theorem is a Frobenius trace calculation that uses properties of the Hermitian Springer action proved in §4.6.
5.1 Density function for torsion sheaves
Following Remark 2.4, for any Hermitian torsion sheaf \((\mathcal {Q},h)\in \operatorname{Herm}_{2d}(X'/X)(k)\), we may define the density polynomial \(\operatorname{Den}(T,\mathcal {Q})\) using the ChoYamauchi formula as follows. Let \(\mathcal {Q}_{v}\) be the summand of \(\mathcal {Q}\) supported over \(v\in X\), we define
where
Here \(I^{\perp}\) is the orthogonal of \(I\) under the Hermitian form on \(\mathcal {Q}_{v}\), so in other words the sum is over all subsheaves of \(\mathcal {Q}_{v}\) that are isotropic under \(h_{v}=h_{\mathcal {Q}_{v}}\), and we write \(\mathfrak {m}_{v}()\) to emphasize the dependence of \(\mathfrak {m}(a;T)\) on \(F'_{v}/F_{v}\) (see Definition 2.2). The functions \(\ell '_{v}()\) and \(t'_{v}()\) are the functions \(\ell '()\) and \(t'()\) defined in §2.4 for \(F'_{v}/F_{v}\).
Expanding the product into a summation, we see
Given an injective Hermitian map \(a \colon \mathcal {E}\hookrightarrow \sigma ^{*} \mathcal {E}^{\vee}\), we have a Hermitian structure on the torsion sheaf \(\operatorname{coker}(a)\) as follows. Applying \(\sigma ^{*}R\underline {\operatorname{Hom}}(,\omega _{X'})\) to the short exact sequence
yields a short exact sequence
Since \(\sigma ^{*}a^{\vee}=a\), we may identify (5.2) and (5.3) and get an isomorphism \(h_{\mathcal {Q}}: \mathcal {Q}\stackrel {\sim }{\to }\sigma ^{*}\mathcal {Q}^{\vee}\). Using this, we may restate Theorem 2.8 as follows.
Theorem 5.1
Let ℰ be a rank \(n\) vector bundle over \(X'\) and \(a:\mathcal {E}\hookrightarrow \sigma ^{*}\mathcal {E}^{\vee}\) be an injective Hermitian map. Then
5.2 Density sheaves
We will define a graded perverse sheaf on \(\operatorname{Herm}_{2d}(X'/X)\) whose Frobenius trace at \(\mathcal {Q}\) recovers \(\operatorname{Den}(T,\mathcal {Q})\). We will suppress \(X'/X\) from the notations.
For \(0 \leq i \leq d\), let \(\operatorname{Herm}_{i,2d}\) be the stack classifying
We have the following maps
Here \(\overleftarrow {f}_{i}\) takes \((\mathcal {I},(\mathcal {Q},h))\) to \((\mathcal {Q},h)\), \(\overrightarrow {f}'_{i}\) takes it to ℐ and \(\overrightarrow {f}''_{i}\) takes it to \(\mathcal {I}^{\bot}/\mathcal {I}\) with the Hermitian structure induced from \(h\).
Recall the perverse sheaf \(\operatorname{HSpr}_{d}\) on \(\operatorname{Herm}_{2d}\) from Definition 4.7(2). It is obtained from the Springer sheaf on \(\operatorname{Herm}_{2d}\) by taking \((\mathbf {Z}/2\mathbf {Z})^{d}\)invariants, and \(\operatorname{HSpr}_{2d}\) carries an action of \(S_{d}\).
Definition 5.2
We define the following graded virtual perverse sheaves on \(\operatorname{Herm}_{2d}\). (In the notation below, the degree of the formal variable \(T\) encodes the grading.)

(1)
\(\mathfrak {P}_{d}(T)=\bigoplus _{j=0}^{d}(1)^{j}(\operatorname{HSpr}_{d})^{(S_{j}\times S_{dj}, \operatorname {sgn}_{j}\boxtimes \mathbf {1})}T^{j}\).

(2)
\(\mathcal {K}^{\operatorname{Eis}}_{d}(T)=\bigoplus _{i=0}^{d} R\overleftarrow {f}_{i,!}R\overrightarrow {f}^{*}_{i}( \overline {\mathbf {Q}}_{\ell}\cdot T^{i}\boxtimes \mathfrak {P}_{di}(T))\).
Theorem 5.3
For any \(\mathcal {Q}\in \operatorname{Herm}_{2d}(k)\), we have
Proof
By the GrothendieckLefschetz trace formula, we have
Comparing with the expansion (5.1) for \(\operatorname{Den}(T,\mathcal {Q})\), it suffices to prove, changing notation \(di\) to \(d\) and \(\mathcal {I}^{\bot}/\mathcal {I}\) to \(\mathcal {Q}\), that
This will be proved in Proposition 5.7. □
The rest of the section is devoted to the proof of (5.4). The idea is to relate the Frobenius trace to a similar Frobenius trace of a graded perverse sheaf on \(\operatorname{Coh}_{d}(X)\) using results from §4.6, and then calculate the latter explicitly.
5.3 Comparison of two graded Frobenius modules
For \((\mathcal {Q},h)\in \operatorname{Herm}_{2d}(k)\), write
Here \(\overline{\operatorname {sgn}}_{j}\) is the inflation of the sign representation of \(S_{j}\) under \(W_{j}\twoheadrightarrow S_{j}\). We view \(\mathfrak {P}_{\mathcal {Q}}(T)\) as a \(\mathbf {Z}\)graded virtual Frobenius module, with the \(\mathbf {Z}\)grading indicated by the power of \(T\). Let \(\mathcal {Q}^{\flat}\in \operatorname{Coh}_{d}(X)(k)\) be in the isomorphism class that corresponds to \((\mathcal {Q},h)\) under the bijection in Lemma 4.12. Define
Define the Frobenius traces
The goal is to get a relationship between \(P_{\mathcal {Q}}(T)\) and \(P_{\mathcal {Q}^{\flat}}(T)\). Note that \(\mathfrak {P}_{\mathcal {Q}^{\flat}}(T)\) is a special case of \(\mathfrak {P}_{\mathcal {Q}}(T)\) when the double cover \(X'=X\sqcup X\) (and \(\mathcal {Q}\) is the direct sum of \(\mathcal {Q}^{\flat}\) on one copy of \(X\) and \(\mathcal {Q}^{\flat ,\vee}\) on the other). We shall apply Proposition 4.13 to express \(P_{\mathcal {Q}}(T)\) in terms of \(P_{\mathcal {Q}^{\flat}}(T)\). For this we need to calculate the decomposition of \(\mathfrak {P}_{\mathcal {Q}^{\flat}}(T)\) given in (4.6).
Recall notations \(Z, Z', \Sigma (Z)\) and \(\Sigma (Z')\) from §4.4. First we show that \(\mathfrak {P}_{\mathcal {Q}^{\flat}}(T)\) factorizes according to the support of \(\mathcal {Q}\) in \(X\). Let \(Z\) be the set of closed points in \(Z\). For \(v\in Z\), let \(\mathcal {Q}^{\flat}_{v}\) denote the direct summand of \(\mathcal {Q}^{\flat}\) whose support is over \(v\).
Lemma 5.4
We have a natural isomorphism of graded \(\operatorname{Fr}\)modules
In particular,
Proof
Choose any \(y\in \Sigma (Z)\) and let \(y\) be the resulting map \(\{1,2,\ldots , d\}\to Z\). Let \(S_{y}=\operatorname{Stab}_{S_{d}}(y)\), then \(S_{y}\cong \prod _{v\in Z}S_{I_{v}}\), where \(I_{v}=y^{1}(v)\). Applying Corollary 3.7 to \(\mathcal {Q}^{\flat}\) and to each \(\mathcal {Q}^{\flat}_{v}\) gives an isomorphism of \((S_{d},\operatorname{Fr})\)modules
Here the factor \(S_{I_{v}}\) of \(S_{y}\) acts on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}_{v}}})\) by the Springer action.
Write \(H_{v}=\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}_{v}}})\) as a \((S_{I_{v}},\operatorname{Fr})\)module. Let \(d_{v}=\# I_{v}\). By Mackey theory, the \((S_{j}\times S_{dj}, \operatorname {sgn}_{j}\boxtimes 1)\)isotypical part of the right side of (5.5) is a direct sum over double cosets \((S_{j}\times S_{dj})\backslash S_{d}/S_{y}\), which can be identified with the set of functions \(i: Z\to \mathbf {Z}_{\ge 0}\), \(v\mapsto i_{v}\le d_{v}\), such that \(\sum _{v}i_{v}=j\). The stabilizer of the \(S_{j}\times S_{dj}\)action on the orbit indexed by \(i\) is isomorphic to \(\prod _{v\in Z}S_{i_{v}}\times S_{d_{v}i_{v}}\) (where the \(S_{i_{v}}\) factor lies in \(S_{j}\), \(S_{d_{v}i_{v}}\) lies in \(S_{dj}\), and \(S_{i_{v}}\times S_{d_{v}i_{v}}\) is naturally a subgroup of \(S_{I_{v}}\cong S_{d_{v}}\)). The contribution of the summand indexed by \(i\) is
This implies that
□
Let \(\mathcal {Q}_{v}\) be the direct summand of \(\mathcal {Q}\) supported over \(v\). Then \(\mathcal {Q}_{v}\) and \(\mathcal {Q}^{\flat}_{v}\) correspond under the bijection in Lemma 4.12.
For any Frobenius module \(M\) with integral weights, let \(\operatorname{Gr}^{W}_{i}M\) be the pure of weight \(i\) part of \(M\). This notation applies also to graded Frobenius modules by taking \(\operatorname{Gr}^{W}_{i}\) on each graded piece.
Proposition 5.5
Let \(\mathcal {Q}\in \operatorname{Herm}_{2d}(k)\).

(1)
We have
$$ \mathfrak {P}_{\mathcal {Q}}(T)\cong \bigotimes _{v\in Z}\mathfrak {P}_{\mathcal {Q}_{v}}(T), \quad P_{ \mathcal {Q}}(T)=\prod _{v\in Z}P_{\mathcal {Q}_{v}}(T). $$(5.6) 
(2)
If \(v\in Z\) is split in \(X'\), then
$$ P_{\mathcal {Q}_{v}}(T)=P_{\mathcal {Q}^{\flat}_{v}}(T). $$(5.7) 
(3)
If \(v\in Z\) is inert in \(X'\), then
$$ P_{\mathcal {Q}_{v}}(T)=\sum _{i}(1)^{i}\operatorname{Tr}(\operatorname{Fr}, \operatorname{Gr}^{W}_{2i\deg (v)} \mathfrak {P}_{\mathcal {Q}^{\flat}_{v}}(T)). $$(5.8)Moreover, \(\operatorname{Tr}(\operatorname{Fr}, \operatorname{Gr}^{W}_{i}\mathfrak {P}_{\mathcal {Q}^{\flat}_{v}}(T))=0\) if \(i\) is not a multiple of \(2\deg (v)\).
Proof
We will use the notations from §4.6.1. For each \(y\in \Sigma (Z)\), the summand \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}}(y)})\cong \otimes _{x\in Z} \mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}_{x}}})\) is graded by multidegree \(\mathbf {i}: Z\to \mathbf {Z}_{\ge 0}\)
We define \(\mathrm {H}^{2\mathbf {i}}({\mathcal {B}_{\mathcal {Q}^{\flat}}})\) as the direct sum of \(\mathrm {H}^{2\mathbf {i}}({\mathcal {B}_{\mathcal {Q}^{\flat}}(y)})\) over all \(y\in \Sigma (Z)\). Then each \(\mathrm {H}^{2\mathbf {i}}({\mathcal {B}_{\mathcal {Q}^{\flat}}})\) is stable under \(S_{d}\). Accordingly, \(\mathfrak {P}_{\mathcal {Q}^{\flat}}(T)\) decomposes into the direct sum of \(\mathfrak {P}^{2\mathbf {i}}_{\mathcal {Q}^{\flat}}(T)\), which is by definition \(\oplus _{j=0}^{d}(1)^{j}\mathrm {H}^{2\mathbf {i}}({\mathcal {B}_{\mathcal {Q}^{\flat}}})^{(S_{j} \times S_{dj}, \operatorname {sgn}_{j}\boxtimes 1)}T^{j}\). Let \(\mathbf {i}_{v}\) be the restriction of \(\mathbf {i}\) to those \(xv\), then under the factorization isomorphism in Lemma 5.4 we have
(1) Recall the involution \(\theta \) on \(\mathrm {H}^{*}({\mathcal {B}_{\mathcal {Q}^{\flat}}})\) in Proposition 4.13. Using the above notation, we see that \(\theta \) acts on \(\mathrm {H}^{2\mathbf {i}}({\mathcal {B}_{\mathcal {Q}^{\flat}}})\) by \(\prod _{v\in  Z_{i}}(1)^{\mathbf {i}(x_{v})}\) (where \(x_{v}\in Z\) is a chosen geometric point over \(v\), as in §4.6.1). Because of (5.9), the action of \(\theta \) on \(\mathfrak {P}_{\mathcal {Q}^{\flat}}(T)\) factorizes as the tensor product of the similarlydefined \(\theta _{v}\) on each \(\mathfrak {P}_{\mathcal {Q}^{\flat}_{v}}(T)\). By Proposition 4.13, \(\mathfrak {P}_{\mathcal {Q}}(T)\) is the Frobenius module obtained by modifying the Frobenius action on \(\mathfrak {P}_{\mathcal {Q}^{\flat}}(T)\) by composing with \(\theta \). By Lemma 5.4, this modified Frobenius structure on \(\mathfrak {P}_{\mathcal {Q}^{\flat}}(T)\) is the tensor product of the similarly modified Frobenius modules \(\mathfrak {P}_{\mathcal {Q}^{\flat}_{v}}(T)\), which in turn are isomorphic to \(\mathfrak {P}_{\mathcal {Q}_{v}}(T)\) by Proposition 4.13. This implies (5.6).
(2) From the definition we see that if \(v\) is split, then \(\theta _{v}\) is the identity on \(\mathfrak {P}_{\mathcal {Q}^{\flat}_{v}}(T)\). Hence \(\mathfrak {P}_{\mathcal {Q}_{v}}(T)=\mathfrak {P}_{\mathcal {Q}^{\flat}_{v}}(T)\) and \(P_{\mathcal {Q}_{v}}(T)=P_{\mathcal {Q}^{\flat}_{v}}(T)\).
(3) Since the cohomology of \(\mathcal {B}_{\mathcal {Q}^{\flat}_{v}}\) is pure by [9, Corollary 2.3(3)], we see that \(\operatorname{Gr}^{W}_{i}\mathfrak {P}_{\mathcal {Q}^{\flat}_{v}}(T)\) is the sum of \(\mathfrak {P}^{2\mathbf {i}_{v}}_{\mathcal {Q}^{\flat}_{v}}(T)\) where \(\mathbf {i}_{v}: \{x\in Z: xv\}\to \mathbf {Z}_{\ge 0}\) satisfies \(\sum _{xv}2\mathbf {i}_{v}(x)=i\). The action of Frobenius sends \(\mathfrak {P}^{2\mathbf {i}_{v}}_{\mathcal {Q}^{\flat}_{v}}(T)\) to \(\mathfrak {P}^{2F^{*}\mathbf {i}_{v}}_{\mathcal {Q}^{\flat}_{v}}(T)\), where \(F^{*}\mathbf {i}_{v}\) means precomposing \(\mathbf {i}_{v}\) with Frobenius. Therefore only constant multidegrees (i.e., constant functions \(\mathbf {i}_{v}\)) contribute to the Frobenius trace of \(\mathfrak {P}_{\mathcal {Q}^{\flat}_{v}}(T)\). This implies \(\operatorname{Tr}(\operatorname{Fr}, \operatorname{Gr}^{W}_{i}\mathfrak {P}_{\mathcal {Q}^{\flat}_{v}}(T))=0\) unless \(i\) is a multiple of \(2\deg (v)\).
By the discussion above,
Since \(v\) is inert, \(\theta _{v}\) acts by \((1)^{i}\) on \(\mathfrak {P}^{(2i,2i,\ldots ,2i)}_{\mathcal {Q}^{\flat}_{v}}(T)\), hence
On the other hand, \(\operatorname{Tr}(\operatorname{Fr}, \operatorname{Gr}^{W}_{2i\deg (v)}\mathfrak {P}_{\mathcal {Q}^{\flat}_{v}}(T))\) is the sum of \(\operatorname{Tr}(\operatorname{Fr}, \mathfrak {P}^{2\mathbf {i}_{v}}_{\mathcal {Q}^{\flat}_{v}}(T))\) with total degree \(\sum _{xv}2\mathbf {i}_{v}(x)=2i\deg (v)\). Since only constant multidegrees contribute to the trace, we again conclude
Combining the above identities we get (5.8). □
5.4 Calculation of \(P_{\mathcal {Q}^{\flat}}(T)\) and \(P_{\mathcal {Q}}(T)\)
Let \(\mathcal {Q}^{\flat}\in \operatorname{Coh}_{d}(X)(k)\) with support \(Z\subset X(\overline{k})\). For each \(v\in Z\) recall \(t_{v}(\mathcal {Q}^{\flat})\) from (2.5) for the local field \(F_{v}\).
Proposition 5.6
For \(\mathcal {Q}^{\flat}\in \operatorname{Coh}_{d}(X)(k)\), we have
Proof
We write \(\operatorname{Coh}_{d}(X)\) simply as \(\operatorname{Coh}_{d}\) in the proof. For \(0\le j\le d\), consider the correspondence
Here \(\operatorname{Coh}_{j,d}\) classifies pairs \((\mathcal {Q}_{j}\subset \mathcal {Q})\) of torsion sheaves of length \(j\) and \(d\) respectively, and the map \(r\) sends \((\mathcal {Q}_{j}\subset \mathcal {Q})\) to \((\mathcal {Q}_{j}, \mathcal {Q}/\mathcal {Q}_{j})\). We claim that
Indeed, consider the following diagram with Cartesian parallelogram
Here \(\pi _{j}=\pi _{j}^{\operatorname{Coh}}\), etc. The composition \(p\circ \pi _{j,d}=\pi _{d}=\pi ^{\operatorname{Coh}}_{d}\). By the proper base change theorem, we get
This isomorphism is \(S_{j}\times S_{dj}\)equivariant by checking easily over the open substack \(\operatorname{Coh}_{d}^{\circ }\). Taking \((S_{j}\times S_{dj},\operatorname {sgn}_{j}\boxtimes 1)\)isotypical parts of both sides we get (5.10).
By Lemma 5.4 it remains to compute \(P_{\mathcal {Q}^{\flat}_{v}}(T)\) for each \(v\). Therefore we may assume \(\mathcal {Q}^{\flat}=\mathcal {Q}^{\flat}_{v}\). By (5.10) and the GrothendieckLefschetz trace formula we have
By Proposition 3.8, \((\operatorname{St}_{\deg (v)j})_{\mathcal {R}}\) is zero unless \(\mathcal {R}\cong k_{v}^{\oplus j}\), in which case the Frobenius trace is \((1)^{j(\deg (v)1)}q_{v}^{j(j1)/2}\). Let \(\mathcal {Q}_{v}[\varpi ]\) be the kernel of the action of a uniformizer \(\varpi \) at \(v\). Note \(V:=\mathcal {Q}^{\flat}_{v}[\varpi ]\) has dimension \(t=t_{v}(\mathcal {Q}^{\flat})\) over \(k_{v}\). Then we only need to sum over \(k_{v}\)subspaces ℛ of \(V\). The above sum becomes
Recall that the “\(q\)binomial theorem” says that \(q_{v}^{j(j1)/2} \# \operatorname{Gr}(j, V)(k_{v})\) is the coefficient of \(x^{j}\) in \((1+x)(1+q_{v}x)\ldots (1+q_{v}^{t1}x)\). Making the change of variables \(x=T^{\deg (v)}\), we get
as desired. □
Now we are ready to prove (5.4).
Proposition 5.7
For \(\mathcal {Q}\in \operatorname{Herm}_{2d}(X'/X)(k)\) with support \(Z\), we have
Proof
By (5.6) it suffices to treat the case \(\mathcal {Q}\) is supported over a single place \(v\). Let \(\mathcal {Q}^{\flat}\in \operatorname{Coh}_{d}(X)(k)\) be the corresponding point. If \(v\) is split, we have \(P_{\mathcal {Q}}(T)=P_{\mathcal {Q}^{\flat}}(T)\) by (5.7), and the formula follows from Proposition 5.6.
If \(v\) is inert, let \(t=t'_{v}(\mathcal {Q})=t_{v}(\mathcal {Q}^{\flat})\). From the form of \(P_{\mathcal {Q}^{\flat}}(T)\) computed in Proposition 5.6, which is valid for any extension of \(k\), the trace of the pure weight pieces of \(\mathfrak {P}_{\mathcal {Q}^{\flat}}(T)\) are separated by different powers of \(q_{v}\), i.e., \(q_{v}^{i}\operatorname{Tr}(\operatorname{Fr}, \operatorname{Gr}^{W}_{2i\deg (v)}\mathfrak {P}_{\mathcal {Q}^{\flat}}(T))\) is the coefficient of \(q_{v}^{i}\) in \(\prod _{j=0}^{t1}(1q_{v}^{j}T^{\deg (v)})\). By (5.8),
which is what we want because \(\eta (\varpi _{v})=1\) in this case. □
Part II. The geometric side
6 Moduli of Hermitian shtukas
In this section we introduce some of the fundamental geometric objects in our story, in particular the moduli stacks of unitary (also called Hermitian) shtukas, which play an analogous role to that of unitary Shimura varieties in the work of KudlaRapoport.
6.1 Hermitian bundles
We adopt the notation of §1.3, and in particular for the remainder of the paper enforce the assumption that \(X\) is proper, and \(\nu \colon X' \rightarrow X\) is a finite étale double cover (possibly trivial).
Definition 6.1
A rank \(n\) Hermitian (also called unitary) bundle on \(X \times S\) with respect to \(\nu :X'\to X\) is a vector bundle ℱ of rank \(n\) on \(X' \times S\), equipped with an isomorphism \(h \colon \mathcal {F} \xrightarrow{\sim} \sigma ^{*} \mathcal {F}^{\vee}\) such that \(\sigma ^{*} h^{\vee} = h\). We refer to \(h\) as the Hermitian structure on ℱ.
We denote by \(\operatorname{Bun}_{U(n)}\) the moduli stack of rank \(n\) unitary bundles on \(X\), which sends a test scheme \(S\) to the groupoid of rank \(n\) unitary bundles on \(X \times S\). The notation is justified by the following remark.
Remark 6.2
There is an equivalence of categories between the groupoid of Hermitian bundles on \(X \times S\), and the groupoid of \(G\)torsors for the group scheme \(G = U(n)\) over \(X\) defined as
Indeed, we choose a square root \(\omega ^{1/2}_{X}\) of \(\omega _{X}\) (which exists over \(k = \mathbf {F}_{q}\) by [27, p.291, Theorem 13]). Then \(\mathcal {F}_{1}:=\nu ^{*}\omega ^{1/2}_{X'}\) is equipped with the canonical Hermitian structure \(h_{1}: \mathcal {F}_{1}\cong \sigma ^{*}\mathcal {F}_{1}\cong \sigma ^{*}\mathcal {F}^{\vee}_{1}\), and \((\mathcal {F}_{n},h_{n}):=(\mathcal {F}_{1},h_{1})^{\oplus n}\) is a rank \(n\) Hermitian bundle on \(X\) whose automorphism group scheme is \(U(n)\). To a Hermitian bundle \((\mathcal {F},h)\) on \(X \times S\), \(\underline{\operatorname{Isom}}_{X\times S}((\mathcal {F}_{n}\boxtimes \mathcal {O}_{S},h_{n}\boxtimes \operatorname{Id}), (\mathcal {F},h))\) (the scheme of unitary isometries) is a right torsor for \(U(n)\) over \(X\times S\). Conversely, for a right \(U(n)\)torsor \(\mathcal {G}\) over \(X\times S\), the contracted product \(\mathcal {G}\stackrel {U(n)}{\times }\mathcal {F}_{n}\) is a Hermitian bundle on \(X \times S\).
6.2 Hecke stacks
We now define some particular Hecke correspondences for \(\operatorname{Bun}_{U(n)}\).
Definition 6.3
Let \(r \geq 0\) be an integer. The Hecke stack \(\operatorname{Hk}_{U(n)}^{r}\) has as \(S\)points the groupoid of the following data:

(1)
\(x_{i}' \in X'(S)\) for \(i = 1, \ldots , r\), with graphs denoted by \(\Gamma _{x'_{i}} \subset X' \times S\).

(2)
A sequence of vector bundles \(\mathcal {F}_{0}, \ldots , \mathcal {F}_{r}\) of rank \(n\) on \(X' \times S\), each equipped with Hermitian structure \(h_{i} \colon \mathcal {F}_{i} \xrightarrow{\sim} \sigma ^{*} \mathcal {F}_{i}^{ \vee}\).

(3)
Isomorphisms \(f_{i} \colon \mathcal {F}_{i1}_{X' \times S  \Gamma _{x'_{i}}  \Gamma _{ \sigma (x'_{i})}} \xrightarrow{\sim} \mathcal {F}_{i}_{X' \times S  \Gamma _{x'_{i}}\Gamma _{\sigma (x'_{i})}}\), for \(1 \leq i \leq r\), compatible with the Hermitian structures, with the following property: there exists a rank \(n\) vector bundle \(\mathcal {F}^{\flat}_{i1/2}\) and a diagram of vector bundles
such that \(\operatorname{coker}(f_{i}^{\leftarrow})\) is locally free of rank 1 over \(\Gamma _{x'_{i}}\), and \(\operatorname{coker}(f_{i}^{\rightarrow})\) is locally free of rank 1 over \(\Gamma _{\sigma (x'_{i})}\). In particular, \(f_{i}^{\leftarrow}\) and \(f_{i}^{\rightarrow}\) are invertible upon restriction to \(X' \times S  \Gamma _{x'_{i}}\Gamma _{\sigma (x'_{i})}\), and the composition
$$ \mathcal {F}_{i1}_{X' \times S  \Gamma _{x'_{i}}\Gamma _{\sigma (x'_{i})}} \xrightarrow{(f_{i}^{\leftarrow})^{1}} \mathcal {F}_{i1/2}^{\flat}_{X' \times S  \Gamma _{x'_{i}}\Gamma _{\sigma (x'_{i})}} \xrightarrow{f_{i}^{\rightarrow}} \mathcal {F}_{i} _{X' \times S  \Gamma _{x'_{i}}\Gamma _{\sigma (x'_{i})}} $$agrees with \(f_{i}\).
Remark 6.4
Condition (3) above is equivalent to asking for the existence of a diagram
such that \(\operatorname{coker}(h_{i}^{\leftarrow})\) is flat of length 1 over \(\Gamma _{\sigma (x'_{i})}\), and \(\operatorname{coker}(h_{i}^{\rightarrow})\) is flat of length 1 over \(\Gamma _{x'_{i}}\). In particular, \(h_{i}^{\leftarrow}\) and \(h_{i}^{\rightarrow}\) are invertible upon restriction to \(X' \times S  \Gamma _{x'_{i}}\Gamma _{\sigma (x'_{i})}\), and the composition
agrees with \(f_{i}\).
Definition 6.5
Terminology for modifications of vector bundles
Given two vector bundles ℱ and \(\mathcal {F}'\) on \(X' \times S\), we will refer to an isomorphism between ℱ and \(\mathcal {F}'\) on the complement of a relative Cartier divisor \(D\subset X'\times S\) as a “modification” between ℱ and \(\mathcal {F}'\), and denote such a modification by \(\mathcal {F} \dashrightarrow \mathcal {F}'\). Given \(x,y \in X'(S)\), we say that the modification is “lower” of length 1 at \(x\) and “upper” of length 1 at \(y\) if it is as in Definition 6.3 (3), i.e. if there exists a diagram
such that \(\operatorname{coker}(f^{\leftarrow})\) is flat of length 1 over \(\Gamma _{x}\), and \(\operatorname{coker}(f^{\rightarrow})\) is flat of length 1 over \(\Gamma _{y}\), and \(\mathcal {F} \dashrightarrow \mathcal {F}'\) agrees with the composition
The condition admits a reformulation as in Remark 6.4.
6.3 Hermitian shtukas
For a vector bundle ℱ on \(X' \times S\), we denote by \({}^{\tau }\mathcal {F} := (\operatorname{Id}_{X'} \times \operatorname{Fr}_{S})^{*} \mathcal {F}\). If ℱ has a Hermitian structure \(h \colon \mathcal {F} \xrightarrow{\sim} \sigma ^{*} \mathcal {F}^{\vee}\), then \({}^{\tau }\mathcal {F}\) is equipped with the Hermitian structure \({}^{\tau }h\); we may suppress this notation when we speak of the “Hermitian bundle” \({}^{\tau }\mathcal {F}\).
Definition 6.6
Let \(r \geq 0\) be an integer. We define \(\operatorname{Sht}_{U(n)}^{r}\) by the Cartesian diagram
A point of \(\operatorname{Sht}_{U(n)}^{r}\) will be called a “\(U(n)\)shtuka”.
Concretely, the \(S\)points of \(\operatorname{Sht}_{U(n)}^{r}\) are given by the groupoid of the following data:

(1)
\(x'_{i} \in X'(S)\) for \(i = 1, \ldots , r\), with graphs denoted \(\Gamma _{x'_{i}} \subset X \times S\). These are called the legs of the shtuka.

(2)
A sequence of vector bundles \(\mathcal {F}_{0}, \ldots , \mathcal {F}_{n}\) of rank \(n\) on \(X' \times S\), each equipped with a Hermitian structure \(h_{i} \colon \mathcal {F}_{i} \xrightarrow{\sim} \sigma ^{*} \mathcal {F}_{i}^{ \vee}\).

(3)
Isomorphisms \(f_{i} \colon \mathcal {F}_{i1}_{X' \times S  \Gamma _{x'_{i}}  \Gamma _{ \sigma (x'_{i})}} \xrightarrow{\sim} \mathcal {F}_{i}_{X' \times S  \Gamma _{x'_{i}} \Gamma _{\sigma (x'_{i})}}\) compatible with the Hermitian structure, which as modifications of the underlying vector bundles on \(X' \times S\) are lower of length 1 at \(x'_{i}\) and upper of length 1 at \(\sigma (x'_{i})\).

(4)
An isomorphism \(\varphi \colon \mathcal {F}_{r} \cong {}^{\tau }\mathcal {F}_{0}\) compatible with the Hermitian structure.
Lemma 6.7
The stack \(\operatorname{Sht}_{U(n)}^{r}\) is empty if and only if \(r\) is odd.
Proof
We first treat the case \(n=1\). Let \(\operatorname{Nm}_{X'/X}: \operatorname{Pic}_{X'}\to \operatorname{Pic}_{X}\) be the norm map. Then \(\operatorname{Bun}_{U(1)}\cong \operatorname{Nm}^{1}_{X'/X}(\omega _{X})\), hence it is a torsor under \(\operatorname{Prym}(X'/X)=\ker (\operatorname{Nm}_{X'/X})\). Moreover, \(\operatorname{Sht}_{U(1)}^{r}\) fits into a Cartesian square
with the bottom horizontal map sending \(D \mapsto \mathcal {O}(D  \sigma D )\). If \(X'\) is geometrically connected, then the stack \(\operatorname{Prym}(X'/X)\) has two connected components, and by a result of Wirtinger, explained in [21, §2], the bottom horizontal map lands in the identity component if and only if \(r\) is even. If \(X'\) is geometrically disconnected (i.e. it is either \(X \coprod X\) or \(X_{k'}\)), then we have \(\pi _{0}(\operatorname{Prym}(X'/X)_{\overline {k}}) \cong \mathbf {Z}\), the Lang map lands in (therefore surjects onto, by Lang’s Theorem) the identity component, and the bottom horizontal map hits the identity component if and only if \(r\) is even. This shows that, in all cases, \(\operatorname{Sht}_{U(1)}^{r}\) is empty if and only if \(r\) is odd.
For general \(n\), taking determinant of a hermitian shtuka gives a map \(\operatorname{Sht}^{r}_{U(n)}\to \operatorname{Sht}^{r}_{U(1)}\). From this we see that if \(r\) is odd, then \(\operatorname{Sht}_{U(n)}^{r}\) is empty for any \(n\) since \(\operatorname{Sht}^{r}_{U(1)}\) is empty.
On the other hand if \(r\) is even, then \(\operatorname{Sht}_{U(1)}^{r}\) is nonempty. If \(n>1\), from an \(S\) point of \(\operatorname{Sht}_{U(1)}^{r}\), we can produce an \(S\)point of \(\operatorname{Sht}_{U(n)}^{r}\) by formation of direct sum with (the base change to \(X\times S\) of) a unitary bundle of rank \(n1\) on \(X\) (e.g. we can take \((\mathcal {F}_{n1},h_{n1})\) from Remark 6.2). □
6.4 Geometric properties
Lemma 6.8
The stack \(\operatorname{Bun}_{U(n)}\) is smooth and equidimensional.
Proof
The standard tangent complex argument, cf. [7, Prop. 1]. □
Lemma 6.9
(1) The projection map \((\operatorname{pr}_{X}, \operatorname{pr}_{r}):\operatorname{Hk}_{U(n)}^{r} \rightarrow (X')^{r} \times \operatorname{Bun}_{U(n)}\) recording \(\{x_{i}\}\) and \((\mathcal {F}_{r},h_{r})\) is smooth of relative dimension \(r(n1)\).
(2) \(\operatorname{Sht}_{U(n)}^{r}\) is a DeligneMumford stack locally of finite type. The map \(\operatorname{Sht}_{U(n)}^{r} \rightarrow (X')^{r}\) is smooth, separated, equidimensional of relative dimension \(r(n1)\).
Proof
The statements about \(\operatorname{Sht}_{U(n)}^{r}\) being locally finite type and separated are wellknown properties of moduli of \(G\)shtukas for general \(G\) [24, Proposition 2.16 and Theorem 2.20].^{Footnote 3}
Part (2) follows from (1) by [14, Lemma 2.13].
So it suffices to check (1). As a selfcorrespondence of \(\operatorname{Bun}_{U(n)}\), \(\operatorname{Hk}^{r}_{U(n)}\) is the \(r\)fold composition of \(\operatorname{Hk}^{1}_{U(n)}\). This allows us to reduce to the case \(r=1\). In this case, the map \((\operatorname{pr}_{X},\operatorname{pr}_{1}): \operatorname{Hk}^{1}_{U(n)}\to X'\times \operatorname{Bun}_{U(n)}\) exhibits \(\operatorname{Hk}^{1}_{U(n)}\) as a \(\mathbf {P}^{n1}\)bundle whose fiber over \((x', \mathcal {F}_{1},h_{1})\) classifies hyperplanes in \(\mathcal {F}_{1,\sigma (x')}\). Indeed, a hyperplane in \(\mathcal {F}_{1, \sigma (x')}\) determines a lower modification at \(\sigma (x')\), and the upper modification at \(x'\) is then determined from the lower modification by the Hermitian structure. This shows that \((\operatorname{pr}_{X},\operatorname{pr}_{1})\) is smooth, separated and equidimensional of relative dimension \((n1)\) in the case \(r=1\), and the general case follows. □
7 Special cycles: basic properties
In this section we define special cycles over the moduli stacks of hermitian shtukas, and construct corresponding cycle classes. The latter task is rather subtle, as the cycles are in most cases of a highly “derived” nature, with their “virtual dimension” differing significantly from their actual dimension.
7.1 Special cycles
Definition 7.1
Let ℰ be a rank \(m\) vector bundle on \(X'\).
We define the stack \(\mathcal {Z}_{\mathcal {E}}^{r}\) whose \(S\)points are the groupoid of the following data:

A \(U(n)\)shtuka with \((\{x'_{1}, \ldots , x'_{r}\}, \{\mathcal {F}_{0}, \ldots , \mathcal {F}_{r}\}, \{f_{1}, \ldots , f_{r}\}, \varphi ) \in \operatorname{Sht}_{U(n)}^{r}(S)\).

Maps of coherent sheaves \(t_{i} \colon \mathcal {E} \boxtimes \mathcal {O}_{S} \rightarrow \mathcal {F}_{i}\) on \(X'\times S\) such that the isomorphism \(\varphi \colon \mathcal {F}_{r} \cong {}^{\tau }\mathcal {F}_{0}\) intertwines \(t_{r}\) with \({}^{\tau }t_{0}\), and the maps \(t_{i1}, t_{i}\) are intertwined by the modification \(f_{i} \colon \mathcal {F}_{i1} \dashrightarrow \mathcal {F}_{i}\) for each \(i = 1, \ldots , r\), i.e. the diagram below commutes.
In the sequel, when writing such diagrams we will usually just omit the “\(\boxtimes \mathcal {O}_{S}\)” factor from the notation.
We will call the \(\mathcal {Z}_{\mathcal {E}}^{r}\) (or their connected components) special cycles of corank \(m\) (with \(r\) legs).
There is an evident map \(\mathcal {Z}_{\mathcal {E}}^{r} \rightarrow \operatorname{Sht}_{U(n)}^{r}\) projecting to the data in the first bullet point. When \(\operatorname{rank} \mathcal {E}=1\), the \(\mathcal {Z}_{\mathcal {E}}^{r} \) are function field analogues (with multiple legs) of the KudlaRapoport divisors introduced in [12, 13].
7.2 Indexing via Hermitian maps
Definition 7.2
Let \(\mathcal {A}^{\mathrm{all}}_{\mathcal {E}}(k)\) be the \(k\)vector space of Hermitian maps^{Footnote 4}\(a \colon \mathcal {E} \rightarrow \sigma ^{*} \mathcal {E}^{\vee}\) such that \(\sigma (a)^{\vee} =a\). Let \(\mathcal {A}_{\mathcal {E}}(k)\subset \mathcal {A}^{\mathrm{all}}_{\mathcal {E}}(k)\) be the subset where the map \(a \colon \mathcal {E} \rightarrow \sigma ^{*} \mathcal {E}^{\vee}\) is injective (as a map of coherent sheaves).
Let \((\{x'_{i}\}, \{\mathcal {F}_{i}\}, \{f_{i}\}, \varphi , \{t_{i}\}) \in \mathcal {Z}_{\mathcal {E}}^{r}(S)\). By the compatibilities between the \(t_{i}\) in the definition of \(\mathcal {Z}_{\mathcal {E}}^{r}\), the compositions
agree for each \(i\), and (7.1) for \(i=r\) also agrees with the Frobenius twist of (7.1) for \(i=0\). Hence (7.1) for every \(i\) gives the same map \(\mathcal {E} \boxtimes \mathcal {O}_{S} \rightarrow \mathcal {E}^{\vee} \boxtimes \mathcal {O}_{S}\), which moreover must come by pullback from \(\mathcal {A}^{\mathrm{all}}_{\mathcal {E}}(k)\). This defines a map \(\mathcal {Z}_{\mathcal {E}}^{r} \rightarrow \mathcal {A}^{\mathrm{all}}_{\mathcal {E}}(k)\). For \(a \in \mathcal {A}^{\mathrm{all}}_{\mathcal {E}}(k)\), we denote by \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) the fiber of \(\mathcal {Z}^{r}_{\mathcal {E}}\) over \(a\). We have
Definition 7.3
For \(a\in \mathcal {A}_{\mathcal {E}}(k)\), let \(D_{a}\) be the effective divisor on \(X\) such that \(\nu ^{1}(D_{a})\) is the divisor of the Hermitian map \(\det (a): \det (\mathcal {E})\to \sigma ^{*}\det (\mathcal {E})^{\vee}\).
Definition 7.4
For any \(a \in \mathcal {A}_{\mathcal {E}}^{\operatorname{all}}(k)\), we define:

\(\mathcal {Z}_{\mathcal {E}}^{r}(a)^{\circ}\subset \mathcal {Z}^{r}_{\mathcal {E}}(a)\) to be the open substack classifying \((\{x'_{i}\}, \{\mathcal {E}\xrightarrow {t_{i}}\mathcal {F}_{i}\})\) with the additional condition that all \(t_{i}\) are injective when restricted to \(X'_{\overline{s}}\) for any geometric point \(\overline{s}\) of the test scheme \(S\).

\(\mathcal {Z}_{\mathcal {E}}^{r}(a)^{*}\subset \mathcal {Z}^{r}_{\mathcal {E}}(a)\) to be the open substack classifying \((\{x'_{i}\}, \{\mathcal {E}\xrightarrow {t_{i}}\mathcal {F}_{i}\})\) with the additional condition that all \(t_{i}\) are nonzero when restricted to \(X'_{\overline{s}}\) for any geometric point \(\overline{s}\) of the test scheme \(S\).
Note that if \(\operatorname{rank}(\mathcal {E})=1\), then the inclusion \(\mathcal {Z}_{\mathcal {E}}^{r}(a)^{\circ} \hookrightarrow \mathcal {Z}_{\mathcal {E}}^{r}(a)^{*}\) is an isomorphism, and both include isomorphically into \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) unless \(a=0\).
7.3 Finiteness properties
We next establish that the projection map \(\mathcal {Z}_{\mathcal {E}}^{r}(a) \rightarrow \operatorname{Sht}_{U(n)}^{r}\) is finite, which will eventually allow us to construct cycle classes on \(\operatorname{Sht}_{U(n)}^{r}\) associated to \(\mathcal {Z}_{\mathcal {E}}^{r}(a) \).
Proposition 7.5
Let ℰ be any vector bundle of rank \(m\) on \(X'\) and let \(a \in \mathcal {A}^{\mathrm{all}}_{\mathcal {E}}(k)\). Then the projection maps \(\mathcal {Z}_{\mathcal {E}}^{r}(a) \rightarrow \operatorname{Sht}_{U(n)}^{r}\) and \(\mathcal {Z}_{\mathcal {E}}^{r}(a)^{*} \rightarrow \operatorname{Sht}_{U(n)}^{r}\) are both finite.
Proof
Note that \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) has a closed substack where the map \(t \colon \mathcal {E}\rightarrow \mathcal {F}\) is 0, which projects isomorphically to \(\operatorname{Sht}_{U(n)}^{r}\). The complement of this closed substack is \(\mathcal {Z}_{\mathcal {E}}^{r}(a)^{*}\), so it suffices to show the finiteness of \(\mathcal {Z}_{\mathcal {E}}^{r}(a)^{*} \rightarrow \operatorname{Sht}_{U(n)}^{r}\). We will show that it is proper and quasifinite. First we establish the properness. It suffices to show this locally on the target, so we pick a HarderNarasimhan polygon \(P\) for \(\operatorname{Bun}_{U(n)}\) and consider the truncation \(\operatorname{Bun}_{U(n)}^{\leq P}\). Define \(\operatorname{Sht}_{U(n)}^{r, \leq P} \) to be the open substack of \(\operatorname{Sht}_{U(n)}^{r}\) obtained as the pullback of \(\operatorname{Bun}_{U(n)}^{\leq P} \hookrightarrow \operatorname{Bun}_{U(n)}\) via the tautological projection \(\operatorname{pr}_{0}: \operatorname{Sht}_{U(n)}^{r} \rightarrow \operatorname{Bun}_{U(n)}\) recording \(\mathcal {F}_{0}\), and \(\mathcal {Z}_{\mathcal {E}}^{r, \leq P}(a) \hookrightarrow \mathcal {Z}_{\mathcal {E}}^{r}(a)\) the analogous pullback.
We can then pick a sufficiently antiample vector bundle \(\mathcal {E}'\) of rank \(m\) on \(X'\) and an injection \(\iota \colon \mathcal {E}' \hookrightarrow \mathcal {E} \) so that the stack \(\underline {\operatorname{Hom}}(\mathcal {E}', )^{\leq P}\) parametrizing \(\{(\mathcal {F} \in \operatorname{Bun}_{U(n)}^{\leq P}, t \in \operatorname{Hom}(\mathcal {E}', \mathcal {F})) \}\) forms a vector bundle over \(\operatorname{Bun}_{U(n)}^{\leq P}\), with respect to the obvious projection map. Let \(a' := (\sigma ^{*} \iota ^{\vee}) \circ a \circ \iota \colon \mathcal {E}' \rightarrow \mathcal {E}^{\prime \,\vee }\). Then we have a closed embedding \(\mathcal {Z}_{\mathcal {E}}^{r, \leq P}(a) \hookrightarrow \mathcal {Z}_{\mathcal {E}'}^{r, \leq P}(a')\) cut out by the condition that the map \(t \colon \mathcal {E}' \rightarrow \mathcal {F}\) factors through \(\iota \), which fits into a commutative diagram
It suffices to show the open substack \(\mathcal {Z}_{\mathcal {E}'}^{r, \leq P}(a')^{*}\) defined by the condition that \(t_{0}\ne 0\) fiberwise over the test scheme is proper over \(\operatorname{Sht}_{U(n)}^{r, \leq P} \). We can factorize this map as the composition of two maps in the diagram below:
where \(j\) is determined by the map \(\mathcal {Z}_{\mathcal {E}'}^{r, \leq P}(a')^{*} \rightarrow \mathbf {P}(\underline {\operatorname{Hom}}( \mathcal {E}', )^{\leq P})\) sending
The map \(\operatorname{pr}_{2}\) is a projective bundle by design, so in order to establish that \(\operatorname{pr}_{2} \circ j\) is proper it suffices to show that \(j\) is finite. Indeed, since data of all the \(t_{i}\) is determined by \(t_{0}\), the analogous map \(\mathcal {Z}_{\mathcal {E}'}^{r, \leq P}(a')^{*} \rightarrow \underline {\operatorname{Hom}}(\mathcal {E}', )^{ \leq P} \times _{\operatorname{Bun}_{U(n)}^{\leq P}} \operatorname{Sht}_{U(n)}^{r, \leq P}\) is a closed embedding. The requirement \(t_{0} = {}^{\tau }t_{0}\) in the definition of \(\mathcal {Z}_{\mathcal {E}'}^{r}\) therefore implies that the map \(j\) is a \(k^{\times}\)torsor onto its image, which is a closed substack of \(\mathbf {P}(\underline {\operatorname{Hom}}(\mathcal {E}', )^{\leq P})\). This completes the proof of properness.
It remains to show that \(\mathcal {Z}_{\mathcal {E}}^{r}(a) \rightarrow \operatorname{Sht}_{U(n)}^{r}\) is quasifinite. Since the map is already established to be proper, it suffices by [23, Tag 01TC] to check that the fibers over fieldvalued points are finite. Let
be such a point valued in a field \(\kappa \). Its fiber in \(\mathcal {Z}_{\mathcal {E}}^{r}(a)(\kappa )\) consists of \(\{t_{i} \colon \mathcal {E}\rightarrow \mathcal {F}_{i} \}_{0\le i\le r}\) fitting into commutative diagrams
such that \(\sigma ^{*} t_{i}^{\vee} \circ h_{i}\circ t_{i} = a \in \mathcal {A}_{ \mathcal {E}}^{\operatorname{all}}(\kappa )\) for each \(i = 0, \ldots , r\). We want to show that there are finitely many possibilities for such \(t_{i} \in \mathrm {H}^{0}({X'_{\kappa}, \mathcal {E}^{\vee}_{\kappa} \otimes \mathcal {F}_{i}})\).
The situation can be abstracted to the following semilinear algebra problem.
Lemma 7.6
Suppose that \(\kappa \) is any field over \(k\), and we have finitedimensional \(\kappa \)vector spaces \(V_{1}, V_{2} \subset V\) with an injective \(\operatorname{Fr}\)semilinear map \(\tau \colon V_{1} \hookrightarrow V_{2}\).
Then the set \(\{ x \in V_{1} \colon \tau (x) = x \in V\}\) is finite.
We assume Lemma 7.6 for the moment and use it to conclude the proof of Proposition 7.5. We apply it to the situation above with \(V_{1} := \operatorname{Hom}_{X'_{\kappa}} (\mathcal {E}_{\kappa}, \mathcal {F}_{0})\), \(V_{2} := \operatorname{Hom}_{X'_{\kappa}}( {}^{\tau }\mathcal {E}_{\kappa}, {}^{\tau }\mathcal {F}_{0})\), which are both viewed as subspaces of
by the obvious inclusion. The map \(V_{1} \rightarrow V_{2}\) is the twist by \(\tau \). Then Lemma 7.6 shows that there are finitely many possibilities for \(t_{0}\) since \(\tau (t_{0})=t_{0}\). The other \(t_{i}\) are determined by \(t_{0}\) (if they exist) because the \(t_{i}\) as well as the modifications \(\mathcal {F}_{i} \dashrightarrow \mathcal {F}_{i+1}\) are all isomorphisms over an open subset of \(X'_{\kappa}\). □
Proof of Lemma 7.6
By replacing \(\kappa \) with an algebraic closure, it suffices to consider the case when \(\kappa \) is algebraically closed. Let us call a subspace \(V_{1}' \subset V_{1}\) “\(\tau \)fixed” if \(\tau (V_{1}') = V_{1}' \subset V\). Since a sum of \(\tau \)fixed subspaces is evidently \(\tau \)fixed, there is a welldefined largest \(\tau \)fixed subspace \(V^{\circ}_{1} \subset V_{1}\). It is a sub \(\kappa \)vector space of \(V_{1}\), hence necessarily finitedimensional. Since \(\tau :V_{1}\to V_{2}\) is injective, the restriction of \(\tau \) to \(V_{1}^{\circ}\) is a \(\operatorname{Fr}\)semilinear bijection. The set \(\{ x \in V_{1} \colon \tau (x) = x \in V\}\) is evidently contained in \((V_{1}^{\circ})^{\tau}\), which is an \(k\)form of \(V_{1}^{\circ}\) (because \(\kappa \) is algebraically closed) and therefore finitedimensional over \(k\). □
7.4 Variation with ℰ
Let \(\mathcal {E}',\mathcal {E}\) be two vector bundles (with possibly distinct ranks) on \(X'\) and \(s \colon \mathcal {E}' \to \mathcal {E}\) be a map of coherent sheaves. Given \(a \colon \mathcal {E} \to \sigma ^{*} \mathcal {E}^{\vee}\) in \(\mathcal {A}^{\mathrm{all}}_{\mathcal {E}}(k)\), let \(a' = (\sigma ^{*} s^{\vee} ) \circ a \circ s \colon \mathcal {E}' \to \sigma ^{*} (\mathcal {E}')^{\vee}\) be the corresponding element in \(\mathcal {A}^{\mathrm{all}}_{\mathcal {E}'}(k)\). Therefore, composing with \(s\) defines a map
sending
The following lemma follows directly from definitions.
Lemma 7.7
If \(\mathcal {E}=\mathcal {E}_{1}\oplus \mathcal {E}_{2}\), and \(a_{i}\in \mathcal {A}^{\mathrm{all}}_{\mathcal {E}_{i}}(k)\) for \(i=1,2\), then there is a canonical isomorphism
where the union runs over all Hermitian maps \(a:\mathcal {E}\to \sigma ^{*}\mathcal {E}^{\vee}\) whose restriction to \(\mathcal {E}_{i}\) is \(a_{i}\) (for \(i=1,2\)). The map from the right side to the left is given by \((z_{\iota _{1}},z_{\iota _{2}})\), where \(\iota _{i}:\mathcal {E}_{i}\hookrightarrow \mathcal {E}\) is the inclusion.
Lemma 7.8
Under the notations of the beginning of this subsection,

(1)
If \(s:\mathcal {E}'\to \mathcal {E}\) is generically surjective, then \(z_{s}:\mathcal {Z}^{r}_{\mathcal {E}}(a)\to \mathcal {Z}^{r}_{\mathcal {E}'}(a')\) is a closed embedding.

(2)
Suppose that \(s\) is generically an isomorphism (in particular ℰ and \(\mathcal {E}'\) have the same rank). Let \(D_{s}\subset X'\) be the divisor of \(\det (s)\). Then the restriction of \(z_{s}\) over \((X\backslash \nu (D_{s}))^{r}\)
$$ \mathcal {Z}^{r}_{\mathcal {E}}(a)_{(X\backslash \nu (D_{s}))^{r}}\subset \mathcal {Z}^{r}_{\mathcal {E}'}(a')_{(X \backslash \nu (D_{s}))^{r}} $$is open and closed. Here we write \(\mathcal {Z}^{r}_{\mathcal {E}}(a)_{(X\backslash \nu (D_{s}))^{r}}\) for the preimage of \((X\backslash \nu (D_{s}))^{r}\) under the leg map \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\to \operatorname{Sht}^{r}_{U(n)}\to X^{\prime \,r}\xrightarrow {\nu ^{r}} X^{r}\).
Proof
(1) Let \(\overline{\mathcal {E}}'=\mathcal {E}'/\ker (s)\xrightarrow {\overline{s}}\mathcal {E}\), equipped with the induced Hermitian map \(\overline{a}' \colon \overline{\mathcal {E}}' \rightarrow \sigma ^{*} (\overline{\mathcal {E}}')^{\vee}\). Then \(s^{*}\) factors as \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\xrightarrow {z_{\overline{s}}}\mathcal {Z}_{\overline{\mathcal {E}}'}^{r}(\overline{a}')\subset \mathcal {Z}^{r}_{\mathcal {E}'}(a')\), the latter being evidently a closed embedding. Therefore it suffices to show \(z_{\overline{s}}\) is a closed embedding. We thus reduce to the case \(s\) is generically an isomorphism.
Let \(D\) be an effective divisor on \(X'\) such that \(\mathcal {E}(D)\hookrightarrow \mathcal {E}'\xrightarrow {s}\mathcal {E}\). Let \(\mathcal {F}^{\mathrm{univ}}\) be the universal Hermitian bundle over \(X'\times \operatorname{Bun}_{U(n)}\), and \(\mathcal {F}^{\mathrm{univ}}_{D}\) its restriction to \(D\times \operatorname{Bun}_{U(n)}\). Let \(\mathcal {V}_{D}=\operatorname{pr}_{2*}\underline{\operatorname{Hom}}(\operatorname{pr}_{1}^{*}\mathcal {E}(D)_{D},\mathcal {F}^{\mathrm{univ}}_{D})\), where \(\operatorname{pr}_{1},\operatorname{pr}_{2}\) are the projections of \(D\times \operatorname{Bun}_{U(n)}\) to the two factors. Then \(\mathcal {V}_{D}\) is a vector bundle of rank equal to \(n\operatorname{rank}(\mathcal {E})\deg (D)\) over \(\operatorname{Bun}_{U(n)}\). Let \(\mathcal {V}_{i,D}\) be the pullback of \(\mathcal {V}_{D}\) over \(\mathcal {Z}^{r}_{\mathcal {E}'}(a')\) via the map \(\operatorname{pr}_{i}: \mathcal {Z}^{r}_{\mathcal {E}'}(a')\to \operatorname{Bun}_{U(n)}\). Then \(\mathcal {V}_{i,D}\) has a section \(v_{i}\) whose value at \((\{x'_{j}\}, \{\mathcal {F}_{j}\}, \{t_{j}: \mathcal {E}'\to \mathcal {F}_{j}\})\in \mathcal {Z}^{r}_{ \mathcal {E}'}(a')\) is the restriction of \(t_{i}\) to \(\mathcal {E}(D)_{D}\to \mathcal {F}_{i}_{D}\). Then \(t_{i}\) extends to ℰ if and only if \(v_{i}\) vanishes at the point \((\{x'_{j}\}, \{\mathcal {F}_{j}\}, \{t_{j}\})\). This identifies \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) as the common zero locus of the sections \((v_{i})_{0\le i\le r1}\) of the vector bundles \(\mathcal {V}_{i,D}\) over \(\mathcal {Z}^{r}_{\mathcal {E}'}(a')\), hence closed in \(\mathcal {Z}^{r}_{\mathcal {E}'}(a')\).
(2) By (1), it remains to show the openness of \(z_{s}\) when restricted to \((X\backslash \nu (D_{s}))^{r}\). Let \(U'=X'\backslash \operatorname{supp}(D_{s}+\sigma D_{s})\). Let \(\operatorname{Sht}^{r}_{U(n),D_{s}}\) be the moduli stack of Hermitian shtukas \((\{x'_{i}\}, \{\mathcal {F}_{i}\})\) of rank \(n\) with legs in \(U^{\prime \,r}\), and trivializations of \(\mathcal {F}_{i}_{D_{s}}\) (as a vector bundle over \(D_{s}\) of rank \(n\)) compatible with the shtuka structures. Then \(\lambda : \operatorname{Sht}^{r}_{U(n),D_{s}}\to \operatorname{Sht}^{r}_{U(n)}_{U^{\prime \,r}}\) is a \(\operatorname{GL}_{n}(\mathcal {O}_{D_{s}})\)torsor. Let \(\mathcal {Z}^{r}_{\mathcal {E}}(a)_{D_{s}}\) and \(\mathcal {Z}^{r}_{\mathcal {E}'}(a')_{D_{s}}\) be the base changes of \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) and \(\mathcal {Z}^{r}_{\mathcal {E}'}(a')\) along \(\lambda \). Since \(\mathcal {Z}^{r}_{\mathcal {E}'}(a')_{D_{s}}\to \mathcal {Z}^{r}_{\mathcal {E}'}(a')_{U^{\prime \,r}}\) is finite étale surjective, it suffices to show that the inclusion \(\mathcal {Z}^{r}_{\mathcal {E}}(a)_{D_{s}}\hookrightarrow \mathcal {Z}^{r}_{\mathcal {E}'}(a')_{D_{s}}\) is open. Using the trivializations of \(\mathcal {F}_{i}_{D_{s}}\), we get an evaluation map
where the target is a discrete set. Then \(\mathcal {Z}^{r}_{\mathcal {E}}(a)_{D_{s}}\) is the preimage of the image of
under \(\operatorname{ev}_{D_{s}}\). Indeed, a map \(\mathcal {E}'\to \mathcal {F}_{i}\) extends to \(\mathcal {E}\to \mathcal {F}_{i}\) if and only if \(\mathcal {E}'_{D_{s}}\to \mathcal {F}_{i}_{D_{s}}\) vanishes on \(\ker (\mathcal {E}'_{D_{s}}\to \mathcal {E}_{D_{s}})\) (this can be checked locally using elementary divisors). Since the target of \(\operatorname{ev}_{D_{s}}\) is discrete, \(\mathcal {Z}^{r}_{\mathcal {E}}(a)_{D_{s}}\subset \mathcal {Z}^{r}_{\mathcal {E}'}(a')_{D_{s}}\) is open and closed. □
7.5 Corank 1 special cycles
A special role is played by the case \(m=1\), i.e. where ℰ is a line bundle on \(X'\), because it is only in this case that we can appropriately control the dimension of the cycles \(\mathcal {Z}_{\mathcal {E}}^{r}\). We will write \(\mathcal {L} := \mathcal {E}\) to emphasize that it is a line bundle.
Note that in this case \(a \in \mathcal {A}_{\mathcal {L}}(k)\) if and only if \(a \neq 0\).
Proposition 7.9
We have \(\dim \mathcal {Z}_{\mathcal {L}}^{r}(a)^{\circ} \leq r(n1)\).
This is established later, in Proposition 9.1 (for \(a\ne 0\)) and Proposition 9.5 (for \(a=0\)), as a consequence of a more refined study of the geometry of \(\mathcal {Z}_{\mathcal {L}}^{r}(a)^{\circ}\).
Remark 7.10
One can show that when \(a\ne 0\), in fact \(\mathcal {Z}_{\mathcal {L}}^{r}(a)^{\circ}\) is LCI of pure dimension \(r(n1)\). This will appear in a future paper; it relies on some ideas from [4]. This fact will not be used in the present paper, but it may be psychologically helpful.
Definition 7.11
For \(a \in \mathcal {A}^{\operatorname{all}}_{\mathcal {L}}(k)\), we define \([ \mathcal {Z}_{\mathcal {L}}^{r}(a)^{\circ} ] \in \operatorname{Ch}_{r(n1)}( \mathcal {Z}_{\mathcal {L}}^{r}(a)^{ \circ})\) to be the cycle class of the union of the irreducible components of \(\mathcal {Z}_{\mathcal {L}}^{r}(a)^{\circ} \) with dimension \(r(n1)\), throwing away the irreducible components of dimension \(< r(n1)\). (According to Remark 7.10, there are no such components to be thrown away at least when \(a\ne 0\), but we neither prove nor use this in the present paper.)
7.6 Corank \(n\) special cycles
In this paper we are mainly concerned with the case where the rank of ℰ is \(m=n\). The following proposition contains basic geometric information about \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\).
Lemma 7.12
Let ℰ be a vector bundle on \(X'\) of rank \(n\), and \(a\in \mathcal {A}_{\mathcal {E}}(k)\). Then the map \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\to X^{\prime \,r}\) recording the legs has image in \((\operatorname{supp}\nu ^{1}(D_{a}))^{r}\).
Proof
Let \((\{x'_{i}\}, \{\mathcal {F}_{i}\}, \{t_{i}\})\) be a geometric point of \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\). For each \(1\le i\le r\), the Hermitian map \(a\) factorizes as \(\mathcal {E}\hookrightarrow \mathcal {F}^{\flat}_{i1/2}\hookrightarrow \mathcal {F}_{i1}\hookrightarrow \sigma ^{*}\mathcal {E}^{ \vee}\), we see that \(x'_{i}\) (the support of \(\mathcal {F}_{i1}/\mathcal {F}^{\flat}_{i1/2}\)) is in the support of \(\sigma ^{*}\mathcal {E}^{\vee}/a(\mathcal {E})\), i.e., \(x'_{i}\in \operatorname{supp}\nu ^{1}(D_{a})\). □
Proposition 7.13
Let ℰ be a vector bundle on \(X'\) of rank \(n\), and \(a\in \mathcal {A}_{\mathcal {E}}(k)\). Then \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) is a proper scheme over \(k\) that depends only on the torsion sheaf \(\mathcal {Q}=\operatorname{coker}(a)=\sigma ^{*}\mathcal {E}^{\vee}/\mathcal {E}\) together with the Hermitian structure \(\overline{a}\) on \(\mathcal {Q}\) induced from \(a\) (see §4.1for the notion of Hermitian structure on a torsion sheaf).
The proof involves a few ideas not yet introduced, and will be given later in §8.4.1.
The next goal is to equip the proper scheme \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) with a 0cycle class in its Chow group. The “virtual dimension” of \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) is at most zero, for if ℰ is a direct sum of line bundles \(\mathcal {L}_{1}\oplus \cdots \oplus \mathcal {L}_{n}\), then \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) is contained in the intersection of \(Z^{r}_{\mathcal {L}_{i}}(a_{ii})\) for \(1\le i\le r\), each having codimension at least \(r\) in \(\operatorname{Sht}^{r}_{U(n)}\) by Proposition 7.9 (which can be shown to be an equality, cf. Remark 7.10). However the actual dimension of \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) can be strictly positive. Our task is to find the correct virtual fundamental class of \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\).
7.7 Intersection theory on stacks
Recall the discussion of intersection theory on DeligneMumford stacks from [28, Appendix A]. Let \(Y\) be a smooth, separated, locally finite type DeligneMumford stack over \(k\) of pure dimension \(d\). Let \(Y_{1}, \ldots , Y_{n}\) be DeligneMumford stacks with maps \(f_{i}: Y_{i}\to Y\). Then there is an intersection product
For \(\zeta _{i}\in \operatorname{Ch}_{*}(Y_{i})\), the intersection product \(\zeta _{1}\cdot _{Y}\cdots \cdot _{Y}\zeta _{n}\) is defined as the Gysin pullback of the external product \(\zeta _{1}\times \cdots \times \zeta _{n}\in \operatorname{Ch}_{*}(Y_{1}\times \cdots \times Y_{n})\) along the diagonal map \(\Delta : Y\to Y^{n}\), which is a regular embedding of codimension \(d(n1)\).
7.8 Intersection problem: the case of a direct sum of line bundles
We now formulate the cycle classes which enter into our intersection problem. We first consider the case ℰ a direct sum of \(m\) line bundles on \(X'\),
Let \(a \in \mathcal {A}_{\mathcal {E}}(k)\). We write \(a\) as an \(m\times m\)matrix with entries \(a_{ij}\in \operatorname{Hom}(\mathcal {L}_{j}, \sigma ^{*}\mathcal {L}^{\vee}_{i})\).
Let
In Definition 7.11 we defined a fundamental class \([\mathcal {Z}_{\mathcal {L}}^{r}(a)^{\circ}] \in \operatorname{Ch}_{r(n1)}(\mathcal {Z}_{ \mathcal {L}}^{r}(a)^{\circ})\). Applying the intersection product construction in §7.7 for \(Y=\operatorname{Sht}^{r}_{U(n)}\) (the hypotheses apply by Lemma 6.9 (2)), we obtain a class
Let \(\mathcal {A}_{\mathcal {E}}^{\operatorname{all}}(a_{11},\ldots , a_{mm})(k)\) be the finite set of Hermitian maps \(a:\mathcal {E}\to \sigma ^{*}\mathcal {E}^{\vee}\) (not assumed to be injective) such that its restriction to \(\mathcal {L}_{i}\) is \(a_{ii}\) for \(i=1,\ldots , m\). By Lemma 7.7, there is a map
such that the fiber over \(a \in \mathcal {A}_{\mathcal {E}}(k)\) identifies with \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\). Since \(a\) is injective, the image of \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\to \mathcal {Z}^{r}_{\mathcal {L}_{i}}(a_{ii})\) lies in \(\mathcal {Z}^{r}_{\mathcal {L}_{i}}(a_{ii})^{\circ}\). In particular,
is open and closed. Restricting (7.3) to \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) gives a cycle class
Remark 7.14
Our notation suggests that \(\zeta ^{r}_{\mathcal {L}_{1},\ldots , \mathcal {L}_{m}}(a)\) (as a cycle class on \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\)) depends, at least a priori, on the decomposition of ℰ into a direct sum of line bundles \(\mathcal {L}_{1},\ldots , \mathcal {L}_{m}\). However, we will show later in Theorem 10.1 that, at least when \(m=n\), it only depends on ℰ, and is equal to the cycle class \([\mathcal {Z}^{r}_{\mathcal {E}}(a)]\) that we will define for general rank \(n\) bundle ℰ.
7.9 Intersection problem: \(m=n\) and ℰ arbitrary
To define a 0cycle \([ \mathcal {Z}_{\mathcal {E}}^{r}(a)] \) for general rank \(n\) vector bundle ℰ on \(X'\), we need to make some auxiliary choice first; eventually we will show that the definition is independent of the choice.
Definition 7.15
Let ℰ be a rank \(n\) vector bundle over \(X'\) and \(a\in \mathcal {A}_{\mathcal {E}}(k)\). A good framing of \((\mathcal {E},a)\) is an \(n\)tuple \((s_{i}: \mathcal {L}_{i}\to \mathcal {E})_{1\le i\le n}\) of \(\mathcal {O}_{X'}\)linear maps from line bundles \(\mathcal {L}_{i}\in \operatorname{Pic}(X')\) satisfying:

(1)
The map \(s=\oplus s_{i}: \mathcal {E}':=\oplus _{i=1}^{n}\mathcal {L}_{i}\to \mathcal {E}\) is injective.

(2)
Let \(D_{s}\) be the divisor of the nonzero map \(\det (s): \otimes _{i=1}^{n}\mathcal {L}_{i}\to \det \mathcal {E}\). Then \(\nu (D_{s})\) (image in \(X\)) is disjoint from \(D_{a}\) (see Definition 7.3).
Lemma 7.16
For any rank \(n\) bundle ℰ on \(X'\) and \(a\in \mathcal {A}_{\mathcal {E}}(k)\), there exists a good framing for \((\mathcal {E},a)\) in the sense of Definition 7.15.
Proof
For notational convenience we give the argument for \(X'\) connected; the case \(X'=X\coprod X\) can be proved with obvious changes.
We strengthen condition (2) on \(s:\oplus _{i=1}^{n}\mathcal {L}_{i}\to \mathcal {E}\) slightly by asking \(\nu (D_{s})\) to avoid a prescribed divisor \(D_{0}\) on \(X\), instead of \(D_{a}\). We prove the existence of \(s\) satisfying this stronger condition by induction on \(n\).
The base case \(n=1\) is trivial: take \(\mathcal {L}_{1}=\mathcal {E}\).
For the inductive step, start by picking any saturated line bundle \(\mathcal {L}_{1}\hookrightarrow \mathcal {E}\). Then \(\mathcal {E}_{n1} := \mathcal {E}/\mathcal {L}_{1}\) is a vector bundle of rank \(n1\). By induction hypothesis we may pick \(\overline{s}: \oplus _{i=2}^{n}\mathcal {L}'_{i} \hookrightarrow \mathcal {E}_{n1}\) satisfying the conditions of Definition 7.15 and such that \(\nu (D_{\overline{s}})\) avoids the given divisor \(D_{0}\). Let \(D_{2},\ldots , D_{n}\) be effective divisors on \(X'\) such that

(1)
\(\nu (D_{2}),\ldots , \nu (D_{n})\) are disjoint from \(\nu (D_{0})\), and

(2)
\(\deg \mathcal {L}'_{i}D_{i}+2g'2< \deg \mathcal {L}_{1}\) for \(i=2,\ldots , n\).
Let \(\mathcal {L}_{i}=\mathcal {L}'_{i}(D_{i})\). By the inequality above we see that \(\operatorname{Ext}^{1}(\mathcal {L}_{i}, \mathcal {L}_{1}) = 0\), so the map \(\overline{s}_{i}: \mathcal {L}_{i} \hookrightarrow \mathcal {E}_{n1}=\mathcal {E}/\mathcal {L}_{1}\) lifts to a map \(s_{i}: \mathcal {L}_{i} \hookrightarrow \mathcal {E}\), \(i=2,\ldots , n\).
Now we have an injection \(s: \oplus _{i=1}^{n}\mathcal {L}_{i}\hookrightarrow \mathcal {E}\) whose divisor \(D_{s}\) satisfies \(D_{s}=D_{\overline{s}}+D_{2}+\cdots +D_{n}\). Since \(\nu (D_{2}),\ldots , \nu (D_{n}), \nu (D_{\overline{s}})\) are disjoint from \(D_{0}\) by construction, the same is true for \(\nu (D_{s})\). □
Corollary 7.17
If \(s:\mathcal {E}'=\oplus _{i=1}^{n}\mathcal {L}_{i}\hookrightarrow \mathcal {E}\) is a good framing, then the map (7.2) realizes \(\mathcal {Z}_{\mathcal {E}}(a)\) as an open and closed subscheme of \(\mathcal {Z}_{\mathcal {E}'}(a')\).
Proof
Closedness is proved in Lemma 7.8(1). Only the openness requires an argument. By the definition of a good framing, \(\nu (D_{s})\) is disjoint from \(D_{a}\), and therefore disjoint from all legs of all points of \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) by Lemma 7.12. Let \(U'=X'\backslash \operatorname{supp}(D_{s}+\sigma D_{s})\), then \(\mathcal {Z}^{r}_{\mathcal {E}}(a)=\mathcal {Z}^{r}_{\mathcal {E}}(a)_{U^{\prime \,r}}\). By Lemma 7.8(2), the inclusion
is open, hence the inclusion \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\hookrightarrow \mathcal {Z}^{r}_{\mathcal {E}'}(a')\) is open. □
DefinitionProposition 7.18
Let ℰ be a vector bundle of rank \(n\) over \(X'\) and \(a\in \mathcal {A}_{\mathcal {E}}(k)\). Let \(s:\oplus _{i=1}^{n}\mathcal {L}_{i}\hookrightarrow \mathcal {E}\) be a good framing of \((\mathcal {E},a)\). Let
Here we are using Corollary 7.17to make sense of the restriction, as it implies that \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) is a union of connected components of \(\mathcal {Z}^{r}_{\mathcal {E}'}(a')\). Then the cycle class \([\mathcal {Z}^{r}_{\mathcal {E}}(a)]\) thus defined is independent of the good framing \(s:\mathcal {E}'=\oplus _{i=1}^{n}\mathcal {L}_{i}\hookrightarrow \mathcal {E}\).
The independence of good framing will be proved in Theorem 10.1 after some preparation in §8. The idea is to construct another 0cycle class on \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) without making auxiliary choices (which is done by introducing Hitchin shtukas), and show that the two constructions agree.
By Proposition 7.13, \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) is proper over \(k\), therefore the degree of the 0cycle of \([\mathcal {Z}^{r}_{\mathcal {E}}(a)]\in \operatorname{Ch}_{0}(\mathcal {Z}^{r}_{\mathcal {E}}(a))\) is a welldefined number in \(\mathbf {Q}\). The main problem we are concerned with in this Part is to determine \(\deg [ \mathcal {Z}_{\mathcal {E}}^{r}(a)] \in \mathbf {Q}\).
8 Hitchintype moduli spaces
In this section we introduce certain “Hitchintype moduli stacks” which will help to analyze the special cycles. In particular, we will be able to use these to give an alternative construction of the cycle classes associated to special divisors, that is manifestly independent of auxiliary choices.
8.1 Hitchin stacks
Until §8.4, we fix an arbitrary positive integer \(m\).
Definition 8.1
The Hitchin stack \(\mathcal {M}^{\mathrm{all}}(m,n)\) (sometimes denoted \(\mathcal {M}^{\mathrm{all}}\) when \(m,n\) are understood) has \(S\)points the groupoid consisting of the following data.

ℰ a rank \(m\) vector bundle on \(X' \times S\).

ℱ a rank \(n\) vector bundle on \(X' \times S\), equipped with a Hermitian map \(h \colon \mathcal {F} \xrightarrow{\sim} \sigma ^{*} \mathcal {F}^{\vee}\).

A map of underlying coherent sheaves \(t \colon \mathcal {E} \rightarrow \mathcal {F}\) over \(X'\times S\).
We define \(\mathcal {M}(m,n)\subset \mathcal {M}^{\mathrm{all}}\) (sometimes denoted ℳ when \(m,n\) are understood) to be the open substack where the map \(t\) base changes to an injective map on \(X'_{\overline {s}}\) for each geometric point \(\overline {s} \rightarrow S\).
Let us emphasize that both ℰ and \((\mathcal {F},h)\) are varying in this definition. We will usually suppress the dependence on \(m,n\) from the notation.
8.2 Hitchin base
Definition 8.2
We define the following two versions of the Hitchin base.

(1)
\(\mathcal {A}^{\mathrm {all}}(m)\) (sometimes denoted \(\mathcal {A}^{\mathrm {\operatorname{all}}}\) when \(m\) is understood) to be the stack whose \(S\)points is the groupoid of the following data:

ℰ a rank \(m\) vector bundle on \(X'\times S\);

\(a \colon \mathcal {E} \rightarrow \sigma ^{*} \mathcal {E}^{\vee}\) is a map of coherent sheaves on \(X'\times S\) such that \(\sigma (a^{\vee}) = a\).


(2)
We define \(\mathcal {A}\subset \mathcal {A}^{\mathrm {all}}\) to be the open substack where \(a\colon \mathcal {E} \rightarrow \sigma ^{*} \mathcal {E}^{\vee}\) is injective after base change to \(X'_{\overline {s}}\) for every geometric point \(\overline {s} \rightarrow S\).
Definition 8.3
For integers \(1 \leq m \leq n\), we define the Hitchin fibration for \(\mathcal {M}^{\mathrm{all}} = \mathcal {M}^{\mathrm{all}}(m,n)\) to be the map \(f \colon \mathcal {M}^{\mathrm{all}} \rightarrow \mathcal {A}^{\mathrm {all}}\) sending \((\mathcal {E}, (\mathcal {F},h), t)\) to the composition
Remark 8.4
In general the Hitchin fibration does not send \(\mathcal {M}(m,n)\) to \(\mathcal {A}(m)\) even when \(m\le n\). However, in the special case \(m=n\), the Hitchin map does send \(\mathcal {M}(n,n)\) to \(\mathcal {A}(n)\) because when \(t:\mathcal {E}\to \mathcal {F}\) is generically injective, it is generically an isomorphism for rank reasons, hence the induced Hermitian map \(a\) on ℰ is generically nondegenerate.
8.3 Hitchin shtukas
We now discuss a notion of shtukas for Hitchin stacks. Throughout, \(\mathcal {M}= \mathcal {M}(m,n)\).
Definition 8.5
Hecke stacks for Hitchin spaces
For \(r \geq 0\), we define \(\operatorname{Hk}_{\mathcal {M}^{\operatorname{all}}}^{r}\) to be the stack whose \(S\)points are given by the groupoid of the following data:

(1)
\((\{x'_{i}\}_{1\le i\le r},\{(\mathcal {F}_{i},h_{i})\}_{0\le i\le r})\in \operatorname{Hk}^{r}_{U(n)}(S)\).

(2)
A vector bundle ℰ of rank \(m\) on \(X' \times S\).

(3)
Maps \(t_{i} \colon \mathcal {E} \rightarrow \mathcal {F}_{i}\) fitting into the commutative diagram
We define the open substack \(\operatorname{Hk}_{\mathcal {M}}^{r} \subset \operatorname{Hk}_{\mathcal {M}^{\operatorname{all}}}^{r}\) by the condition that \(t_{0} \) base changes to an injective map along every geometric point \(\overline {s} \rightarrow S\) (equivalently, every \(t_{i}\) has this property). Let \(\operatorname{pr}_{i}: \operatorname{Hk}^{r}_{\mathcal {M}}\to \mathcal {M}\) (resp. \(\operatorname{pr}_{i}^{\operatorname{all}} \colon \operatorname{Hk}^{r}_{\mathcal {M}^{\operatorname{all}}} \to \mathcal {M}^{\operatorname{all}}\)) be the map recording \((\mathcal {E}, \mathcal {F}_{i}, h_{i}, t_{i})\), for \(0\le i\le r\).
Definition 8.6
Shtukas for Hitchin stacks
For \(r \geq 0\), we define \(\operatorname{Sht}_{\mathcal {M}^{\operatorname{all}}}^{r}\) as the fibered product
and the open substack \(\operatorname{Sht}_{\mathcal {M}}^{r} \subset \operatorname{Sht}_{\mathcal {M}^{\operatorname{all}}}^{r}\) as the fibered product
Explicitly, the stack \(\operatorname{Sht}_{\mathcal {M}^{\operatorname{all}}}^{r}\) parametrizes diagrams of the form below, with notation as in Definition 8.5,
and \(\operatorname{Sht}_{\mathcal {M}}^{r}\) is the open substack where \(t_{0}\) base changes to an injective map along every geometric point \(\overline {s} \rightarrow S\) (equivalently, the same property holds for every \(t_{i}\)).
In particular, \(\mathcal {E} \xrightarrow{\sim} {}^{\tau }\mathcal {E}\) is a shtuka with no legs, exhibiting ℰ as arising from a rank \(m\) vector bundle on \(X'\), i.e. coming from \(\operatorname{Bun}_{\operatorname{GL}'_{m}}(k)\). Note that in (8.2), ℰ is not fixed, so the automorphisms of ℰ are present in the functor of points of \(\operatorname{Sht}_{\mathcal {M}^{\operatorname{all}}}^{r}\). Therefore, if we define
then \(\operatorname{Sht}_{\mathcal {M}^{\operatorname{all}}}^{r}\) decomposes as a disjoint union of special cycles
This decomposition can be refined. The compositions \(f \circ \operatorname{pr}_{i} \colon \operatorname{Hk}_{\mathcal {M}^{\operatorname{all}}}^{r} \rightarrow \mathcal {A}^{ \mathrm {all}}\) all coincide, and they induce a map
Here \(\mathcal {A}^{\mathrm {all}}(k)\) is the groupoid of pairs \(\{\mathcal {E}, \mathcal {E}\xrightarrow{a} \sigma ^{*} \mathcal {E}^{\vee}\}\). Let us write \(\operatorname{Aut}(a) := \operatorname{Aut}(\mathcal {E}\xrightarrow{a} \sigma ^{*} \mathcal {E}^{\vee}) \subset \operatorname{Aut}(\mathcal {E})\). Then each \(\mathcal {E}\xrightarrow{a} \sigma ^{*} \mathcal {E}^{\vee}\) defines a map \(a: \operatorname{Spec\,}k \to B \operatorname{Aut}(a)(k)\hookrightarrow \mathcal {A}^{\operatorname{all}}(k)\), the latter map being an openclosed inclusion, and we have a Cartesian square
In particular, \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\) is a finite étale cover of an openclosed substack of \(\operatorname{Sht}_{\mathcal {M}^{\operatorname{all}}}^{r}\) isomorphic to \([\mathcal {Z}_{\mathcal {E}}^{r}(a)/\operatorname{Aut}(a)(k)]\). Similarly, restricting to the injective locus (see Definition 7.4) we have a Cartesian square
If \(m=n\) and \(a\in \mathcal {A}(k)\), then we have \(\mathcal {Z}_{\mathcal {E}}^{r}(a)^{\circ}=\mathcal {Z}_{\mathcal {E}}^{r}(a)\) and we can replace \(\mathcal {A}^{\mathrm {all}}(k)\) by \(\mathcal {A}(k)\) in the above diagram (see Remark 8.4).
8.4 From vector bundles to torsion sheaves
For the rest of the section, we concentrate on the case \(m=n\). In this case, we will relate \(\mathcal {M}= \mathcal {M}(n,n)\) to the moduli stack of Hermitian torsion sheaves introduced in §4.1. We introduce the following abbreviated notations.
Definition 8.7
Let \(d\in \mathbf {Z}_{\ge 0}\).

(1)
Let \(\mathcal {M}_{d} = \mathcal {M}(n,n)_{d}\) be the openclosed substack of \(\mathcal {M}= \mathcal {M}(n,n)\) consisting of \((\mathcal {E}\xrightarrow {t}\mathcal {F})\) where \(d=\frac{\deg \mathcal {E}^{\vee}\deg (\mathcal {E})}{2}=\chi (X',\mathcal {E})\).

(2)
Let \(\mathcal {A}_{d} = \mathcal {A}(n)_{d}\) be the openclosed substack of \(\mathcal {A}= \mathcal {A}(n)\) consisting of \((\mathcal {E},a)\) where \(d=\frac{\deg \mathcal {E}^{\vee}\deg (\mathcal {E})}{2}=\chi (X',\mathcal {E})\).
By Remark 8.4, the Hitchin map for \(\mathcal {M}^{\mathrm{all}}= \mathcal {M}^{\mathrm{all}}(n,n)\) restricts to a map
When \(d\) is understood, we abbreviate \(f\) for \(f_{d}\).
Recall that \(\operatorname{Herm}_{2d}=\operatorname{Herm}_{2d}(X'/X)\) is the moduli stack of length \(2d\) torsion coherent sheaves \(\mathcal {Q}\) on \(X'\) equipped with a Hermitian structure \(h_{\mathcal {Q}} \colon \mathcal {Q}\xrightarrow{\sim} \sigma ^{*}\mathcal {Q}^{\vee}\), where \(\mathcal {Q}^{\vee} := \underline {\operatorname{Ext}}^{1}(\mathcal {Q}, \omega _{X'})\) such that \(\sigma ^{*} h_{\mathcal {Q}}^{\vee}=h_{\mathcal {Q}}\). Alternatively we may think of \(h_{\mathcal {Q}}\) as the datum of a perfect pairing
where \(\omega _{X',F'}\) is the constant Zariski sheaf of rational differential form on \(X'\). The Hermitian condition is equivalent to \(h'_{\mathcal {Q}}(u,v)=\sigma ^{*}h'_{\mathcal {Q}}(v,u)\) for local sections \(u,v\) of \(\mathcal {Q}\).
In §4.5 we have also introduced the moduli stack \(\operatorname{Lagr}_{2d}=\operatorname{Lagr}_{2d}(X'/X)\) classifying \((\mathcal {Q},h_{\mathcal {Q}}, \mathcal {L})\) where \((\mathcal {Q},h_{\mathcal {Q}})\in \operatorname{Herm}_{2d}\) and \(\mathcal {L}\subset \mathcal {Q}\) is a Lagrangian subsheaf.
There is a canonical map \(g: \mathcal {A}_{d} \rightarrow \operatorname{Herm}_{2d}\) sending \((\mathcal {E},a)\) to the torsion sheaf \(\mathcal {Q}=\sigma ^{*} \mathcal {E}^{\vee}/\mathcal {E}\) together with the Hermitian structure \(h_{\mathcal {Q}}\) defined in §5.1.
We have a map \(g_{\mathcal {M}}: \mathcal {M}_{d} \rightarrow \operatorname{Lagr}_{2d}\) sending \((\mathcal {E} \xrightarrow {t} \mathcal {F}) \in \mathcal {M}_{d}\) to the torsion Hermitian sheaf \((\mathcal {Q}=\sigma ^{*} \mathcal {E}^{\vee}/\mathcal {E}, h_{\mathcal {Q}})\) constructed above together with the Lagrangian \(\mathcal {L}=\mathcal {F}/\mathcal {E}\).
Lemma 8.8
The maps defined above fit into a Cartesian diagram
Proof
Given \(a: \mathcal {E} \rightarrow \sigma ^{*} \mathcal {E}^{\vee}\) that is injective, the datum of a subsheaf \(\mathcal {L}\subset \sigma ^{*} \mathcal {E}^{\vee}/\mathcal {E}\) is the same as a coherent sheaf ℱ such that \(\mathcal {E} \subset \mathcal {F} \subset \sigma ^{*} \mathcal {E}^{\vee}\). It is easy to see that ℒ is Lagrangian if and only if ℱ is selfdual under the Hermitian map \(a\). □
Corollary 8.9
The Hitchin fibration \(f_{d} \colon \mathcal {M}_{d} \rightarrow \mathcal {A}_{d}\) is proper.
Proof
Apply Lemma 8.8 and the fact that \(\upsilon _{2d}\) is proper. □
Lemma 8.10
The map \(\upsilon _{2d} \colon \operatorname{Lagr}_{2d} \rightarrow \operatorname{Herm}_{2d}\) is small.
Proof
The map \(\pi ^{\operatorname{Herm}}_{2d}\) from §4.2 factors as
Since \(\lambda _{2d}\) is surjective and \(\pi ^{\operatorname{Herm}}_{2d}\) is small by Proposition 4.5, we get the desired statement. □
8.4.1 Proof of Proposition 7.13
Let \(\operatorname{Lagr}(\mathcal {Q})\) be the moduli space of Lagrangian subsheaves of \(\mathcal {Q}\). Let \(\operatorname{Hk}^{r}_{\operatorname{Lagr}(\mathcal {Q})}\) be its Hecke version, classifying points \(\{x'_{i}\}_{1\le i\le r}\) of \(X'\) and chains of Lagrangian subsheaves of \(\mathcal {Q}\)
where the dashed arrow \(f_{i}\) are modifications at \(x'_{i}\cup \sigma (x'_{i})\), similar to those in Definition 6.3. There is a natural map \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\to \operatorname{Hk}^{r}_{\operatorname{Lagr}(\mathcal {Q})}\) sending a point \((\{x'_{i}\}, \{t_{i}: \mathcal {E}\to \mathcal {F}_{i}\})\) of \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) to the collection of (necessarily Lagrangian) subsheaves \(\overline{\mathcal {F}}_{i}=\operatorname{coker}(t_{i})\subset \mathcal {Q}=\sigma ^{*}\mathcal {E}^{\vee}/\mathcal {E}\). This map fits into a Cartesian diagram
Now both \(\operatorname{Lagr}(\mathcal {Q})\) and \(\operatorname{Hk}^{r}_{\operatorname{Lagr}(\mathcal {Q})}\) are proper schemes over \(k\), hence the same is true for \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\). The diagram also makes it clear that \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) only depends on \((\mathcal {Q},\overline{a})\).
8.5 Smoothness
Lemma 8.11
The map
sending \((\mathcal {E}\xrightarrow {t}\mathcal {F},h)\) to \((\operatorname{coker}(t), (\mathcal {F},h))\) is smooth of relative dimension \(dn\). In particular, \(\mathcal {M}_{d}\) is smooth of pure dimension \(dn+n^{2}(g1)\).
Proof
Consider the stack \(\mathcal {M}'_{d}\) classifying \((\mathcal {T},\mathcal {F},h, s)\) where \(\mathcal {T}\in \operatorname{Coh}_{d}(X')\), \((\mathcal {F},h)\in \operatorname{Bun}_{U(n)}\) and an \(\mathcal {O}_{X'}\)linear map \(s:\mathcal {F}\to \mathcal {T}\). Let \(\beta '_{d}: \mathcal {M}'_{d}\to \operatorname{Coh}_{d}(X') \times \operatorname{Bun}_{U(n)}\) be the natural map. Due to the vanishing of \(\operatorname{Ext}^{1}(\mathcal {F},\mathcal {T})\), \(\beta '_{d}\) exhibits \(\mathcal {M}'_{d}\) as a vector bundle over \(\operatorname{Coh}_{d}(X') \times \operatorname{Bun}_{U(n)}\) of rank \(dn=\dim \operatorname{Hom}_{X'}(\mathcal {F},\mathcal {T})\). Now \(\mathcal {M}_{d}\) is the open substack of \(\mathcal {M}'_{d}\) where \(s\) is surjective. Therefore \(\beta _{d}\) is also smooth of relative dimension \(dn\). □
Proposition 8.12
The map \(g \colon \mathcal {A}_{d} \rightarrow \operatorname{Herm}_{2d}\) is smooth.
Proof
We have a map \(s^{\operatorname{Lagr}}_{2d} \colon \operatorname{Lagr}_{2d}\to X'_{d}\) sending \((\mathcal {Q},h_{\mathcal {Q}},\mathcal {L})\) to the divisor of ℒ. Recall also the map \(s^{\operatorname{Herm}}_{2d}: \operatorname{Herm}_{2d}\to X_{d}\) sending \((\mathcal {Q},h_{\mathcal {Q}})\) to the descent of the divisor of \(\mathcal {Q}\) to \(X\).
Recall in §4.5 we introduced the open subset
Let \(\operatorname{Lagr}_{2d}^{\diamondsuit } \subset \operatorname{Lagr}_{2d}\) and \(\mathcal {M}_{d}^{\diamondsuit } \subset \mathcal {M}_{d}\) be the preimages of \((X'_{d})^{\diamondsuit }\) under the maps \(s^{\operatorname{Lagr}}_{2d}\) and \(s^{\operatorname{Lagr}}_{2d}\circ g_{\mathcal {M}}\).
We claim that both squares in the diagram
are Cartesian. The left square is Cartesian by definition. Now we show that the right square is Cartesian. Let \((\mathcal {Q},h_{\mathcal {Q}})\in \operatorname{Herm}_{2d}\), \(D' \in (X_{d}')^{\diamondsuit }\) lying over \(D=s^{\operatorname{Herm}}_{2d}(\mathcal {Q},h_{\mathcal {Q}})\). Since \(D'\cap \sigma (D')=\varnothing \), there is a unique Lagrangian subsheaf \(\mathcal {L}\subset \mathcal {Q}\) supported on the support of \(D'\), namely \(\mathcal {L}=\mathcal {Q}_{\operatorname{supp}D'}\). This gives the unique point \((\mathcal {Q},h_{\mathcal {Q}}, \mathcal {L})\in \operatorname{Lagr}^{\diamondsuit }_{2d}\) mapping to \((\mathcal {Q},h_{\mathcal {Q}})\in \operatorname{Herm}_{2d}\) and \(D'\in (X'_{d})^{\diamondsuit }\).
Note that the map \((X'_{d})^{\diamondsuit } \rightarrow X_{d}\) is faithfully flat: it is clearly surjective, and the map \(\nu _{d}: X'_{d} \rightarrow X_{d}\) is a finite morphism between smooth schemes, hence flat. We will show that \(\mathcal {M}_{d}^{\diamondsuit } \rightarrow \operatorname{Lagr}_{d}^{\diamondsuit }\) is smooth. By fppf descent it then follows that \(\mathcal {A}_{d} \rightarrow \operatorname{Herm}_{d}\) is also smooth.
Recall from §4.5 that \(\varepsilon '_{d}: \operatorname{Lagr}_{2d}\to \operatorname{Coh}_{d}(X')\) (recording only ℒ) restricts to an isomorphism \(\operatorname{Lagr}_{2d}^{\diamondsuit }\stackrel {\sim }{\to }\operatorname{Coh}_{d}(X')^{\diamondsuit }:=\operatorname{Coh}_{d}(X')_{(X_{d}')^{ \diamondsuit }}\). Therefore it suffices to show that the composition \(\mathcal {M}_{d}^{\diamondsuit }\xrightarrow {g_{\mathcal {M}}}\operatorname{Lagr}_{2d}^{\diamondsuit }\xrightarrow {\varepsilon '_{d}} \operatorname{Coh}_{d}(X')^{ \diamondsuit }\) is smooth. This follows from the smoothness of \(\mathcal {M}_{d}\to \operatorname{Coh}_{d}(X')\) proved in Lemma 8.11. □
Corollary 8.13
The Hitchin fibration \(f \colon \mathcal {M}_{d} \rightarrow \mathcal {A}_{d}\) is small. The complex \(Rf_{*}\overline {\mathbf {Q}}_{\ell}\) is a shifted perverse sheaf that is the middle extension from any dense open substack of \(\mathcal {A}_{d}\).
Proof
By the smoothness of \(g\) in Proposition 8.12 and the Cartesian diagram in Lemma 8.8, the smallness of \(f\) follows from that of \(\upsilon _{2d}: \operatorname{Lagr}_{2d}\to \operatorname{Herm}_{2d}\), which is proved in Lemma 8.10. □
8.6 Cycle class from Hitchin shtukas
In this subsection we take \(m=n\), so \(\mathcal {M}= \mathcal {M}(n,n)\) and \(\mathcal {A}= \mathcal {A}(n)\). Now consider the Hitchin shtukas for \(\mathcal {M}_{d}\subset \mathcal {M}\). Let \(N=\dim \mathcal {M}_{d}\). By Corollary 8.13, \(\dim \mathcal {A}_{d}=N\). The Cartesian diagram (8.1) restricts to a Cartesian diagram
We would like to define a 0cycle class on \(\operatorname{Sht}_{\mathcal {M}_{d}}^{r}\) as the Gysin pullback of a cycle on \(\operatorname{Hk}^{r}_{\mathcal {M}_{d}}\) along the Frobenius graph of \(\mathcal {M}_{d}\). Although the virtual dimension of \(\operatorname{Hk}^{r}_{\mathcal {M}_{d}}\) is the same as \(\dim \mathcal {M}_{d}\), its actual dimension may be larger. For this reason we have to define in a roundabout way a virtual fundamental cycle on \(\operatorname{Hk}^{r}_{\mathcal {M}_{d}}\) of the virtual dimension by relating it to \(\operatorname{Hk}^{1}_{\mathcal {M}_{d}}\), which we show is smooth below.
Lemma 8.14
The stack \(\operatorname{Hk}^{1}_{\mathcal {M}_{d}}\) is smooth and equidimensional of the same dimension as \(\mathcal {M}_{d}\).
Proof
Let \((x',\mathcal {F}_{0} \dashrightarrow \mathcal {F}_{1}) \in \operatorname{Hk}_{U(n)}^{1}\). Let \(\mathcal {F}^{\flat}:=\mathcal {F}^{\flat}_{1/2}=\mathcal {F}_{0} \cap \mathcal {F}_{1}\) as in Definition 6.3. The generically compatible Hermitian structures on \(\mathcal {F}_{0}\) and \(\mathcal {F}_{1}\) equip this intersection with a Hermitian structure \(h^{\flat}: \mathcal {F}^{\flat} \hookrightarrow \sigma ^{*} (\mathcal {F}^{\flat})^{\vee}\) whose cokernel has length 1 at \(x'\) and at \(\sigma (x')\). We call such a bundle almost Hermitian with defect at the ordered pair of conjugate points \((x',\sigma (x'))\). Conversely, given \((\mathcal {F}^{\flat},h^{\flat})\) almost Hermitian with defect at \((x', \sigma (x'))\), one can uniquely recover \(\mathcal {F}_{0}\) (resp. \(\mathcal {F}_{1}\)) as the upper modification of \(\mathcal {F}^{\flat}\) at \(x'\) (resp. \(\sigma (x')\)) inside \(\sigma ^{*}(\mathcal {F}^{\flat})^{\vee}\).
Let \(\operatorname{Bun}^{\flat}_{U(n)}\) be the moduli stack parametrizing \((x'\in X', \mathcal {F}^{\flat}, h^{\flat})\) where \((\mathcal {F}^{\flat}, h^{\flat})\) is an almost Hermitian bundle with defect at \((x',\sigma (x'))\). The discussion in the previous paragraph shows that there is an isomorphism \(\operatorname{Hk}^{1}_{U(n)}\stackrel {\sim }{\to }\operatorname{Bun}^{\flat}_{U(n)}\) over \(X'\). Let \(\mathcal {M}^{\flat}_{d}\) be the moduli stack of \((x',\mathcal {E}\xrightarrow {t}\mathcal {F}^{\flat}, h^{\flat})\) where \((\mathcal {F}^{\flat}, h^{\flat})\) is almost Hermitian with defect at \((x',\sigma (x'))\), ℰ is a vector bundle on \(X'\) of rank \(n\) and \(\chi (X',\mathcal {E})=d\), and \(t\) is injective. Then we have an isomorphism \(\operatorname{Hk}^{1}_{\mathcal {M}_{d}}\cong \mathcal {M}^{\flat}_{d}\).
We have a natural map
sending \((x',\mathcal {E}\xrightarrow {t}\mathcal {F}^{\flat}, h^{\flat})\) to \((\operatorname{coker}(t), (x',\mathcal {F}^{\flat}, h^{\flat}))\). The same argument as Lemma 8.11 shows that \(\beta ^{\flat}_{d}\) exhibits \(\mathcal {M}^{\flat}_{d}\) as an open substack in a vector bundle of rank \(n(d1)\) over \(\operatorname{Coh}_{d1}(X')\times \operatorname{Bun}^{\flat}_{U(n)}\). Now \(\dim \operatorname{Coh}_{d1}(X')=0\) and \(\dim \operatorname{Bun}^{\flat}_{U(n)}=\dim \operatorname{Hk}^{1}_{U(n)}=\dim \operatorname{Bun}_{U(n)}+n\) by Lemma 6.9(1). Therefore \(\operatorname{Hk}^{1}_{\mathcal {M}_{d}}\cong \mathcal {M}^{\flat}_{d}\) is of pure dimension \(\dim \operatorname{Bun}_{U(n)}+dn\), which is the same as \(\dim \mathcal {M}_{d}\) by Lemma 8.11. □
Definition 8.15
For any stack \(S\) over \(k\) we define a morphism
by the formula \(\Phi ^{r}_{S}(\xi _{0},\ldots , \xi _{r})=(\xi _{0},\xi _{1},\xi _{1}, \xi _{2},\xi _{2},\ldots , \xi _{r1}, \xi _{r}, \xi _{r}, \operatorname{Fr}( \xi _{0}))\). When \(r\) is fixed in the context, we simply write \(\Phi _{S}\).
We rewrite \(\operatorname{Sht}^{r}_{\mathcal {M}_{d}}\) as the fiber product
Here the vertical map \((\operatorname{pr}_{0},\operatorname{pr}_{1})^{r}\) sends \((h_{1},\ldots , h_{r})\in (\operatorname{Hk}^{1}_{\mathcal {M}_{d}})^{r} \) to \((\operatorname{pr}_{0}(h_{1}), \operatorname{pr}_{1}(h_{1}), \ldots , \operatorname{pr}_{0}(h_{r}),\operatorname{pr}_{1}(h_{r})) \in (\mathcal {M}_{d})^{2r}\), while \(\Delta \) is the diagonal map.
Definition 8.16
We define a 0cycle classes \([\operatorname{Sht}^{r}_{\mathcal {M}_{d}}]\in \operatorname{Ch}_{0}(\operatorname{Sht}^{r}_{\mathcal {M}_{d}})\) as the image of the fundamental class of \((\operatorname{Hk}^{1}_{\mathcal {M}_{d}})^{r} \times \mathcal {M}_{d}\) (which is smooth of the same dimension as \((\mathcal {M}_{d})^{r+1}\) by Lemma 8.14) under the refined Gysin map along \(\Phi _{\mathcal {M}_{d}}:(\mathcal {M}_{d})^{r+1}\to (\mathcal {M}_{d})^{2r+2}\) (which is defined since \(\mathcal {M}_{d}\) is smooth and equidimensional by Lemma 8.11; see [10, Theorem 2.1.12(xi)])
9 Special cycles of corank one
In this section we prove geometric properties of the special cycles \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) when \(m= \operatorname{rank} \mathcal {E}= 1\) (where the number field analogues are called “KudlaRapoport divisors”). In particular, we show that for \(a\ne 0\), \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) are local complete intersections of dimension \(r\) less than \(\operatorname{Sht}^{r}_{U(n)}\). When \(a=0\), we show that \(\mathcal {Z}^{r}_{\mathcal {E}}(a)^{*}\) has dimension at most \(\dim \operatorname{Sht}^{r}_{U(n)}r\). These geometric properties are proved by studying stratifications introduced and analyzed in §9.1 and §9.2.
9.1 Stratification of the special cycle when \(m=1\)
In this section we fix a line bundle ℒ on \(X'_{k}\) and \(a\in \mathcal {A}_{\mathcal {L}}(k)\), i.e., \(a: \mathcal {L}\to \sigma ^{*}\mathcal {L}^{\vee}\) is nonzero Hermitian. We now define a stratification of \(\mathcal {Z}:=\mathcal {Z}_{\mathcal {L}}^{r}(a)_{\overline{k}}\) and estimate the dimension of each stratum.
When \(n=1\), we have \(\dim \mathcal {Z}=0\). Indeed, Lemma 7.12 implies that there is a 0dimensional closed subscheme \(D \subset (X')^{r}\) such that the morphism \(\mathcal {Z}\rightarrow \operatorname{Sht}_{U(1)}^{r}\) takes values in the preimage of \(D\) under the leg map \(\operatorname{Sht}_{U(1)}^{r} \rightarrow (X')^{r}\). This preimage is 0dimensional by Lemma 6.9 (2). As \(Z \rightarrow \operatorname{Sht}_{U(1)}^{r}\) is finite (by Proposition 7.5) with 0dimensional image, we conclude that \(\dim \mathcal {Z}= 0\).
Therefore in the rest of this subsection we will assume \(n\ge 2\).
For each \((\mathcal {L}\xrightarrow {t_{i}}\mathcal {F}_{i})_{0\le i\le r}\in \mathcal {Z}(\overline{k})\) with legs \((x'_{i})_{1\le i\le r}\in X'(\overline{k})^{r}\), let \(D_{i}\) (\(0\le i\le r\)) be the divisor on \(X'_{\overline{k}}\) such that \(t_{i}: \mathcal {L}(D_{i})\hookrightarrow \mathcal {F}_{i}\) is saturated. For each \(1\le i\le r\), we have one of the four cases:
 (0):

\(D_{i}=D_{i1}\);
 (+):

\(D_{i}=D_{i1}+\sigma (x'_{i})\);
 (−):

\(D_{i}=D_{i1}x'_{i}\);
 (±):

\(D_{i}=D_{i1}x'_{i}+\sigma (x'_{i})\).
Since the composition \(\mathcal {L}\xrightarrow {t_{i}}\mathcal {F}_{i}\xrightarrow {h}\sigma ^{*}\mathcal {F}_{i}^{\vee} \xrightarrow {\sigma ^{*}t_{i}^{\vee}} \sigma ^{*}\mathcal {L}^{\vee}\) is equal to \(a\), we see that \(D_{i}+\sigma (D_{i})\) is a subdivisor of the divisor of \(a\). Therefore \(\nu (D_{i})\le D_{a}\) as divisors on \(X(\overline {k})\).
9.1.1 Indexing set for strata
Consider the set \(\mathfrak {D}\) of sequences of effective divisors \((D_{i})_{0\le i\le r}\) on \(X'_{\overline {k}}\) satisfying

\(\nu (D_{i})\le D_{a}\) for all \(0\le i\le r\).

For each \(1\le i\le r\), the pair \((D_{i1}, D_{i})\) falls into one of the cases \((0),(+),(),(\pm )\) above for some \(x'_{i}\in X'(\overline{k})\).

\(D_{r}={}^{\tau }D_{0}\).
It is clear that \(\mathfrak {D}\) is a finite set. This will be the index set for our stratification of \(\mathcal {Z}\).
9.1.2 Definition of strata
Fix \(D_{\bullet }=(D_{i})_{0\le i\le r}\in \mathfrak {D}\). Let \(I_{0} :=\{1\le i\le rD_{i}=D_{i1}\}\). Similarly we define \(I_{+}, I_{}\) and \(I_{\pm}\) as the set of those \(i\) such that \((D_{i1}, D_{i})\) falls into case \((+),()\) and \((\pm )\) respectively. Let \(\mathcal {Z}[D_{\bullet }]\) be the substack of \(\mathcal {Z}_{\overline {k}}\) classifying
such that every \(t'_{i}\) is saturated. Let
be the map recording those \(x'_{i}\) for \(i\in I_{0}\). Note that for \(i\in I_{+}\cup I_{}\cup I_{\pm}\), \(x'_{i}\) is determined by \(D_{\bullet }\).
Proposition 9.1
Let \(n \geq 2\).

(1)
The substacks \(\mathcal {Z}[D_{\bullet }]\) for \(D_{\bullet }\in \mathfrak {D}\) give a partition of \(\mathcal {Z}\).

(2)
Each geometric fiber of \(\pi [D_{\bullet }]\) has dimension \(\le (n1)I_{+}+(n2)I_{0}\).

(3)
We have \(\dim \mathcal {Z}[D_{\bullet }]\le r(n1)\). The equality can only be achieved when \(I_{0}=\{1,2,\ldots , r\}\), i.e., all \(D_{i}\) are equal to the same divisor of \(X'\), which is then necessarily defined over \(k\). ^{Footnote 5}
Proof
(1) Each geometric point of \(z\in \mathcal {Z}\) defines a (unique) point \(D_{\bullet }\in \mathfrak {D}\) by taking the zero divisor of \(t_{i}\), and then \(z\in \mathcal {Z}[D_{\bullet }]\) by definition.
(2) Let \(\mathcal {H}[D_{\bullet }]\) be the substack of the fiber of \((\operatorname{Hk}^{r}_{\mathcal {M}})_{\overline {k}}\) over \((\mathcal {L},a)\in \mathcal {A}_{d}(k)\) classifying data \((\{x'_{i}\},\mathcal {L}\xrightarrow {t_{i}}\mathcal {F}_{i})\) such that \(t_{i}\) extends to a map \(t'_{i}: \mathcal {L}(D_{i})\to \mathcal {F}_{i}\) which is saturated. Note that for \(i\in I_{+}\cup I_{}\cup I_{\pm}\), \(x'_{i}\) is determined by \(D_{\bullet }\). Let \(\mathcal {M}[D_{i}]\) be the substack of the fiber of \(\mathcal {M}(1,n)_{\overline {k}}\) over \((\mathcal {L},a)\in \mathcal {A}(1,n)(k)\) classifying maps \(t:\mathcal {L}\to \mathcal {F}\) that extend to a saturated map \(t': \mathcal {L}(D_{i})\to \mathcal {F}\). Then we have a Cartesian diagram of stacks over \(\overline {k}\)
Note since \(D_{r}={}^{\tau }D_{0}\), the Frobenius morphism sends \(\mathcal {M}[D_{0}]\) to \(\mathcal {M}[D_{r}]\).
Let
be the projection \(p_{r}\) and the map recording \(x'_{i}\) for \(i\in I^{0}\).
Claim 9.2
The map \(\Pi [D_{\bullet }]\) is smooth and representable of relative dimension \((n1)I_{+}+(n2)I_{0}\).
Assuming the claim, we finish the proof of (2). Indeed, we may apply Lemma 9.3 below to the Cartesian diagram (9.1) to conclude, except that \(\mathcal {M}[D_{0}]\) is generally not a scheme. To remedy, we may restrict to a finite type open substack \(\mathcal {M}[D_{0}]^{\le P}\subset \mathcal {M}[D_{0}]\) by bounding HarderNarasimhan polygon of \((\mathcal {F},h)\), and impose level structures on the Hermitian bundle \((\mathcal {F},h)\) at a closed point \(x\in X\) to arrive at a scheme \(\widetilde {\mathcal {M}}[D_{0}]^{\le P}\) which is a torsor over \(\mathcal {M}[D_{0}]^{\le P}\) by an algebraic group \(H\). Truncating and imposing the same level structures to \(\mathcal {H}[D_{\bullet }]\) gives a scheme \(\widetilde {\mathcal {H}}[D_{\bullet }]^{\le P}\) (with legs away from \(x\)) such that \(\widetilde {\mathcal {H}}[D_{\bullet }]^{\le P}/H=\mathcal {H}[D_{\bullet }]^{\le P}\). Let \(\widetilde {\mathcal {Z}}[D_{\bullet }]^{\le P}\) be defined by a Cartesian diagram similar to (9.1), with \(\mathcal {H}[D_{\bullet }]\) replaced by \(\widetilde {\mathcal {H}}[D_{\bullet }]^{\le P}\) and \(\mathcal {M}[D_{i}]\) replaced by \(\widetilde {\mathcal {M}}[D_{i}]^{\le P}\) for \(i=0,r\). We apply Lemma 9.3 to conclude that the fibers of \(\widetilde {\mathcal {Z}}[D_{\bullet }]^{\le P}\to X^{\prime \,I_{0}}_{\overline {k}}\) have dimension \(\le (n1)I_{+}+(n2)I_{0}\). Now \(\widetilde {\mathcal {Z}}[D_{\bullet }]^{\le P}/H(k)\stackrel {\sim }{\to }\mathcal {Z}[D_{\bullet }]^{\le P}\) and for varying \(P\) and \(x\), \(\mathcal {Z}[D_{\bullet }]^{\le P}\) cover \(\mathcal {Z}[D_{\bullet }]\), hence the same dimension estimate holds for \(\mathcal {Z}[D_{\bullet }]\).
It remains to prove the claim.
For \(r\ge j\ge 0\), let \(\mathcal {H}_{\ge j}\) be the moduli stack defined similarly to \(\mathcal {H}[D_{\bullet }]\) but classifying saturated maps \(\{t_{i}: \mathcal {L}(D_{i}) \to \mathcal {F}_{i}\}_{j\le i\le r}\) (lying over \(a\)) only for \(i\) in the indicated range. We can factorize \(\Pi [D_{\bullet }]\) as
The desired smoothness and relative dimension claims would follow from the following four statements:
 (\(H0\)):

If \(i\in I_{0}\), then \(\Pi _{i}\) exhibits \(\mathcal {H}_{\ge i1}\) as an open substack in a \(\mathbf {P}^{n2}\)bundle over \(\mathcal {H}_{\ge i}\).
 (\(H+\)):

If \(i\in I_{+}\), then \(\Pi _{i}\) exhibits \(\mathcal {H}_{\ge i1}\) as an open substack in a \(\mathbf {P}^{n1}\)bundle over \(\mathcal {H}_{\ge i}\).
 (\(H\)):

If \(i\in I_{}\), then \(\Pi _{i}\) is an isomorphism.
 (\(H\pm \)):

If \(i\in I_{\pm}\), then \(\Pi _{i}\) is an open immersion.
We next establish each of these statements.
Proof of (\(H0\)). When \(i\in I_{0}\), \(D_{i1}=D_{i}\). We write the modification \(\mathcal {F}_{i1}\dashrightarrow \mathcal {F}_{i}\) as
Here both arrows have cokernel of length one supported at the labelled points. Such modifications of \(\mathcal {F}_{i}\) are parametrized by a hyperplane \(H\) in the fiber \(\mathcal {F}_{i}_{\sigma (x'_{i})}\). The requirement that \(t_{i}:\mathcal {L}(D_{i})\to \mathcal {F}_{i}\) should land in \(\mathcal {F}^{\flat}_{i1/2}\) is equivalent to the (closed) condition that \(H\) should contain the line given by the image of \(\mathcal {L}(D_{i})_{\sigma (x'_{i})}\). This cuts out a \(\mathbf {P}^{n2}\) in the space of hyperplanes \(H \subset \mathcal {F}_{i}_{\sigma (x'_{i})}\). The further requirement that \(t_{i1}:\mathcal {L}(D_{i})\to \mathcal {F}^{\flat}_{i1/2}\to \mathcal {F}_{i1}\) be saturated is an open condition.
This argument globalizes in the evident way, exhibiting that \(\Pi _{i}\) as an open substack in a \(\mathbf {P}^{n2}\)bundle. This applies similarly for the analogous arguments below for the other cases.
Proof of (\(H+\)). When \(i\in I_{+}\), we have \(D_{i1}=D_{i}\sigma (x'_{i})\). We write the modification of \(\mathcal {F}_{i}\) as in (9.2). This time the choice of the \(\mathcal {F}_{i}\leftarrow \mathcal {F}^{\flat}_{i1/2}\) is the open subset of those hyperplanes \(H\in \mathbf {P}(\mathcal {F}_{i}_{\sigma (x'_{i})})\) that do not contain the image of \(\mathcal {L}(D_{i})_{\sigma (x'_{i})}\). The requirement that \(\mathcal {L}(D_{i1})=\mathcal {L}(D_{i}\sigma (x'_{i}))\to \mathcal {F}^{\flat}_{i1/2}\to \mathcal {F}_{i1}\) be saturated at \(x'_{i}\) imposes a further open condition.
Proof of (\(H\)). When \(i\in I_{}\), we have \(D_{i1}=D_{i}+x'_{i}\). We write the modification as
where both arrows have cokernel of length one supported at the labelled points. Now \(t_{i}: \mathcal {L}(D_{i})\to \mathcal {F}_{i}\) is required to extend to \(\mathcal {L}(D_{i}+x'_{i})\to \mathcal {F}^{\sharp }_{i1/2}\). This determines the upper modification \(\mathcal {F}_{i}\to \mathcal {F}^{\sharp }_{i1/2}\) uniquely, which in turn determines the lower modification \(\mathcal {F}^{\sharp }_{i1/2}\leftarrow \mathcal {F}_{i1}\) as well. We get a map \(t'_{i1}: \mathcal {L}(D_{i1})=\mathcal {L}(D_{i}+x'_{i})\to \mathcal {F}^{\sharp }_{i1/2}\). We claim that \(t'_{i1}\) automatically lands in \(\mathcal {F}_{i1}\). Indeed, the claim is equivalent to saying that under the pairing between \(\mathcal {F}_{i}_{x'_{i}}\) and \(\mathcal {F}_{i}_{\sigma (x'_{i})}\), the images of \(t_{i}(x'_{i})\) and \(t_{i}(\sigma (x'_{i}))\) pair to zero. The latter statement is equivalent to saying that the induced Hermitian map \(a'_{i}: \mathcal {L}(D_{i})\to \sigma ^{*}(\mathcal {L}(D_{i}))^{\vee}\) vanishes at \(x'_{i}\). But we know that the divisor of \(a'_{i}\) is \(\nu ^{*}D_{a}D_{i}\sigma (D_{i})\). Since \(D_{i1}=D_{i}+x'_{i}\) satisfies \(\nu (D_{i1})\le D_{a}\) by assumption, we see that \(\nu ^{*}D_{a}D_{i}\sigma (D_{i})\ge x'_{i}+\sigma (x'_{i})\), hence \(a'_{i}\) is guaranteed to vanish at \(x'_{i}\) and \(\sigma (x'_{i})\). This shows that there is a unique lifting of any point of \(\mathcal {H}_{\ge i}\) to \(\mathcal {H}_{\ge i1}\), hence \(\Pi _{i}\) is an isomorphism.
Proof of (\(H\pm \)). When \(i\in I_{\pm}\), we have \(D_{i}+x'_{i}=D_{i1}+\sigma (x'_{i})\). We write the modification as in (9.3). As in the case (\(H\)), the requirement that \(t_{i}: \mathcal {L}(D_{i})\to \mathcal {F}_{i}\) should extend to \(\mathcal {L}(D_{i}+x'_{i})\to \mathcal {F}^{\sharp }_{i1/2}\) determines the modification. Then we automatically get a map \(t_{i1}: \mathcal {L}(D_{i1})=\mathcal {L}(D_{i}+x'_{i}\sigma (x'_{i}))\to \mathcal {F}_{i1}\); the requirement that \(t_{i1}\) be saturated is an open condition. Therefore \(\Pi _{i}\) is an open immersion in this case.
(3) By (2) we have
Equality holds only if \(I_{}\) and \(I_{\pm}\) are empty. However, by degree reasons we have \(I_{+}=I_{}\), so in the equality case we must have \(I_{+}=\varnothing \) as well. We conclude that equality can only be achieved if \(I_{0}=\{1,2,\ldots , r\}\); in other words, all \(D_{i}\) must be the same. In particular, since \(D_{r} = {}^{\tau }D_{0}\), this forces \(D_{0}\) to be defined over \(k\). □
The Lemma below, a slight variant of [14, Lemma 2.13], was used above.
Lemma 9.3
Variant of [14, Lemma 2.13]
Let \(W,Z, T\) be schemes of finite type over \(\overline {k}\). Let \(Z^{(1)}\) be the Frobenius twist of \(Z\) (i.e., the pullback of \(Z\) under the \(q\)Frobenius \(\operatorname{Spec\,}\overline {k}\to \operatorname{Spec\,}\overline {k}\)). Let \(\widetilde {h} = (h_{1}, h_{T}) \colon W \rightarrow Z^{(1)} \times T\) be smooth of relative dimension \(d\), and \(h_{0}\colon W \rightarrow Z\) be an arbitrary map. Define \(V\) as the fibered product
Then each fiber of the composition map \(V \rightarrow W\xrightarrow {h_{T}}T\) has dimension \(\le d\).
Proof
Restricting \(W\) over a point \(t\in T(\overline {k})\), we reduce to the case \(T\) itself is the point \(\operatorname{Spec\,}\overline {k}\). We may assume \(Z=\operatorname{Spec\,}R\) where \(R=\overline{k}[x_{1},\ldots , x_{l}]/I\). Let \(R^{(1)}=\overline{k}\otimes _{\overline{k}}R\cong \overline{k}[\xi _{1},\ldots , \xi _{l}]/I^{(1)}\) be the base change of \(R\) under \(\operatorname{Fr}_{q}\), where \(\xi _{i}=1\otimes x_{i}\). Since \(h_{1}: W\to Z^{(1)}\) is smooth of relative dimension \(d\), by Zariski localizing we may assume \(W=\overline{k}[\xi _{1},\ldots , \xi _{l}, y_{1},\ldots , y_{m+d}]/(I^{(1)}, r_{1},\ldots , r_{m})\), with \((\frac {\partial r_{i}}{\partial y_{j}})_{j=1}^{m}\) having rank \(m\) (\(r_{i}\in \overline{k}[\xi _{1},\ldots , \xi _{l}, y_{1},\ldots , y_{m+d}]\)). Under \(h_{0}: W\to Z\), the coordinates \(x_{i}\) of \(Z\) pullback to functions \(\overline{f}_{i}\) on \(W\), \(1\le i\le l\). We lift \(f_{i}\) to polynomials \(f_{i}\in \overline{k}[\xi _{1},\ldots , \xi _{l}, y_{1},\ldots , y_{m+d}]\).
By definition, \(V\) has the form
where \(g_{i}=\xi _{i}f_{i}^{q}\). In particular, \(V\) is a closed subscheme of
The Jacobian matrix for the defining equations of \(U\) has the form
which evidently has rank \(l+m\). Hence \(U\) is smooth of dimension \(d\). Since \(V\hookrightarrow U\), \(\dim V\le d\). □
9.2 The case \(m=1\) and \(a=0\)
We keep the notations from §9.1. In this subsection we extend the discussion in §9.1 to the case \(a=0\). Fix a line bundle \(\mathcal {L}\in \operatorname{Pic}(X')\). Recall from Definition 7.4 that \(\mathcal {Z}_{\mathcal {L}}^{r}(0)^{\circ}\) is the moduli stack classifying hermitian shtukas \((\{x'_{i}\},\{\mathcal {F}_{i}\})\) together with compatible maps \(\{\mathcal {L}\xrightarrow {t_{i}}\mathcal {F}_{i}\})\) with \(t_{i}\) injective (fiberwise over the test scheme \(S\)) and the image of \(t_{i}\) being isotropic. In this subsection, let
If \(n=1\) then \(\mathcal {Z}_{\mathcal {L}}^{r}(0)^{\circ}=\varnothing \). We always assume \(n\ge 2\) below.
9.2.1 Indexing set for strata
Let \(I_{0}\sqcup I_{+}\sqcup I_{}\) be a partition of \(\{1,2,\ldots , r\}\) such that \(I_{+}=I_{}\). We denote this partition simply by \(I_{\bullet }\). For any \(N\in \mathbf {Z}_{\ge 0}\), define \(\mathfrak {D}(N;I_{\bullet })\) to be the moduli space of sequences of effective divisors \((D_{i})_{0\le i\le r}\) on \(X'_{\overline{k}}\) such that

(1)
\(\deg (D_{0})\le N\).

(2)
For \(?=0,+\) or −, and \(i\in I_{?}\), the pair \((D_{i1},D_{i})\) belongs to the corresponding Case (?) listed in the beginning of §9.1 (for some \(x'_{i}\in X'_{\overline{k}}\) in the case ? = + or −).

(3)
\(D_{r}={}^{\tau }D_{0}\).
We have a map recording the points \(x'_{i}\) for \(i\in I_{+}\cup I_{}\):
Lemma 9.4
The map \(\pi _{+}: \mathfrak {D}(N;I_{\bullet })\to (X'_{\overline {k}})^{I_{+}}\) is quasifinite.
Proof
For a fixed geometric point \((x'_{i})_{i\in I_{+}} \in (X'_{\overline {k}})^{I_{+}}(\overline {k})\), its fiber in \(\mathfrak {D}(N;I_{\bullet })\) consists of \((D_{0}, \{x'_{i}\}_{i\in I_{}})\) such that \(\deg D_{0}\le N\) and
Let \(v\in X'\) be a closed point that intersects \(\operatorname{supp}D_{0}\). If \(\deg (v)>N\), then \(D_{0}\) cannot contain all geometric points over \(v\) and hence there exists a geometric point \(yv\) such that \(y\in \operatorname{supp}{}^{\tau }D_{0}\) but \(y\notin \operatorname{supp}D_{0}\). By (9.4), \(y=\sigma (x'_{i})\) for some \(i\in I_{+}\). Therefore points in \(D_{0}\) are either over closed points of degree \(\le N\), or in the Galois orbit of \(\sigma (x'_{i})\) for some \(i\in I_{+}\). This leaves finitely many possibilities for \(D_{0}\), hence for \(\{x'_{i}\}_{i\in I_{}}\) as well. □
9.2.2 Definition of strata
For a partition \(I=(I_{0},I_{+},I_{})\) of \(\{1,2,\ldots , r\}\), define \(\mathcal {Z}[N;I_{\bullet }]\) to be the stack classifying
such that \(\{D_{i}\}\in \mathfrak {D}(N;I_{\bullet })\) with image \(\{x'_{i}\}_{i\in I_{?}}\) under \(\pi _{?}\) (? = +,−), and \(t_{i}\) extends to a saturated embedding \(\mathcal {L}(D_{i})\hookrightarrow \mathcal {F}_{i}\). We have a map
The following is the analog of Proposition 9.1 when \(a=0\).
Proposition 9.5
Let \(n \geq 2\).

(1)
For varying \(N\in \mathbf {Z}_{\ge 0}\) and partitions \(I_{\bullet }\) of \(\{1,2,\ldots , r\}\) such that \(I_{+}=I_{}\), the substacks \(\mathcal {Z}[N;I_{\bullet }]\) give a partition of \(\mathcal {Z}\).

(2)
The fibers of the map \(\pi [N;I_{\bullet }]\) have dimension \(\le (n1)I_{+}+(n2)I_{0}\).

(3)
We have \(\dim \mathcal {Z}[N;I_{\bullet }]\le r(n1)\). Moreover, when \(n\ge 3\), the equality can only be achieved when \(I_{0}=\{1,2,\ldots , r\}\), i.e., all \(D_{i}\) are equal to the same divisor of \(X'\) defined over \(k\).
Proof
(1) is similar to Proposition 9.1(1), except we have to argue that the strata are only nonempty for \(I_{+}=I_{}\), and that Case \((\pm )\) cannot appear for points in \(\mathcal {Z}=\mathcal {Z}^{r}_{\mathcal {L}}(0)^{\circ}\). The first statement follows from the assumption that \(D_{r} = {}^{\tau }D_{0}\) has the same degree as \(D_{0}\). For the second statement, suppose Case \((\pm )\) happens for the modification
and let \(H\subset \mathcal {F}_{i1,x'_{i}}\) be the hyperplane that is the image of \(\mathcal {F}_{i1/2}^{\flat}\). Then \(H^{\bot}\subset \mathcal {F}_{i1,\sigma (x'_{i})}\) is the line along which the upper modification \(\mathcal {F}_{i1/2}^{\flat} \hookrightarrow \mathcal {F}_{i}\) is performed. Let \(\ell _{x'_{i}}\) (resp. \(\ell _{\sigma (x'_{i})}\)) be the image of \(\mathcal {L}(D_{i1})\to \mathcal {F}_{i1}\) at \(x'_{i}\) (resp. at \(\sigma (x'_{i})\)). Since the image of \(\mathcal {L}(D_{i1})\) is isotropic (because \(a=0\)), \((\ell _{x'_{i}},\ell _{\sigma (x'_{i})})=0\) under the pairing between \(\mathcal {F}_{i1,x'_{i}}\) and \(\mathcal {F}_{i1,\sigma (x'_{i})}\). The condition \(D_{i}+x'_{i}=D_{i1}+\sigma (x'_{i})\) happens only if \(\ell _{x'_{i}}\not \subset H\) and \(\ell _{\sigma (x'_{i})}=H^{\bot}\). This contradicts the fact that \((\ell _{x'_{i}},\ell _{\sigma (x'_{i})})=0\).
(2) is proved in the same way as Proposition 9.1(2).
(3) Applying (2) and Lemma 9.4 we get
Since \(n\ge 2\), we have \(n\le 2(n1)\), therefore the above is \(\le (n1)(I_{0}+2I_{+})=r(n1)\). When \(n\ge 3\), we have strict inequalities \(n<2(n1)\), so equality can only be achieved when \(I_{+}\), hence \(I_{}\), are all empty. □
10 Comparison of two cycle classes
The goal in this section is to show the following theorem.
Theorem 10.1
Let \(\mathcal {E}\in \operatorname{Bun}_{\operatorname{GL}_{n}'}(k)\) and \(a\in \mathcal {A}_{\mathcal {E}}(k)\). Let \(s:\mathcal {E}'=\oplus _{i=1}^{n}\mathcal {L}_{i}\hookrightarrow \mathcal {E}\) be a good framing of \((\mathcal {E},a)\) in the sense of Definition 7.15. Let \(a':\mathcal {E}'\to \sigma ^{*}(\mathcal {E}')^{\vee}\) be the Hermitian map induced from \(a\). Then we have an equality in the Chow group
Here the restriction on the RHS is via (8.3), noting that \(\mathcal {Z}_{\mathcal {E}}^{r}(a)^{\circ}=\mathcal {Z}_{\mathcal {E}}^{r}(a)\), and the class \([\operatorname{Sht}^{r}_{\mathcal {M}(n,n)}]\) is as in Definition 8.16.
In particular, the cycle class \([\mathcal {Z}^{r}_{\mathcal {E}}(a)]\) as in DefinitionProposition 7.18is welldefined (i.e., independent of the choice of a good framing).
Below we consider the case where \(X'\) is geometrically connected. At the end of this section (§10.5) we comment on how to modify the argument in the case \(X'=X\coprod X\) or \(X'=X_{k'}\), where \(k'/k\) is the quadratic extension.
10.1 First reductions
For a vector bundle ℰ on \(X'\) let \(\mu _{\min}(\mathcal {E})\in \mathbf {Q}\) be the smallest slope that appears in the HarderNarasimhan filtration of ℰ. For ℰ of rank \(n\) and \(a\in \mathcal {A}_{\mathcal {E}}(k)\), a good framing \(s:\oplus _{i=1}^{n}\mathcal {L}_{i}\hookrightarrow \mathcal {E}\) for \((\mathcal {E},a)\) is called very good if it satisfies the additional condition

(3)
\(\mu _{\min}(\mathcal {E})>\max \{\deg \mathcal {L}_{i}+2g'1\}_{1\le i\le n}\).
Most of the work in this section will be devoted to proving the slightly weaker statement below.
Theorem 10.2
Suppose \(X'\) is connected. Then the identity (10.1) holds if \(s:\mathcal {E}'=\oplus _{i=1}^{n}\mathcal {L}_{i}\hookrightarrow \mathcal {E}\) is a very good framing of \((\mathcal {E},a)\).
Lemma 10.3
Theorem 10.2implies Theorem 10.1.
Proof
Choose effective divisors \(D_{i}\) on \(X'\) (\(1\le i\le n\)) such that \(\nu (D_{1}+\cdots +D_{n})\) is multiplicityfree and disjoint from \(\nu ^{1}(D_{a})\). Let \(\mathcal {L}'_{i}=\mathcal {L}_{i}(D_{i})\). When the \(D_{i}\)’s have sufficiently large degree, the resulting map
is a very good framing. Let \(a''\) be the induced Hermitian map \(\oplus \mathcal {L}'_{i} \rightarrow \sigma ^{*} (\oplus \mathcal {L}'_{i})^{\vee}\). By Theorem 10.2 we have
Therefore, to prove (10.1) it suffices to show
Let \(U'\) be the complement of \(\cup _{i=1}^{n}\operatorname{supp}(D_{i}+\sigma D_{i})\) in \(X'\). By construction, \(U'\) contains \(\nu ^{1}(D_{a})\), therefore \(\mathcal {Z}_{\mathcal {E}}^{r}(a)_{U^{\prime \,r}}=\mathcal {Z}_{\mathcal {E}}^{r}(a)\) by Lemma 7.12. Let \(\zeta _{i}=[\mathcal {Z}^{r}_{\mathcal {L}_{i}}(a'_{ii})^{\circ}]_{U^{\prime \,r}}\in \operatorname{Ch}_{r(n1)}( \mathcal {Z}^{r}_{\mathcal {L}_{i}}(a'_{ii})^{\circ}_{U^{\prime \,r}})\). Similarly define \(\zeta '_{i}\) using \((\mathcal {L}'_{i}, a''_{ii})\). Then it suffices to show the equality
where the intersection products are taken over \(\operatorname{Sht}^{r}_{U(n)}_{U^{\prime \,r}}\). Applying Lemma 7.8 to each injection \(\mathcal {L}'_{i}\hookrightarrow \mathcal {L}_{i}\), we see that \(\mathcal {Z}^{r}_{\mathcal {L}_{i}}(a'_{ii})_{U^{\prime \,r}}\hookrightarrow \mathcal {Z}^{r}_{\mathcal {L}'_{i}}(a''_{ii})_{U^{\prime \,r}}\) is open and closed. Therefore \(\mathcal {Z}^{r}_{\mathcal {L}_{i}}(a'_{ii})^{\circ}_{U^{\prime \,r}}\hookrightarrow \mathcal {Z}^{r}_{\mathcal {L}'_{i}}(a''_{ii})^{ \circ}_{U^{\prime \,r}}\) is open. This shows that the fundamental class \(\zeta _{i}\) is the open restriction of \(\zeta '_{i}\) to \(\mathcal {Z}^{r}_{\mathcal {L}_{i}}(a'_{ii})^{\circ}_{U^{\prime \,r}}\). The equality (10.2) then follows. □
10.2 Auxiliary moduli spaces
Let \(\underline{d}=(d_{i})_{1\le i\le n}\in \mathbf {Z}^{n}_{\ge 0}\) and \(e\in \mathbf {Z}_{\ge 0}\). Write \(d=\sum d_{i}\).
Recall that \(\mathcal {M}_{e}\subset \mathcal {M}(n,n)\) is the openclosed substack where \(\chi (X',\mathcal {E})=e\). Let \(\mathcal {M}_{\underline{d}}\) be the moduli stack classifying \((\{\mathcal {L}_{i}\}_{1\le i\le n}, (\mathcal {F},h),\{t'_{i}:\mathcal {L}_{i}\to \mathcal {F}\}_{1 \le i\le n})\) where

\(\mathcal {L}_{i}\) is a line bunde on \(X'\) with \(\chi (X',\mathcal {L}_{i})=d_{i}\) for \(1\le i\le n\);

\((\mathcal {F},h)\in \operatorname{Bun}_{U(n)}\) satisfying
$$ \mu _{\min}(\mathcal {F})>\max \{d_{i}+3g'2\}_{1\le i\le n}. $$(10.3) 
For each \(1\le i\le n\), \(t'_{i}: \mathcal {L}_{i}\to \mathcal {F}\) is an injective map (fiberwise over the test scheme).
We define \(\operatorname{Hk}^{r}_{\mathcal {M}_{\underline{d}}}\) to be the moduli stack classifying
where \(\mathcal {L}_{i}\) are the same as in \(\mathcal {M}_{\underline{d}}\), each \(\mathcal {F}_{j}\) satisfies the analogue of (10.3) with ℱ replaced by \(\mathcal {F}_{j}\), and fiberwise injective maps \(t'_{ij}:\mathcal {L}_{i}\to \mathcal {F}_{j}\) are compatible with the isomorphisms between \(\mathcal {F}_{j1}\) and \(\mathcal {F}_{j}\) away from \(x'_{i}\).
Lemma 10.4
The stacks \(\mathcal {M}_{\underline{d}}\) and \(\operatorname{Hk}^{1}_{\mathcal {M}_{\underline{d}}}\) are smooth stacks of pure dimension \(dn  (n^{2}2n)(g1)\).
Proof
We first prove the statement for \(\mathcal {M}_{\underline{d}}\). Consider the map \(\gamma _{\underline{d}}: \mathcal {M}_{\underline{d}}\to \prod _{i=1}^{n}\operatorname{Pic}^{d_{i}+g'1}_{X'} \times \operatorname{Bun}_{U(n)}\) sending \((\{\mathcal {L}_{i}\},(\mathcal {F},h),\{t_{i}\})\) to \((\{\mathcal {L}_{i}\},(\mathcal {F},h))\). For \((\mathcal {F},h)\in \operatorname{Bun}_{U(n)}\) and \(\mathcal {L}_{i}\in \operatorname{Pic}^{d_{i}+g'1}_{X'}\), the condition \(\mu _{\min}(\mathcal {F})\ge \max \{d_{i}+3g'2\}_{1\le i\le n}=\max \{ \deg \mathcal {L}_{i}+2g'1\}_{1\le i\le n}\) guarantees that \(\operatorname{Ext}^{1}(\mathcal {L}_{i}, \mathcal {F})=\operatorname{Hom}(\mathcal {F},\mathcal {L}_{i}\otimes \omega _{X'})^{\vee}=0\). Noting that \(\deg \mathcal {F}= n(g'1)\), the RiemannRoch formula implies that \(\gamma _{\underline{d}}\) exhibits \(\mathcal {M}_{\underline{d}}\) as an open substack of a vector bundle of rank \(\dim \operatorname{Hom}(\oplus \mathcal {L}_{i}, \mathcal {F})=n\sum _{i}\deg \mathcal {L}_{i}=dnn^{2}(g'1)\) over the base. In particular, \(\mathcal {M}_{\underline{d}}\) is smooth and equidimensional. Since \(\dim \operatorname{Pic}^{d_{i}+g'1}_{X'}=g'1\) and \(\dim \operatorname{Bun}_{U(n)}=n^{2}(g1)\), we conclude that
The argument for \(\operatorname{Hk}^{1}_{\mathcal {M}_{\underline{d}}}\) is similar. The natural map
exhibits \(\operatorname{Hk}^{1}_{\mathcal {M}_{\underline{d}}}\) as an open substack of a vector bundle of rank \(\dim \operatorname{Hom}(\oplus \mathcal {L}_{i}, \mathcal {F}_{0}\cap \mathcal {F}_{1})=dnn\) (using that \(\deg (\mathcal {F}_{0}\cap \mathcal {F}_{1})=n(g'1)1\)) over the base. Here we need the stronger inequality \(\mu _{\min}(\mathcal {F}_{0})>\max \{d_{i}+3g'2\}\) to guarantee \(\mu _{\min}(\mathcal {F}_{0}\cap \mathcal {F}_{1})\ge \max \{d_{i}+3g'2\}\). In particular, \(\operatorname{Hk}^{1}_{\mathcal {M}_{\underline{d}}}\) is smooth and equidimensional, and
as desired. □
Let \(\mathcal {M}_{\underline{d},e}\) be the moduli stack of \((\{\mathcal {L}_{i}\}_{1\le i\le n}, \mathcal {E}, \mathcal {F},h,s,t)\) where

\(\mathcal {L}_{i}\in \operatorname{Pic}_{X'}\) satisfies \(\chi (X',\mathcal {L}_{i})=d_{i}\) for \(i=1,\ldots , n\);

\(\mathcal {E}\in \operatorname{Bun}_{\operatorname{GL}_{n}'}\) satisfies \(\chi (X',\mathcal {E})=e\);

\((\mathcal {F},h)\in \operatorname{Bun}_{U(n)}\);

\(t: \mathcal {E}\to \mathcal {F}\) is an injective map;

\(s: \oplus _{i=1}^{n}\mathcal {L}_{i}\to \mathcal {E}\) is a very good framing for \((\mathcal {E},a)\), where \(a=\sigma ^{*}t^{\vee} \circ h\circ t\) is the induced Hermitian map on ℰ.
Note that being a very good framing requires \(d_{i}<\mu _{\min}(\mathcal {E})(3g'2)\) for all \(i\), which imposes an open condition on ℰ. We view \(\mathcal {M}_{\underline{d},e}\) as a correspondence
Here \(w_{1}\) records \(\oplus _{i=1}^{n}\mathcal {L}_{i}\xrightarrow {t\circ s}\mathcal {F}\) and \(w_{2}\) records \(\mathcal {E}\xrightarrow {t}\mathcal {F}\).
We denote the Hitchin bases for \(\mathcal {M}_{\underline{d}}, \mathcal {M}_{e}\) and \(\mathcal {M}_{\underline{d},e}\) by \(\mathcal {A}_{\underline{d}}\), \(\mathcal {A}_{e}\) and \(\mathcal {A}_{\underline{d},e}\) respectively. Here \(\mathcal {A}_{\underline{d}}\) parametrizes \((\{\mathcal {L}_{i}\}_{1\le i\le n},a'=(a'_{ij}))\) (where \(a'_{ij}: \mathcal {L}_{j}\to \sigma ^{*}\mathcal {L}_{i}^{\vee}\)) such that \(a':\oplus \mathcal {L}_{i}\to \sigma ^{*}(\oplus \mathcal {L}_{i})^{\vee}\) is an injective Hermitian map. The base \(\mathcal {A}_{e}\) classifies \((\mathcal {E}, a)\) with \(a\) an injective Hermitian map. The base \(\mathcal {A}_{\underline{d},e}\) is the moduli stack of \((\{\mathcal {L}_{i}\}_{1\le i\le n}, \mathcal {E}, s,a)\) where \((\mathcal {E},a)\in \mathcal {A}_{e}\), \(s: \oplus _{i=1}^{n}\mathcal {L}_{i}\to \mathcal {E}\) is a very good framing of \((\mathcal {E},a)\) and \(\mathcal {L}_{i}\in \operatorname{Pic}_{X'}\) with \(\chi (X',\mathcal {L}_{i})=d_{i}\). We view \(\mathcal {A}_{\underline{d},e}\) as a correspondence
We have Hitchin maps
These maps together give a map of correspondences (10.4) to (10.5). Note \(f_{\underline{d}}\) is not necessarily proper because we have imposed an open condition on the minimal slope of ℱ.
Similarly we define the Hecke version \(\operatorname{Hk}^{r}_{\mathcal {M}_{\underline{d} ,e}}\) of \(\mathcal {M}_{\underline{d},e}\) as the moduli stack of \((\{x'_{i}\}_{1\le i\le r}, \{\mathcal {E}\to \mathcal {F}_{i}\}_{0\le i\le r})\in \operatorname{Hk}^{r}_{\mathcal {M}_{e}}\) together with a very good framing \(s:\oplus \mathcal {L}_{i}\hookrightarrow \mathcal {E}\) for \((\mathcal {E},a)\) with \(\chi (X',\mathcal {L}_{i})=d_{i}\). Again we view \(\operatorname{Hk}^{r}_{\mathcal {M}_{\underline{d},e}}\) as a correspondence
Lemma 10.5
The maps \(w_{1},u_{1}\) and \(h_{1}\) are étale.
Proof
We first prove that \(u_{1}\) is étale. Let \((X_{de}\times X_{e})^{\heartsuit }\) be the open subscheme of divisors \((D_{1},D_{2})\in X_{de}\times X_{e}\) such that \(D_{1}\) is multiplicityfree and disjoint from \(D_{2}\); let \((X'_{de}\times X_{e})^{\heartsuit }\) be the preimage of \((X_{de}\times X_{e})^{\heartsuit }\) in \(X'_{de}\times X_{e}\). We have a map \(\alpha : X'_{de}\times X_{e}\to X_{d}\) sending \((D'_{1}, D_{2})\) to \(\nu (D_{1})+D_{2}\). Let \(\alpha ^{\heartsuit }\) be the restriction of \(\alpha \) to \((X'_{de}\times X_{e})^{\heartsuit }\). By factorizing \(\alpha ^{\heartsuit }\) as the composition
we see that \(\alpha ^{\heartsuit }\) is étale. From the definition we have a map
where \(j'_{de}: \mathcal {A}_{\underline{d},e}\to X'_{de}\) sends \((\{\mathcal {L}_{i}\}, \oplus \mathcal {L}_{i}\xrightarrow {s}\mathcal {E},a)\) to \(\operatorname{Div}(s)\) (the divisor of \(\det (s)\)) and \(j_{e}: \mathcal {A}_{\underline{d},e}\to X_{e}\) sends it to \(D_{a}\) (see Definition 7.3). The map \(\mathcal {A}_{\underline{d}}\to X_{d}\) used in the fiber product records the divisor \(D_{a'}\) of the Hermitian map \(a'\) on \(\oplus \mathcal {L}_{i}\). We claim that \(j\) is an open immersion. Indeed, given \((\{\mathcal {L}_{i}\},a')\in \mathcal {A}_{\underline{d}}\) and \((D'_{1}, D_{2})\in (X'_{de}\times X_{e})^{\heartsuit }\) such that \(\nu (D_{1})+D_{2}=D_{a'}\), by the disjointness of \(D'_{1},\sigma (D'_{1})\) and \(\nu ^{1}(D_{2})\), there is one and only one coherent sheaf ℰ such that \(\oplus \mathcal {L}_{i}\subset \mathcal {E}\subset \sigma ^{*}(\oplus \mathcal {L}_{i})^{\vee}\), \(\mathcal {E}/\oplus \mathcal {L}_{i}\) is supported on \(D'_{1}\), and \(\sigma ^{*}(\oplus \mathcal {L}_{i})^{\vee}/\mathcal {E}\) is supported away from \(D'_{1}\). This would give a very good framing of ℰ if the open condition \(\mu _{\min}(\mathcal {E})>\max \{d_{i}+3g'2\}_{1\le i\le n}\) is satisfied. This shows that \(j\) is an open immersion. Since \(\alpha ^{\heartsuit }\) is étale, we conclude that \(\mathcal {A}_{\underline{d},e}\) is étale over \(\mathcal {A}_{\underline{d}}\).
To show \(w_{1}\) is étale, we observe that \(\mathcal {M}_{\underline{d},e}\cong \mathcal {M}_{\underline{d}}\times _{\mathcal {A}_{\underline{d}}}\mathcal {A}_{\underline{d},e}\). Since \(u_{1}\) is étale, so is \(w_{1}\).
Finally, \(\operatorname{Hk}^{r}_{\mathcal {M}_{\underline{d},e}}\) is the open substack of \(\operatorname{Hk}^{r}_{\mathcal {M}_{\underline{d}}}\times _{\mathcal {A}_{\underline{d}}}\mathcal {A}_{\underline{d},e}\) where the legs avoid \(\operatorname{Div}(s)\). Since \(u_{1}\) is étale, so is \(h_{1}\). □
10.3 Auxiliary Hitchin shtukas
We define \(\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d}}}\) and \(\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d},e}}\) as the fiber product
The maps \(w_{i}\) and \(h_{i}\) induce maps
The stack \(\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d},e}}\) decomposes into a disjoint union of openclosed substacks indexed by \((\{\mathcal {L}_{i}\}, \oplus \mathcal {L}_{i}\xrightarrow {s} \mathcal {E}, a)\in \mathcal {A}_{\underline{d},e}(k)\)
Correspondingly, the diagram (10.7) decomposes into the disjoint union indexed by \(\mathcal {A}_{\underline{d},e}(k)\) of diagrams of the form
Here \(\mathcal {Z}^{r}_{\oplus \mathcal {L}_{i}}(a')^{\heartsuit }\subset \mathcal {Z}^{r}_{\oplus \mathcal {L}_{i}}(a')\) (where \(a'\) is the Hermitian map on \(\oplus \mathcal {L}_{i}\) induced from \(a\)) is cut by the open condition \(\mu _{\min}(\mathcal {F}_{j})>\max \{d_{i}+3g'2\}\) for all \(0\le j\le r\). From this description and Corollary 7.17, we see that:
Lemma 10.6
The map \(u_{1}\) (resp. \(u_{2}\)), when restricted to each connected component of \(\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d},e}}\), is an isomorphism onto a connected component of \(\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d}}}\) (resp. \(\operatorname{Sht}^{r}_{\mathcal {M}_{e}}\)).
10.4 Zero cycles on auxiliary Hitchin shtukas
Similar to the definition of \([\operatorname{Sht}^{r}_{\mathcal {M}_{e}}]\) given in §8.6, we define 0cycles supported on \(\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d},e}}\) and \(\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d}}}\) as follows.
We rewrite \(\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d}}}\) as the fiber product
Here \(\Phi _{\mathcal {M}_{\underline{d}}}=\Phi ^{r}_{\mathcal {M}_{\underline{d}}}\) and the vertical maps are defined as in Definition 8.15. By the smoothness of \(\operatorname{Hk}^{1}_{\mathcal {M}_{\underline{d}}}\) and \(\mathcal {M}_{\underline{d}}\) proved in Lemma 10.4 and the dimension calculation there, we define
Similarly, using the Cartesian diagram
and the smoothness and dimension calculations of \(\operatorname{Hk}^{1}_{\mathcal {M}_{\underline{d},e}}\) and \(\mathcal {M}_{\underline{d}, e}\) (which follow from Lemma 10.5 and Lemma 10.4), we define
Lemma 10.7
We have \(u_{1}^{*}[\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d}}}]=[\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d},e}}]\in \operatorname{Ch}_{0}(\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d},e}}) \).
Proof
This is because the maps \(w_{1}\) and \(h_{1}\) are both étale by Lemma 10.5. □
Lemma 10.8
We have \(u_{2}^{*}[\operatorname{Sht}^{r}_{\mathcal {M}_{e}}]=[\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d},e}}]\in \operatorname{Ch}_{0}( \operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d},e}}) \).
Proof
The diagram (10.8) is obtained from (8.6) (for \(\mathcal {M}_{d}\) replaced by \(\mathcal {M}_{e}\)) by base changing termwise along the map of the following two Cartesian diagrams induced by \(u_{2}: \mathcal {A}_{\underline{d},e}\to \mathcal {A}_{e}\):
Note that \(u_{2}: \mathcal {A}_{\underline{d},e}\to \mathcal {A}_{e}\) is smooth since it exhibits \(\mathcal {A}_{\underline{d},e}\) as an open substack of a vector bundle over \(\mathcal {A}_{e}\) (using the condition \(\mu _{\min}(\mathcal {E})>\max \{d_{i}+3g'2\}\)). We conclude by applying Proposition 10.9 below. □
10.4.1 Compatibility of cycle classes under Hitchin base change
To state the next result, we need some notations. Suppose we are given:

stacks \(S,M\) and \(H\) that are locally of finite type over \(k\) and can be stratified into locally closed substacks that are global quotient stacks;

the stack \(M\) is smooth of pure dimension \(N\) with a map \(f:M\to S\);

a map \(\widetilde {h}: H \to S^{r} \times _{\Delta ^{r}, S^{2r}}M^{2r}\) (the fiber product uses the \(r\)fold product of the diagonal \(\Delta ^{r}: S^{r} \to S^{2r}\)).
Define \(h \colon H \rightarrow M^{2r}\) as \(\widetilde {h}\) followed by the second projection. Form the Cartesian square
Let \(u: S'\to S\) be a smooth representable morphism of pure relative dimension \(D\). Let \(M'=M\times _{S}S'\), \(H'=H\times _{S^{r}}S^{\prime \,r}\) with natural maps \(\widetilde {h}': H'\to S^{\prime \,r}\times _{\Delta ^{r}, S^{\prime \,2r}}M^{\prime \,2r}\). Let \(h':H'\to M^{\prime \,2r}\) be the resulting map. Let \(u_{M}: M'\to M\) and \(u_{H}: H'\to H\) be the natural maps. Form the Cartesian square
Since \(S^{r+1}\times _{\Phi _{S}, S^{2r+2}, \Delta ^{r+1}}S^{r+1}=S(k)\), \(\operatorname{Sht}_{H}\) decomposes as
Similarly \(\operatorname{Sht}'_{H}\) decomposes into the disjoint union of \(\operatorname{Sht}_{H}'(s')\) indexed by \(s'\in S'(k)\). Then the natural map \(u_{\operatorname{Sht}}: \operatorname{Sht}'_{H}\to \operatorname{Sht}_{H}\) lifts to an isomorphism
Proposition 10.9
Let \(\zeta \in \operatorname{Ch}_{*}(H)\) and \([M]\) be the fundamental class of \(M\). Then we have
Proof
Consider first the diagram where all squares are Cartesian
Here the top vertical arrows are smooth and representable. By the compatibility of Gysin map with flat pullback [10, Theorem 2.1.12(ix)], we have
Here we recall that \(D\) is the relative dimension of \(u\). We have
where \(S'_{s}=u^{1}(s)\), which is a smooth scheme over \(k\). Factorize \(u_{\operatorname{Sht}}\) as the composition
From (10.11) we see that \(i\) is a regular embedding of codimension \((r+1)D\). Now \(v\) and \(u_{\operatorname{Sht}}\) are both smooth. Applying [10, Theorem 2.1.12(ix)] we have \(u^{*}_{\operatorname{Sht}}()=i^{!}v^{*}\) as maps \(\operatorname{Ch}_{*}(\operatorname{Sht}_{H})\to \operatorname{Ch}_{*}(\operatorname{Sht}'_{H})\). Therefore
On the other hand, consider the following diagram where all squares are Cartesian
Here \(\Phi _{1}\) is the base change of \(\Phi _{S'}\) and \(\Phi _{2}\) is the base change of \(\Phi _{M}\). The outer square of the top rows give (10.9). By the transitivity of Gysin maps, we have
Since \(u^{2r+2}_{M}\) is smooth representable, we have \(\Phi _{2}^{!}u_{H}^{*}(\zeta \times [M])=\Phi _{M}^{!}u_{H}^{*}(\zeta \times [M])\). Hence
Since both \(\Phi _{1}\) and \(i\) are regular embeddings of the same codimension, we have \(\Phi ^{!}_{1}()=i^{!}()\) as maps \(\operatorname{Ch}_{*}(\operatorname{Sht}_{H}\times _{S^{r+1}}S^{\prime \,r+1})\to \operatorname{Ch}_{*(r+1)D}( \operatorname{Sht}'_{H})\), by [10, Theorem 2.1.12(xi)] and [5, Theorem 6.2(c)]. Therefore
Combining (10.10), (10.12) and (10.13) we conclude
□
Lemma 10.10
Let \((\{\mathcal {L}_{i}\}_{1\le i\le n}, a')\in \mathcal {A}_{\underline{d}}(k)\) and \(\mathcal {E}':=\oplus _{i=1}^{n}\mathcal {L}_{i}\). Then we have an equality
For the definition of \(\zeta ^{r}_{\mathcal {L}_{1},\ldots , \mathcal {L}_{n}}(a')\) see §7.8.
Proof
We will apply the Octahedron Lemma [28, Theorem A.10] to a diagram of moduli stacks in our setting. Since the Octahedron Lemma requires certain stacks in question to be DeligneMumford, we need to rigidify our moduli stacks to satisfy these requirements. This is a minor technical issue which we encourage the reader to ignore: it is simply because the Octahedron Lemma in [28] is not stated and proved in the most general form.
Let \(v\in X'\). Let \(P_{v}\) be the moduli space (a scheme!) of line bundles on \(X'\) together with a trivialization of their fibers over \(v\). Let \(\mathbf {G}_{v}=\operatorname{Res}^{k_{v}}_{k}\mathbf {G}_{m}\). Then \(P_{v}\to \operatorname{Pic}_{X'}\) is a \(\mathbf {G}_{v}\)torsor.
Now for each moduli stacks \(\mathcal {M}_{\underline{d}}, \mathcal {A}_{\underline{d}}\) and \(\operatorname{Hk}^{r}_{\dot{\mathcal {M}}_{\underline{d}}}\) that involve an \(n\)tuple of line bundles \(\{\mathcal {L}_{i}\}\), we write \(\dot{\mathcal {M}}_{\underline{d}}, \dot{\mathcal {A}}_{\underline{d}}\) and \(\dot{\operatorname{Hk}}^{r}_{\dot{\mathcal {M}}_{\underline{d}}}\) to mean their rigidified versions where \(\mathcal {L}_{i}\in \operatorname{Pic}_{X'}\) is replaced by \(\dot{\mathcal {L}}_{i}\in P_{v}\). Note that we do not impose any compatibility condition between the rigidifcation on \(\mathcal {L}_{i}\) and the rest of the structures classified by these moduli stacks. Define \(\operatorname{Sht}^{r}_{\dot{\mathcal {M}}_{\underline{d}}}\) using the dotted version of the left one of the Cartesian diagrams in (10.6).
Note that \(\dot{\mathcal {M}}_{\underline{d}}, \dot{\mathcal {A}}_{\underline{d}}\) and \(\dot{\operatorname{Hk}}^{r}_{\dot{\mathcal {M}}_{\underline{d}}}\) are now schemes, and they are \(\mathbf {G}_{v}^{n}\)torsors over their undotted counterparts. The dotted version of Lemma 10.4 remains valid if we add \(n\deg (v)=\dim \mathbf {G}_{v}^{n}\) to the dimensions. Also, \(\operatorname{Sht}^{r}_{\dot{\mathcal {M}}_{\underline{d}}}\simeq \operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d}}}\times _{ \mathcal {A}_{\underline{d}}(k)}\dot{\mathcal {A}}_{\underline{d}}(k)\). Since \(\dot{\mathcal {A}}_{\underline{d}}(k)\to \mathcal {A}_{\underline{d}}(k)\) is surjective, to prove the Lemma, it suffices to prove its dotted version: for any \((\{\dot{\mathcal {L}}_{i}\}, a')\in \dot{\mathcal {A}}_{\underline{d}}(k)\), writing \(\mathcal {E}':=\oplus _{i=1}^{n}\mathcal {L}_{i}\), then there is an open and closed embedding \(\mathcal {Z}^{r}_{\mathcal {E}'}(a')\hookrightarrow \operatorname{Sht}^{r}_{\dot{\mathcal {M}}_{\underline{d}}}\) (using the rigifications \(\dot{\mathcal {L}}_{i}\) of \(\mathcal {L}_{i}\)); then we shall prove
For \(i=1,\ldots , n\), let \(\mathcal {N}_{d_{i}}\) be the open substack of \(\mathcal {M}(1,n)\) consisting of points \((\mathcal {L}\hookrightarrow \mathcal {F},h)\) where \(\chi (X',\mathcal {L})=d_{i}\) and \(\mu _{\min}(\mathcal {F})>d_{i}+3g'2\). Similarly define \(\operatorname{Hk}^{1}_{\mathcal {N}_{d_{i}}}\) and \(\operatorname{Sht}^{r}_{\mathcal {N}_{d_{i}}}\); these are open substacks of \(\operatorname{Hk}^{1}_{\mathcal {M}(1,n)}\) and \(\operatorname{Sht}^{r}_{\mathcal {M}(1,n)}\) respectively.
Let \(\dot{\mathcal {N}}_{d_{i}}, \operatorname{Hk}^{1}_{\dot{\mathcal {N}}_{d_{i}}}\) be the rigidified versions of \(\mathcal {N}_{d_{i}}\) and \(\operatorname{Hk}^{1}_{\mathcal {N}_{d_{i}}}\) where \(\mathcal {L}\in \operatorname{Pic}_{X'}\) is replaced by \(\dot{\mathcal {L}}\in P_{v}\). Let \(\omega _{i}: \dot{\mathcal {N}}_{d_{i}}\to \operatorname{Bun}_{U(n)}\) and \(\widetilde {\omega }_{i}: \operatorname{Hk}^{1}_{\dot{\mathcal {N}}_{d_{i}}}\to \operatorname{Hk}^{1}_{U(n)}\) be the forgetful maps.
We shall apply the Octahedron Lemma [28, Theorem A.10] to the following diagram:
The fiber products of the three columns are
where
Let \(\overline {\mathcal {M}}_{\underline{d}}\) be the moduli stack of \((\{\dot{\mathcal {L}}_{i}\}, \oplus _{i=1}^{n}\mathcal {L}_{i}\xrightarrow {t'} \mathcal {F},h)\) defined similarly as \(\dot{\mathcal {M}}_{\underline{d}}\) but without the condition that \(t'\) be injective, only that \(t'_{i}=t_{\mathcal {L}_{i}}\) be injective. Then \(\dot{\mathcal {M}}_{\underline{d}}\hookrightarrow \overline {\mathcal {M}}_{\underline{d}}\) is open. Similarly define \(\operatorname{Hk}^{1}_{\overline {\mathcal {M}}_{\underline{d}}}\) and \(\operatorname{Sht}^{r}_{\overline {\mathcal {M}}_{\underline{d}}}\). Note that \(\overline {\mathcal {M}}_{\underline{d}}\) is exactly the fiber product of
Similar remarks apply to \(\operatorname{Hk}^{1}_{\overline {\mathcal {M}}_{\underline{d}}}\). Therefore the fiber products of the three rows of (10.15) are
The common fiber product of (10.16) and (10.19) is \(\operatorname{Sht}^{r}_{\overline {\mathcal {M}}_{\underline{d}}}\), which decomposes as a disjoint union over the groupoid \(\mathcal {B}(k)\) of \((\dot{\mathcal {L}}_{i},a'_{ii})_{1\le i\le n}\) where \(\dot{\mathcal {L}}_{i}\in P_{v}(k)\) with \(\chi (X',\mathcal {L}_{i})=d_{i}\) and \(a'_{ii}:\mathcal {L}_{i}\to \sigma ^{*}\mathcal {L}_{i}^{\vee}\) injective Hermitian. For a point \({(\dot{\mathcal {L}}_{i}, a'_{ii})_{1\le i\le n}}\) of \(\mathcal {B}(k)\), we have
where \(\operatorname{Sht}^{r}_{\overline {\mathcal {M}}_{\underline{d}}}_{(\dot{\mathcal {L}}_{i}, a'_{ii})_{1\le i\le n}}\) means the pullback of \(\operatorname{Sht}^{r}_{\overline {\mathcal {M}}_{\underline{d}}}\) to \(\operatorname{Spec\,}k\) along the corresponding \(\operatorname{Spec\,}k \rightarrow \mathcal {B}(k)\).
We check that the assumptions for applying the Octahedron Lemma are satisfied (the numbering below refers to that in [28, Theorem A.10]).

(1)
All members in the diagram (10.15) are smooth and equidimensional. This is clear for \(\operatorname{Bun}_{U(n)}\) and \(\operatorname{Hk}^{1}_{U(n)}\). The same argument as in Lemma 10.4 proves that \(\dot{\mathcal {N}}_{d_{i}}\) and \(\operatorname{Hk}^{1}_{\dot{\mathcal {N}}_{d_{i}}}\) are smooth of pure dimension \(d_{i}n+(n^{2}2n+2)(g1)+\deg (v)\).

(2)
We check that, in forming the fiber products of the middle and bottom rows and the left and middle columns, the intersections are proper intersections with smooth equidimensional outcomes with the expected dimension. Here we use Lemma 6.8 and 6.9 to argue for the left and middle columns. For the rows, the same argument as in Lemma 10.4 proves that \(\overline {\mathcal {M}}_{\underline{d}}\) and \(\operatorname{Hk}^{1}_{\overline {\mathcal {M}}_{\underline{d}}}\) are smooth of the same dimension as \(\dot{\mathcal {M}}_{\underline{d}}\), which is \(dn(n^{2}2n)(g1)+n\deg (v)\). This is the virtual dimension for \(\overline {\mathcal {M}}_{\underline{d}}\) as the fiber product of (10.18), since
$$\begin{aligned} &\sum _{i=1}^{n}\dim \mathcal {N}_{d_{i}}(n1)\dim \operatorname{Bun}_{U(n)} \\ &\quad=\sum _{i=1}^{n} \left (d_{i}n+(n^{2}2n+2)(g1)+\deg (v)\right )n^{2}(n1)(g1) \\ &\quad=dn(n^{2}2n)(g1)+n\deg (v). \end{aligned}$$ 
(3)
We check the fiber products of the top row and right column of (10.15) satisfy the conditions for [28, A.2.10]. The fiber product of the top row is also a proper intersection: this follows from the same calculation as for the middle and bottom rows. The fiber product of the right column is also a proper intersection: this uses the decomposition (10.17) and the calculation of the dimension of \(\mathcal {Z}^{r}_{\mathcal {L}_{i}}(0)^{*}\) in Proposition 9.5 and the dimension of \(\mathcal {Z}^{r}_{\mathcal {L}_{i}}(a'_{ii})^{*}\) in Proposition 9.1.
The only issue is that \((\operatorname{Hk}^{1}_{\overline {\mathcal {M}}_{\underline{d}}})^{r+1}\) may not be a DeligneMumford stack, which was part of the requirement of [28, A.2.10]. However, we argue that this is not really an issue. The proof of the Octahedron Lemma allows the following flexibility: since eventually we only care about the 0cycles restricted to \(\operatorname{Sht}^{r}_{\dot{\mathcal {M}}_{\underline{d}}}\), in the middle steps of forming the fiber products, we may restrict to open substacks as long as the final fiber product contains \(\operatorname{Sht}^{r}_{\dot{\mathcal {M}}_{\underline{d}}}\) and only need check the relevant requirements there. Now in (10.19) we may restrict to the open substack \((\operatorname{Hk}^{1}_{\dot{\mathcal {M}}_{\underline{d}}})^{r+1}\subset (\operatorname{Hk}^{1}_{\overline {\mathcal {M}}_{\underline{d}}})^{r+1}\), which is a scheme.

(4)
The same remark as above shows that it suffices to check that the fiber squares obtained from (10.16) and (10.19), after replacing \(\overline {\mathcal {M}}_{\underline{d}}\) by \(\dot{\mathcal {M}}_{\underline{d}}\), each satisfy the condition [28, A.2.8]. Therefore it suffices to check

\(\operatorname{Sht}^{r}_{\dot{\mathcal {M}}_{\underline{d}}}\) admits a finite flat presentation in the sense of [28, Definition A.1]. This is true because \(\operatorname{Sht}^{r}_{\dot{\mathcal {M}}_{\underline{d}}}\) is a scheme.

The diagonal map \(\Delta : \operatorname{Sht}^{r}_{U(n)}\hookrightarrow \prod _{i=1}^{n}\operatorname{Sht}^{r}_{U(n)}\) is a regular local immersion. This is true because \(\operatorname{Sht}^{r}_{U(n)}\) is a smooth DeligneMumford stack.

The map \(\Phi _{\dot{\mathcal {M}}_{\underline{d}}}: \dot{\mathcal {M}}_{\underline{d}}^{r+1}\to \dot{\mathcal {M}}_{ \underline{d}}^{2r+2}\) is a regular local immersion. This is true because \(\dot{\mathcal {M}}_{\underline{d}}\) is a smooth equidimensional scheme by the dotted version of Lemma 10.4.

The conclusion of the (variant of) Octahedron Lemma says that the following two elements in \(\operatorname{Ch}_{0}(\operatorname{Sht}^{r}_{\overline {\mathcal {M}}_{\underline{d}}})\)
become the same when restricted to \(\operatorname{Sht}^{r}_{\dot{\mathcal {M}}_{\underline{d}}}\). Further restricting to \(\mathcal {Z}_{\mathcal {E}'}^{r}(a')\) we get the desired identity (10.14). □
Proof of Theorem 10.2
Restricting the equality in Lemma 10.10 to \(\mathcal {Z}_{\mathcal {E}}^{r}(a)\), which is open and closed in \(\mathcal {Z}_{\mathcal {E}'}^{r}(a')\) by Corollary 7.17, we get
For fixed \((\oplus \mathcal {L}_{i}\hookrightarrow \mathcal {E},a)\in \mathcal {A}_{\underline{d},e}(k)\), \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) can be viewed as a finite étale cover of an openclosed substack in \(\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d}}},\operatorname{Sht}^{r}_{\mathcal {M}_{\underline{d},e}}\) and \(\operatorname{Sht}^{r}_{\mathcal {M}_{e}}\) by Lemma 10.6. By Lemma 10.7 and Lemma 10.8 we have
Combining this with (10.20) proves the theorem. □
10.5 Proof of Theorem 10.1 for \(X'=X\coprod X\) or \(X_{k'}\)
Here \(k'/k\) is the quadratic extension.
In the case \(X'=X\coprod X\), we have \(\operatorname{Bun}_{U(n)}\cong \operatorname{Bun}_{\operatorname{GL}_{n}}\). We shall identify a Hermitian bundle ℱ on \(X'\) with a pair of vector bundles \((\mathcal {F}_{1},\mathcal {F}_{2})\) equipped with an isomorphism \(\mathcal {F}_{2}\cong \mathcal {F}_{1}^{\vee}\), each living on one copy of \(X\). A vector bundle ℰ on \(X'\) of rank \(n\) corresponds to two rank \(n\) vector bundles \((\mathcal {E}_{1},\mathcal {E}_{2})\), each living on one copy of \(X\). Now \(\mathcal {A}_{\mathcal {E}}(k)\) is the set of injective maps \(a: \mathcal {E}_{1}\to \mathcal {E}^{\vee}_{2}\). A good framing \(s:\oplus _{i=1}^{n}\mathcal {L}_{i}\hookrightarrow \mathcal {E}\) for \((\mathcal {E},a)\) now consists of line bundles \(\mathcal {L}_{i}=(\mathcal {L}_{i,1}, \mathcal {L}_{i,2})\) (\(1\le i\le n\)) satisfying the same conditions in Definition 7.15; it is called very good if it satisfies the additional conditions
 (\(3_{1}\)):

\(\mu _{\min}(\mathcal {E}_{1})>\max \{\deg \mathcal {L}_{i,1}+2g1\}_{1\le i\le n}\), and
 (\(3_{2}\)):

\(\mu _{\min}(\mathcal {E}_{2})>\max \{\deg \mathcal {L}_{i,2}+2g1\}_{1\le i\le n}\).
The same argument of Lemma 10.3 shows that it suffices to prove the analogue of Theorem 10.2, i.e., prove Theorem 10.1 for very good framings.
In both the \(X'=X\coprod X\) and \(X'=X_{k'}\) case, we need to modify the definitions of \(\mathcal {M}_{\underline{d}}\) and \(\mathcal {M}_{\underline{d},e}\) as follows. In the definition of \(\mathcal {M}_{\underline{d},e}\), we use the notion of very good framing just defined over geometric fibers of \(X'_{S}\to S\) (which are of the form \(X_{\overline{s}}\coprod X_{\overline{s}}\)). In the definition of \(\mathcal {M}_{\underline{d}}\), we change the inequality (10.3) to two inequalities over the geometric fibers of \(X'_{S}\to S\)
The same inequalities should be imposed in the definition of \(\mathcal {N}_{d_{i}}\) that appear in the proof of Lemma 10.10. With these changes, the argument for proving Theorem 10.2 goes through.
11 Local intersection number and trace formula
11.1 Local nature of the intersection problem
Recall from Proposition 7.13 that \(\mathcal {Z}^{r}_{\mathcal {E}}(a)\) only depends on the Hermitian torsion sheaf \(\mathcal {Q}=\operatorname{coker}(a)\). In this subsection we show that the 0cycle class \([\mathcal {Z}^{r}_{\mathcal {E}}(a)]\) also only depends on \(\mathcal {Q}\).
Recall the stacks \(\operatorname{Herm}_{2d}=\operatorname{Herm}_{2d}(X'/X)\) and \(\operatorname{Lagr}_{2d}\) from §4. We have a selfcorrespondence \(\operatorname{Hk}^{r}_{\operatorname{Lagr}_{2d}}\) of \(\operatorname{Lagr}_{2d}\) over \(\operatorname{Herm}_{2d}\): it classifies \((\mathcal {Q},h,\{\mathcal {L}_{i}\}_{0\le i\le r})\) where \((\mathcal {Q},h)\in \operatorname{Herm}_{2d}\), \(\mathcal {L}_{i}\subset \mathcal {Q}\) are Lagrangian subsheaves such that \(\mathcal {L}_{i}/(\mathcal {L}_{i}\cap \mathcal {L}_{i1})\) has length one for \(1\le i\le r\). Define the local version \(\operatorname{Sht}^{r}_{\operatorname{Lagr}_{2d}}\) of \(\operatorname{Sht}^{r}_{\mathcal {M}_{d}}\) by the Cartesian diagram
We have a decomposition into openclosed substacks
Lemma 11.1
The stack \(\operatorname{Hk}^{1}_{\operatorname{Lagr}_{2d}}\) is smooth of dimension zero.
Proof
We may identify \(\operatorname{Hk}^{1}_{\operatorname{Lagr}_{2d}}\) with the moduli stack of \((0\subset \mathcal {L}'\subset \mathcal {L}\subset \mathcal {Q}, h)\) where \((\mathcal {L}\subset \mathcal {Q},h)\in \operatorname{Lagr}_{2d}\) and \(\mathcal {L}/\mathcal {L}'\) has length one. Under the local chart for \(\operatorname{Herm}_{2d}\) described in Lemma 4.3, \(\operatorname{Hk}^{1}_{\operatorname{Lagr}_{2d}}\) becomes \([\mathfrak {p}/P]\), where \(P\subset \mathrm {O}_{2d}=\mathrm {O}(V)\) is the parabolic subalgebra stabilizing a pair of subspaces \(L'\subset L\) with \(L\) Lagrangian and \(\dim L'=d1\). This local description implies that \(\operatorname{Hk}^{1}_{\operatorname{Lagr}_{2d}}\) is smooth of dimension zero. □
Rewriting \(\operatorname{Sht}^{r}_{\operatorname{Lagr}_{2d}}\) as the fibered product (cf. Definition 8.15)
we define a 0cycle class
Restricting to \(\mathcal {Z}^{r}_{\mathcal {Q}}\) we get
Recall the maps \(g_{\mathcal {M}}: \mathcal {M}_{d}\to \operatorname{Lagr}_{2d}\) and \(g:\mathcal {A}_{d}\to \operatorname{Herm}_{2d}\) defined in §8.4. We also have a map \(g_{\operatorname{Hk}}: \operatorname{Hk}^{r}_{\mathcal {M}_{d}}\to \operatorname{Hk}^{r}_{\operatorname{Lagr}_{2d}}\) sending \((\{x'_{i}\} , \{\mathcal {E}\xrightarrow {t_{i}}\mathcal {F}_{i}\})\) to \((\mathcal {Q}=\operatorname{coker}(a), h_{\mathcal {Q}}, \{\operatorname{coker}(t_{i})\})\) (\(a\) is the induced Hermitian map on ℰ). The maps \(g_{\operatorname{Hk}}, g_{\mathcal {M}}\) and \(g\) exhibit the diagram (8.5) as the pullback of the diagram (11.1) via the base change \(g: \mathcal {A}_{d}\to \operatorname{Herm}_{2d}\). In particular we have a natural map
For fixed \((\mathcal {E},a)\in \mathcal {A}_{d}(k)\) with image \(\mathcal {Q}=\operatorname{coker}(a)\in \operatorname{Herm}_{2d}(k)\), \(g_{\operatorname{Sht}}\) restricts to an isomorphism to the openclosed subschemes
Proposition 11.2
We have an equality
Proof
Apply Proposition 10.9 to the diagram (11.2), the fundamental class \(\zeta =[(\operatorname{Hk}^{1}_{\operatorname{Lagr}_{2d}})^{r} \times \operatorname{Lagr}_{2d}]\) and the base change map \(u=g: \mathcal {A}_{d}\to \operatorname{Herm}_{2d}\). By Proposition 8.12, \(g\) is smooth. We then have
Since \(g_{\operatorname{Hk}}\) is smooth, \(g_{\operatorname{Hk}}^{*}[(\operatorname{Hk}^{1}_{\operatorname{Lagr}_{2d}})^{r} \times \operatorname{Lagr}_{2d}]=[(\operatorname{Hk}^{1}_{ \mathcal {M}_{d}})^{r} \times \operatorname{Hk}^{1}_{\mathcal {M}_{d}}]\). The above equality then becomes (11.4). □
Combined with Theorem 10.1, we get a local description of the cycle class \([\mathcal {Z}^{r}_{\mathcal {E}}(a)]\):
Corollary 11.3
For any \((\mathcal {E},a)\in \mathcal {A}_{d}(k)\) with image \(\mathcal {Q}=\operatorname{coker}(a)\in \operatorname{Herm}_{2d}(k)\), \([\mathcal {Z}^{r}_{\mathcal {E}}(a)]\) is the same as \([\mathcal {Z}^{r}_{\mathcal {Q}}]\) under the isomorphism (11.3). In particular,
11.2 Sheaves on \(\operatorname{Herm}_{2d}\)
To describe the direct image complex \(Rf_{*}\overline {\mathbf {Q}}_{\ell}\) on \(\mathcal {A}_{d}\), by the Cartesian diagram (8.4), we first need to understand \(R(\upsilon _{2d})_{*}\overline {\mathbf {Q}}_{\ell}\) on \(\operatorname{Herm}_{2d}\).
Lemma 11.4
The perverse sheaf \(R(\upsilon _{2d})_{*}\overline {\mathbf {Q}}_{\ell}\) on \(\operatorname{Herm}_{2d}\) is canonically isomorphic to \((\operatorname{Spr}^{\operatorname{Herm}}_{2d})^{S_{d}}\) (see Proposition 4.5(2)). Here the \(S_{d}\)action on \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}\) is the restriction of the Springer \(W_{d}\)action.
Proof
We have a Cartesian diagram
where \(\varepsilon '_{d}\) sends \((\mathcal {Q},h_{\mathcal {Q}},\mathcal {L})\) to ℒ and \(\widetilde {\varepsilon }_{d}\) sends \((\mathcal {Q}_{1}\subset \cdots \subset \mathcal {Q}_{d}\subset \cdots \subset \mathcal {Q},h)\) to \((\mathcal {Q}_{1}\subset \cdots \subset \mathcal {Q}_{d})\). By proper base change
In particular, \(R\lambda _{2d*}\overline {\mathbf {Q}}_{\ell}\) carries an action of \(S_{d}\). Moreover, the induced \(S_{d}\)action on \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}\cong R(\upsilon _{2d})_{*}R\lambda _{2d*}\overline {\mathbf {Q}}_{\ell}\) is the restriction of the Springer \(W_{d}\)action to \(S_{d}\): this can be easily checked over \(\operatorname{Herm}^{\circ }_{2d}\), and then the statement holds over \(\operatorname{Herm}_{2d}\) since \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}\) is the middle extension from its restriction to \(\operatorname{Herm}^{\circ }_{2d}\) by Proposition 4.5(2). Since \((R\lambda _{2d*}\overline {\mathbf {Q}}_{\ell})^{S_{d}}\cong \varepsilon ^{\prime \,*}_{d}(\operatorname{Spr}_{d,X'})^{S_{d}} \cong \overline {\mathbf {Q}}_{\ell}\), we conclude that \((\operatorname{Spr}^{\operatorname{Herm}}_{2d})^{S_{d}}\cong R(\upsilon _{2d})_{*}(R\lambda _{2d*} \overline {\mathbf {Q}}_{\ell})^{S_{d}}\cong R(\upsilon _{2d})_{*}\overline {\mathbf {Q}}_{\ell}\), as desired. □
It is an elementary exercise to see that \(\operatorname{Ind}_{S_{d}}^{W_{d}}\mathbf {1}\) decomposes into irreducible representations
where \(\rho _{i}=\operatorname{Ind}^{W_{d}}_{(\mathbf {Z}/2\mathbf {Z})^{d}\rtimes (S_{i}\times S_{di})}( \chi _{i}\boxtimes \mathbf {1})\), and \(\chi _{i}: (\mathbf {Z}/2\mathbf {Z})^{d}\rtimes (S_{i}\times S_{di})\to \{\pm 1\}\) is the character that is nontrivial on the first \(i\) factors of \((\mathbf {Z}/2\mathbf {Z})^{i}\), trivial on the rest and trivial on \(S_{i}\times S_{di}\). The decomposition also shows up in [28, §8.1.1].
Recall the notation \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}[\rho ]\) from Definition 4.7.
Corollary 11.5
There is a canonical decomposition
Proof
By Lemma 11.4 and Frobenius reciprocity, we have
The desired decomposition then follows from (11.5). □
Definition 11.6
Define the graded perverse sheaf on \(\operatorname{Herm}_{2d}(X'/X)\)
The fundamental class of the selfcorrespondence \(\operatorname{Hk}^{1}_{\operatorname{Lagr}_{2d}}\) of \(\operatorname{Lagr}_{2d}\) is viewed as a cohomological correspondence of the constant sheaf on \(\operatorname{Lagr}_{2d}\) with itself. It induces an endomorphism (see notation from [28, A.4.1])
Proposition 11.7
The action of \((\upsilon _{2d})_{!}[\operatorname{Hk}^{1}_{\operatorname{Lagr}_{2d}}]\) on \(R(\upsilon _{2d})_{*}\overline {\mathbf {Q}}_{\ell}\) preserves the decomposition (11.6), and it acts on \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}[\rho _{i}]\) by multiplication by \((d2i)\).
Proof
By Proposition 4.5(2), \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}[\rho _{i}]\) is the middle extension from its restriction to \(\operatorname{Herm}^{\circ }_{2d}\), it suffices to prove the same statement on \(\operatorname{Herm}_{2d}^{\circ }\). Let \(I_{d}' \subset X_{d}' \times X'\) be the universal divisor \(\{(D \in X_{d}', y \in X') \colon y \in D\}\). The maps \(\operatorname{pr}(D,y) = D\) and \(q(D,y) = Dy+\sigma (y)\) define the incidence correspondence as in [28, proof of Proposition 8.3],
Now over \(\operatorname{Herm}_{2d}^{\circ }\), the map \(\operatorname{Herm}_{2d}^{\circ }\to X^{\circ }_{d}\) is smooth, and \(\operatorname{Hk}^{1}_{\operatorname{Lagr}_{2d}}\) is the pullback of (11.7). We therefore reduce to checking the statement for the action of \([I'_{d}]\) on the direct image sheaf of \(\nu _{d}: X'_{d}\to X_{d}\), which is done in the proof of [28, Proposition 8.3]. □
11.3 Lefschetz trace formula
We shall give a slight generalization of the Lefschetz trace formula [28, Proposition A.12] expressing the intersection number of a cycle with the graph of Frobenius as a trace. Instead of the graph of Frobenius, we need to intersect along \(\Phi _{M}:M^{r+1}\to M^{2r+2}\). Consider the following situation:

Let \(S\) be an algebraic stack locally of finite type over \(k=\mathbf {F}_{q}\). Assume \(S\) can be stratified by locally closed substacks that are global quotients.

Let \(M\) be a smooth equidimensional stack over \(k=\mathbf {F}_{q}\) of dimension \(N\) with a proper representable map \(f: M\to S\).

For \(1\le i\le r\), let \((\operatorname{pr}^{i}_{0},\operatorname{pr}^{i}_{1}): C_{i}\to M\times _{S}M\) be a selfcorrespondence of \(M\) over \(S\). Assume \(\operatorname{pr}^{i}_{0}\) is proper and representable.
Form the Cartesian diagram
Then \(\operatorname{Sht}_{C}\) decomposes as
where \(\operatorname{Sht}_{C}(s)\) is the fibered product of \(\operatorname{Sht}_{C} \rightarrow S(k)\) against \(s \in S(k)\). For \(s \in S(k)\), we write \(\operatorname{Sht}_{C}_{s}\) for the fibered product
Then \(\operatorname{Sht}_{C}_{s} \rightarrow \operatorname{Sht}_{C}(s)\) is a torsor for \(\operatorname{Aut}(s)\), and in particular a finite étale cover.
Suppose we are given cycle classes
The cycle class \(\operatorname{cl}(\zeta _{i})\in \mathrm {H}^{\mathrm {BM}}_{2N}({C_{i},\overline {\mathbf {Q}}_{\ell}(N)})\) is viewed as a cohomological correspondence between the constant sheaf on \(M\) and itself. Therefore it induces an endomorphism of \(Rf_{!}\overline {\mathbf {Q}}_{\ell}\) which we denote by \(f_{!}\operatorname{cl}(\zeta _{i})\).
Proposition 11.8
For each \(s\in S(k)\) we have
Proof
We first prove the formula when \(S\) is a scheme of finite type. In this case \(M,C_{i}\) are also schemes of finite type over \(k\). Let \(C=C_{1}\times _{M}\times \cdots \times _{M}C_{r}\) be the composition correspondence, with maps \(\operatorname{pr}_{i}: C\to M\) for \(0\le i\le r\). Consider the diagram where all squares are Cartesian
Here
We have \(\Phi _{M}=\Phi _{2}\circ \Phi _{1}\). Let \(\zeta =\Phi _{2}^{!}(\zeta _{1}\times \zeta _{2}\times \cdots \times \zeta _{r} \times [M])\in \operatorname{Ch}_{N}(C)\).
On the one hand, by the transitivity of the Gysin maps,
Applying the Lefschetz trace formula [28, Proposition A.12], we get
One the other hand, by a diagram chase, we see that \(\operatorname{cl}(\zeta )\) is the composition of the cohomological correspondences \(\operatorname{cl}(\zeta _{i})\) (\(1 \le i\le r\)), hence \(f_{!}\operatorname{cl}(\zeta )\in \operatorname{End}(Rf_{!}\overline {\mathbf {Q}}_{\ell})\) is the composition of \(f_{!}\operatorname{cl}(\zeta _{1})\circ f_{!}\operatorname{cl}(\zeta _{2})\circ \cdots \circ f_{!}\operatorname{cl}(\zeta _{r})\). Combining this fact with (11.9) and (11.10) we get the desired formula.
Now consider the general case where \(S\) is a stack locally of finite type over \(k\) and we aim to prove the formula for \(s\in S(k)\). We claim that there exists a scheme \(S'\) of finite type over \(k\) and a smooth map \(u: S'\to S\) such that \(u(S'(k))\) contains \(s\). Indeed, pick any smooth map \(u_{1}: S_{1}\to S\) with \(S_{1}\) a scheme of finite type over \(k\), such that \(s\) is contained in the image of \(u\). Let \(s_{1}\in S_{1}(\mathbf {F}_{q^{m}})\) be a point that maps to \(s\). Let \((S_{1}/S)^{m}\) be the \(m\)fold fibered product of \(S_{1}\) over \(S\), based changed to \(\overline {k}\). We equip \((S_{1}/S)^{m}\) with the \(\operatorname{Fr}\)descent datum given by \((x_{1},\ldots , x_{m})\mapsto (\operatorname{Fr}(x_{m}), \operatorname{Fr}(x_{1}),\ldots , \operatorname{Fr}(x_{m1}))\). This gives a descent of \((S_{1}/S)^{m}\) to a scheme \(S'\) over \(k\) equipped with a map \(u:S'\to S\) which is still smooth since \(u_{1}\) is. Now \(s_{1}\) gives rise to a \(k\)point \(s'=(s_{1},\operatorname{Fr}(s_{1}),\ldots , \operatorname{Fr}^{m1}(s_{1}))\in S'(k)\) such that \(u(s')=s\).
Let \(M'=M\times _{S}S'\), \(C'_{i}=C_{i}\times _{S}S'\) and let \(u_{C_{i}}:C'_{i}\to C_{i}\) be the projection. Define \(\operatorname{Sht}'_{C}\) using the analog of the diagram (11.8) with \(M\) and \(C_{i}\) replaced by \(M'\) and \(C'_{i}\). Then \(\operatorname{Sht}'_{C}\cong \operatorname{Sht}_{C}\times _{S(k)}S'(k)\). For \(s'\in S'(k)\) such that \(u(s')=s\), we get an isomorphism \(\operatorname{Sht}'_{C}_{s'}\stackrel {\sim }{\to }\operatorname{Sht}_{C}_{s}\). Let \(\zeta '_{i}=u_{C_{i}}^{*}\zeta _{i}\). Now we apply Proposition 10.9 to the diagram (11.8) along the base change map \(u:S'\to S\) to get
Restricting to \(\operatorname{Sht}'_{C}_{s'}\cong \operatorname{Sht}_{C}_{s}\) and taking degrees we get
On the other hand, letting \(f':M'\to S'\), by smooth base change we have
Since the right sides of (11.11) and (11.12) are equal by the scheme case that is already proven, the left sides of (11.11) and (11.12) are also equal, proving the proposition in general. □
Recall the graded perverse sheaf \(\mathcal {K}^{\mathrm {Int}}_{d}(T)\) on \(\operatorname{Herm}_{2d}\) from Definition 11.6.
Corollary 11.9
Let \((\mathcal {Q},h_{\mathcal {Q}})\in \operatorname{Herm}_{2d}(k)\). We have
Proof
The first equality is an application of Proposition 11.8 to the case \(S=\operatorname{Herm}_{2d}\), \(M=\operatorname{Lagr}_{2d}\), \(C_{i}=\operatorname{Hk}^{1}_{\operatorname{Lagr}_{2d}}\) and \(\zeta _{i}=[\operatorname{Hk}^{1}_{\operatorname{Lagr}_{2d}}]\). The second equality follows from Proposition 11.7. The third one is a direct calculation. □
Combining Corollary 11.9 with Corollary 11.3 we get:
Corollary 11.10
Let \((\mathcal {E},a)\in \mathcal {A}_{d}(k)\) with image \((\mathcal {Q},h_{\mathcal {Q}})\in \operatorname{Herm}_{2d}(k)\). Then we have
11.4 Symmetry
This subsection is not used in the proof of the main theorem. The graded perverse sheaf \(\mathcal {K}^{\operatorname{Int}}_{d}(T)\) has a palindromic symmetry that we spell out. First, the étale double covering \(\nu :X'\to X\) gives a local system \(\eta _{X'/X}\) on \(X\) with monodromy in \(\pm 1\). It induces a local system \(\eta _{d}\) on \(X_{d}\) with monodromy in \(\pm 1\): its stalk at a divisor \(x_{1}+\cdots +x_{d}\in X_{d}(\overline {k})\) is \(\otimes _{i=1}^{d}(\eta _{X'/X})_{x_{i}}\). Let
where \(s^{\operatorname{Herm}}_{2d}: \operatorname{Herm}_{2d}\to X_{d}\) is the support map. This is a rankone local system on \(\operatorname{Herm}_{2d}\) with monodromy in \(\pm 1\).
Lemma 11.11
We have a canonical isomorphism of perverse sheaves on \(\operatorname{Herm}_{2d}\):
Proof
By Proposition 4.5(2), \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}[\rho _{d}]\) is the middle extension of its restriction to the open dense substack \(\operatorname{Herm}_{2d}^{\circ }\) (preimage of \(X^{\circ }_{d}\)). The same is true for \(\eta _{d}\) because it is a local system and \(\operatorname{Herm}_{2d}\) is smooth. Therefore it suffices to check their equality over \(\operatorname{Herm}_{2d}^{\circ }\), over which both are obtained by pushing out the \(W_{d}\)torsor \((X^{\prime \,d})^{\circ }\to X_{d}^{\circ }\) along the character \(\chi _{d}: W_{d}\to \{\pm 1\}\). □
Lemma 11.12
There is an isomorphism of graded perverse sheaves on \(\operatorname{Herm}_{2d}\)
Proof
The equality amounts to
Both sides are middle extensions from \(\operatorname{Herm}^{\circ }_{2d}\) by Proposition 4.5(2), over which they correspond to representations \(\rho _{di}\) and \(\chi _{d}\otimes \rho _{i}\) of \(W_{d}\). By definition,
Inserting \(\chi _{d}_{W_{i}\times W_{di}}\cong \chi _{i}\boxtimes \chi _{di}\) to the right side above gives
□
Lemma 11.13
If \((\mathcal {Q},h_{\mathcal {Q}})\in \operatorname{Herm}_{2d}(k)\) is the image of some \((\mathcal {E},a)\in \mathcal {A}_{d}(k)\), then \(\operatorname{Tr}(\operatorname{Fr}, \eta ^{\operatorname{Herm}}_{2d}_{\mathcal {Q}})=1\).
Proof
If \(X'/X\) is split, then \(\eta ^{\operatorname{Herm}}_{2d}\) is trivial, and there is nothing to prove. Below we assume \(X'/X\) is nonsplit. The local system \(\eta _{d}\) on \(X_{d}\) is pulled back from a local system \(\eta _{\operatorname{Pic}}\) on \(\operatorname{Pic}_{X}\) via the AbelJacobi map \(\mathrm {AJ}_{d}:X_{d}\to \operatorname{Pic}^{d}_{X}\subset \operatorname{Pic}_{X}\). The Frobenius trace function of \(\eta _{\operatorname{Pic}}\) is the idèle class character
trivial on the image of \(\operatorname{Nm}_{X'/X}: \operatorname{Pic}_{X'}(k)\to \operatorname{Pic}_{X}(k)\). Denote by \(\det _{X}(\mathcal {Q})\) the image of \(\mathcal {Q}\) under \(\operatorname{Herm}_{2d}\to X_{d}\xrightarrow {\mathrm {AJ}_{d}} \operatorname{Pic}^{d}_{X}\). We have \(\eta ^{\operatorname{Herm}}_{2d}_{\mathcal {Q}}\cong \eta _{\operatorname{Pic}}_{\det _{X}(\mathcal {Q})}\) as \(\operatorname{Fr}\)modules. Now \((\mathcal {Q},h_{\mathcal {Q}})\) comes from \((\mathcal {E},a)\), which implies
By [27, p.291, Theorem 13], \(\omega _{X}\) is a square in \(\operatorname{Pic}_{X}(k)\), hence \(\eta _{F'/F}(\omega _{X}^{\otimes n})=1\). Since \(\eta _{F'/F}(\operatorname{Nm}_{X'/X}(\det \mathcal {E}))=1\), we see that \(\eta _{F'/F}(\det _{X}(\mathcal {Q}))=1\), hence \(\operatorname{Tr}(\operatorname{Fr}, \eta ^{\operatorname{Herm}}_{2d}_{\mathcal {Q}})=1\). □
Corollary 11.14
Let \((\mathcal {E},a)\in \mathcal {A}_{d}(k)\) with image \((\mathcal {Q},h_{\mathcal {Q}})\in \operatorname{Herm}_{2d}(k)\). Then \(s\mapsto q^{ds}\operatorname{Tr}(\operatorname{Fr}, \mathcal {K}_{d}^{\mathrm {Int}}(q^{2s})_{\mathcal {Q}})\) is an even function in \(s\). In particular, its odd order derivatives at \(s=0\) vanish.
By Corollary 11.10, this implies \(\deg [\mathcal {Z}^{r}_{\mathcal {E}}(a)]=0\) for \(r\) odd. However, we know from Lemma 6.7 that \(\operatorname{Sht}^{r}_{U(n)}=\varnothing \) when \(r\) is odd, which implies \(\mathcal {Z}^{r}_{\mathcal {E}}(a)=\varnothing \).
Part III. The comparison
12 Matching of sheaves
12.1 Recap
Let
where the notation is as in Theorem 2.8, being a renormalization of the \(a^{\mathrm {th}}\) Fourier coefficient of \(E_{a}(m(\mathcal {E}), s, \Phi )\).
We emphasize that, in keeping with §1.3, \(X\) is proper and \(\nu \colon X' \rightarrow X\) is a finite étale double cover (possibly trivial).
Theorem 12.1
Keep the notations above. Let \((\mathcal {E},a)\in \mathcal {A}_{d}(k)\). Then we have
In the previous parts, we have found sheaves on \(\mathcal {A}_{d}\) which correspond to the two sides of (12.2), in the sense of the functionsheaf dictionary. Let us summarize the situation.
On the analytic side, we proved a formula expressing the nonsingular Fourier coefficient of the Siegel–Eisenstein series in terms of the Frobenius trace of a graded virtual perverse sheaf \(\mathcal {K}_{d}^{\operatorname{Eis}}(T)\) on \(\operatorname{Herm}_{2d}(X'/X)\).
Theorem 12.2
Combination of Theorems 2.8 and 5.3
Let \((\mathcal {E},a)\in \mathcal {A}_{d}(k)\). Then we have
On the geometric side, in Corollary 11.10, we found a formula expressing the degree of the special 0cycle in terms of \(r^{\mathrm {th}}\) derivative of the Frobenius trace of another graded perverse sheaf \(\mathcal {K}^{\operatorname{Int}}_{d}(T)\) on \(\operatorname{Herm}_{2d}(X'/X)\), repeated below:
12.2 Proof of the main theorem
Comparing (12.3) and (12.4), we see that in order to prove Theorem 12.1, it remains to match the graded sheaves \(\mathcal {K}_{d}^{\mathrm {Int}}(T)\) and \(\mathcal {K}_{d}^{\operatorname{Eis}}(T)\) on \(\operatorname{Herm}_{2d}(X'/X)\).
Proposition 12.3
We have \(\mathcal {K}^{\operatorname{Int}}_{d}(T) \cong \mathcal {K}_{d}^{\operatorname{Eis}}(T)\) as graded perverse sheaves on \(\operatorname{Herm}_{2d}(X'/X)\).
Proof
Both sides can be written as \(\operatorname{Spr}^{\operatorname{Herm}}_{2d}[\rho ]\) for some graded virtual representation \(\rho \) of \(W_{d}\). By definition (Definition 11.6), the sheaf \(\mathcal {K}^{\operatorname{Int}}_{d}(T)\) corresponds to
We calculate the (a priori virtual) representation of \(W_{d}\) which corresponds under Springer theory to the \(\mathcal {K}_{d}^{\operatorname{Eis}}(T)\) from Definition 5.2. The operation \(R\overleftarrow {f}_{i,!}R\overrightarrow {f}^{*}_{i}\) corresponds under Springer theory to \(\operatorname{Ind}_{W_{di} \times S_{i}}^{W_{d}}\) and \(\mathfrak {P}_{di}(T)\) corresponds to \(\sum _{j=0}^{di} (1)^{j} \operatorname{Ind}_{W_{j} \times W_{dij}}^{W_{di}} ( \operatorname {sgn}_{j} \boxtimes \mathbf {1}) T^{j}\) (cf. Definition 4.7 for \(\operatorname{HSpr}_{d}\)). Hence \(\mathcal {K}_{d}^{\operatorname{Eis}}(T)\) corresponds to
After reindexing, we have
The desired statement then follows from the Lemma below (whose notation has been reindexed) by comparing each coefficient. □
Lemma 12.4
We have the identity of virtual representations of \(W_{d}\):
Proof
We will prove this by comparing traces of an arbitrary element \(g \in W_{d}\). For \(g \in W_{d}\),
Here, when \(w^{1}gw\in S_{dj} \times W_{j}\), we write \(w^{1}gw=(g',g'')\) for \(g'\in S_{dj}\) and \(g''\in W_{j}\).
Identify \(W_{d}\) with the group of permutations of \(\{\pm 1, \ldots , \pm d\}\) that commute with the involution \(\sigma \) exchanging \(j \leftrightarrow j\) for all \(1 \leq j \leq d\). The subgroup \(S_{dj}\times W_{j}\) is the stabilizer of \(\{1,2,\ldots , dj\}\). Therefore the coset space \(W_{d}/(S_{dj} \times W_{j})\) is in natural bijection with subsets \(J\subset \{\pm 1, \ldots , \pm d\}\) such that \(J=dj\) and \(J\cap (J)=\varnothing \). Let \(\mathfrak {J}_{g}\) be the set of \(J\subset \{\pm 1, \ldots , \pm d\}\) such that \(J=dj\), \(J\cap (J)=\varnothing \) and \(gJ=J\). Let \(g''_{J}\) be the permutation of \(g\) on \(\{\pm 1, \ldots , \pm d\}\backslash (J\cup (J))\). Combining this with (12.5), we obtain
For any \(g \in W_{d}\), the cycle decomposition of \(g\) can be grouped into a decomposition \(g = g_{1} \ldots g_{r}\) (unique up to reordering) where \(g_{i}\) is one of the two forms:

(positive bicycle) \(g_{i}\) is a product of two disjoint cycles \(c_{i}\sigma (c_{i})\) (in particular, no two elements appearing in \(c_{i}\) are negatives of each other).

(negative cycle) \(g_{i}\) is a single cycle invariant under the involution \(\sigma \).
Let \(C^{+}_{g}\) be the set of cycles of \(g\) that are part of a positive bicycle (i.e., \(C_{g}^{+}\) contains both \(c_{i}\) and \(\sigma (c_{i})\) for each positive bicycle \(g_{i}\)). For any \(x\in W_{d}\) we denote by \(\underline{x}\subset \{\pm 1,\ldots , \pm d\}\) the set of elements that are not fixed by \(x\). For a cycle \(c\) we let \(c\) be its length. From this description we see that \(J\in \mathfrak {J}_{g}\) if and only if \(J\) is a union of \(\underline{c}\) for a subset of cycles \(c\in C^{+}_{g}\). In other words, consider the set \(\mathfrak {I}_{g}\) of subsets \(I\subset C^{+}_{g}\) such that \(\underline{I}\) is disjoint from \(\sigma (\underline{I})\). Then we have a bijection \(\mathfrak {I}_{g}\stackrel {\sim }{\to }\mathfrak {J}_{g}\) sending \(I\in \mathfrak {I}_{g}\) to \(J:=\cup _{c\in I}\underline{c}\).
For \(I\in \mathfrak {I}_{g}\), let \(g'_{I}\) be the product of \(g_{i}\) such that \(g_{i}\) contains a cycle in common with \(I\); let \(g''_{I}\) be the product of the remaining \(g_{i}\)’s. The above discussion allows us to rewrite
This sum factorizes as a product over the \(g_{i}\) with individual factors as follows:

For a positive bicycle \(g_{i} = c_{i} \sigma (c_{i})\), the local factor is the sum of three contributions, corresponding to whether \(c_{i}\in I\), \(\sigma (c_{i})\in I\) or neither \(c_{i}\) nor \(\sigma (c_{i})\) is in \(I\). The first two cases each contribute 1. The last case leads to a contribution of \((1)^{c_{i}} \operatorname {sgn}(c_{i})=1\). The total contribution of the factor corresponding to a positive bicycle \(g_{i}\) is therefore \(1+1+(1)=1\).

For each negative cycle \(g_{i}\), since it always appears in \(g''_{I}\), its contribution is \((1)^{g_{i}/2}\overline {\operatorname {sgn}}(g_{i})\). Let \(\overline{g}_{i}\) be the image of \(g_{i}\) in \(S_{d}\), which is a cycle of length \(g_{i}/2\). Then \((1)^{g_{i}/2}\overline {\operatorname {sgn}}(g_{i})=(1)^{\overline{g}_{i}}\operatorname {sgn}(\overline{g}_{i})=1\). Therefore the contribution of the factor corresponding to a negative cycle \(g_{i}\) is −1.
Summarizing, we have found
On the other hand, we have
Indeed, if \(g_{i}=c_{i}\sigma (c_{i})\) is positive, then we have \(\chi _{d}(c_{i})=\chi _{d}(\sigma (c_{i}))=1\) because both \(c_{i}\) and \(\sigma (c_{i})\) can be conjugated into \(S_{d}\). If \(g_{i}\) is negative, then up to conjugacy we may assume \(g_{i}\) is the cyclic permutation \((1,2,\ldots , m, 1,\ldots , m)\) for some \(1\le m\le d\). Then \(g_{i}=(1,1)(1,2,\ldots , m) (1,2,\ldots , m)\), from which we see \(\chi _{d}(g_{i})=1\).
We conclude that the right side of (12.6) is \(\prod _{}\chi _{d}(g_{i})=\chi _{d}(g)\). This completes the proof. □
12.3 The split case \(X'=X\coprod X\)
We make our result more explicit in the split case \(X'=X\coprod X\).
On the analytic side in §2.1, the group \(H_{n}=\operatorname{GL}_{2n,F}\) and \(P_{n}\) is the standard parabolic corresponding to the partition \((n,n)\), with Levi \(M_{n}\simeq \operatorname{GL}_{n,F}\times \operatorname{GL}_{n,F}\). We then have the degenerate principal series
Let \(\mathcal {E}=(\mathcal {E}_{1},\mathcal {E}_{2})\in \operatorname{Bun}_{M_{n}}(k)\simeq \operatorname{Bun}_{\operatorname{GL}_{n}}(k) \times \operatorname{Bun}_{\operatorname{GL}_{n}}(k)\), and let \(a:\mathcal {E}_{1}\to \mathcal {E}_{2}^{\vee}\) be an injective map of \(\mathcal {O}_{X}\)modules. Then by §2.6 the Siegel–Eisenstein series has a welldefined \(a^{\mathrm {th}}\) Fourier coefficient \(E_{a}(m(\mathcal {E}),s,\Phi )\) at \((\mathcal {E}_{1},\mathcal {E}_{2})\). By Theorem 2.8 and 5.1 we have
where \(\mathscr{L} _{n}(s)=\prod _{i=1}^{n}\zeta _{F}(i+2s)\) and, for a torsion \(\mathcal {O}_{X}\)module \(\mathcal {Q}\), the density polynomial is given by
Here see (2.2) for \(\mathfrak {m}_{v}(t_{v};T)\). The normalized Fourier coefficient (12.1) is
Next we come to the geometric side. We have a natural partition
The moduli of hermitian shtukas \(\operatorname{Sht}_{U(n)}^{r}\) defined in §6 is then partitioned into
and there is a natural isomorphism
Here we recall that \(\operatorname{Sht}_{\operatorname{GL}_{n}}^{\mu}\) is the moduli of shtukas for \(\operatorname{GL}_{n}\), cf. [28], whose \(S\)points are given by the groupoid of the following data:

(1)
\(x_{i} \in X(S)\) for \(i = 1, \ldots , r\).

(2)
\(\mathcal {F}_{0}, \ldots , \mathcal {F}_{n}\in \operatorname{Bun}_{\operatorname{GL}_{n}} ( S)\).

(3)
An elementary modification \(f_{i} \colon \mathcal {F}_{i1} \dashrightarrow \mathcal {F}_{i}\) at the graph of \(x_{i}\), which is of upper of length 1 if \(\mu _{i}=+1\) and of lower of length 1 if \(\mu _{i}=1\).

(4)
An isomorphism \(\varphi \colon \mathcal {F}_{r} \cong {}^{\tau }\mathcal {F}_{0}\).
In particular, \(\operatorname{Sht}_{\operatorname{GL}_{n}}^{\mu}\) is empty unless \(\sum _{i=1}^{r} \mu _{i}=0\).
For the special cycle \(\mathcal {Z}_{\mathcal {E}}^{r} \) (cf. Definition 7.1) associated to \(\mathcal {E}=(\mathcal {E}_{1},\mathcal {E}_{2})\) above, we have a partition
where an object in \(\mathcal {Z}^{\mu}_{\mathcal {E}}(S)\) is an object as above in \(\operatorname{Sht}_{\operatorname{GL}_{n}}^{\mu}(S)\) together with maps
such that the diagram commutes
Let \(a:\mathcal {E}_{1}\to \mathcal {E}_{2}^{\vee}\) be a map of \(\mathcal {O}_{X}\)modules. Then \(\mathcal {Z}^{\mu}_{\mathcal {E}}(a)\) is the openclosed subscheme of \(\mathcal {Z}^{\mu}_{\mathcal {E}}\) such that the common composition (12.7) is equal to \(a \boxtimes \operatorname{Id}_{\mathcal {O}_{S}} \).
For an injective \(a:\mathcal {E}_{1}\to \mathcal {E}_{2}^{\vee}\), our §7 shows that \(\mathcal {Z}_{\mathcal {E}}^{\mu}(a)\) is proper over \(\operatorname{Spec\,}k\) and defines a class \([\mathcal {Z}_{\mathcal {E}}^{\mu}(a)]\in \operatorname{Ch}_{0}( \mathcal {Z}_{\mathcal {E}}^{\mu}(a))\) for each \(\mu \in \{\pm 1\}^{r}\). Then our main Theorem asserts
where \(d=(\chi (X,\mathcal {E}_{1})+\chi (X,\mathcal {E}_{2}))\). We remark that \(\deg [\mathcal {Z}_{\mathcal {E}}^{\mu}(a)]\) is not independent of \(\mu \in \{\pm 1\}^{r}\), even if we restrict our attention to those \(\mu \) with \(\sum _{i=1}^{r} \mu _{i} = 0\).
Notes
A map of vector bundles of the form \(a \colon \mathcal {E}\rightarrow \sigma ^{*} \mathcal {E}^{\vee}\) is Hermitian if \(\sigma ^{*} a^{\vee }= a\).
By this we mean a torsion \(\mathcal {O}_{F'}\)module with an \(\mathcal {O}_{F'}/\mathcal {O}_{F}\)Hermitian form.
See also [28, paragraph after Theorem 5.4] for a sketch of the separatedness in a similar situation, which readily adapts here.
We will later in §8.2 introduce a space \(\mathcal {A}^{\mathrm{all}}\) over \(\operatorname{Bun}_{\operatorname{GL}'_{m}}\) for which \(\mathcal {A}^{\mathrm{all}}_{\mathcal {E}}(k)\) is the \(k\)rational points of the fiber over \(\mathcal {E}\in \operatorname{Bun}_{\operatorname{GL}'_{m}}(k)\), justifying the notation.
In this case, \(\mathcal {Z}[D_{\bullet }]\) can be identified with the open substack \(\mathring{\mathcal {Z}}^{r}_{\mathcal {L}(D_{0})}(a')\subset \mathcal {Z}^{r}_{\mathcal {L}(D_{0})}(a')\) (where \(a'\) is the map \(\mathcal {L}(D_{0}) \rightarrow \sigma ^{*} \mathcal {L}(D_{0})^{\vee}\) induced from \(a\)) defined by requiring all the maps \(t'_{i}: \mathcal {L}(D_{0})\to \mathcal {F}_{i}\) be saturated.
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Acknowledgements
We thank Michael Harris, Chao Li, Alan Peng, Zhiyu Zhang, and the anonymous referee for comments on a draft. We thank the referee for helpful suggestions and corrections.
Funding
TF was supported by NSF Postdoctoral Fellowship DMS1902927, NSF Grant DMS2302520, and the Friends of the Institute for Advanced Study. ZY was partially supported by the Packard Fellowship, and the Simons Investigator grant. WZ is partially supported by the NSF grant DMS #1901642 and the Simons Investigator grant.
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Feng, T., Yun, Z. & Zhang, W. Higher Siegel–Weil formula for unitary groups: the nonsingular terms. Invent. math. 235, 569–668 (2024). https://doi.org/10.1007/s0022202301228y
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DOI: https://doi.org/10.1007/s0022202301228y