The Topology of ${{{\mathcal A}}_{g}}$ and Its Compactifications

Abstract

We survey old and new results about the cohomology of the moduli space ${{{\mathcal A}}_{g}}$ of principally polarized abelian varieties of genus g and its compactifications. The main emphasis lies on the computation of the cohomology for small genus and on stabilization results. We review both geometric and representation theoretic approaches to the problem. The appendix provides a detailed discussion of computational methods based on trace formulae and automorphic representations, in particular Arthur’s endoscopic classification of automorphic representations for symplectic groups.

Appendix: Computation of Intersection Cohomology Using the Langlands Program

Olivier Taïbi

CNRS, Unité de mathématiques pures et appliquées, ENS de Lyon, France; olivier.taibi@ens-lyon.fr In this appendix we explain a method for the explicit computation of the Euler characteristics for both the cohomology of local systems ${\mathbb {V}}_{\lambda }$ on ${{{\mathcal A}}_{g}}$ and their intermediate extensions to ${{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}$. Furthermore, we explain how to compute individual intersection cohomology groups in the latter case. The main tools here are trace formulas and results on automorphic representations, notably Arthur’s endoscopic classification of automorphic representations of symplectic groups [9].

We start by explaining in Proposition 3 the direct computation of $e({{{\mathcal A}}_{4}})=9$. This Euler characteristic, as well as $e({{{\mathcal A}}_{g}}, {\mathbb {V}}_{\lambda })$ for g ≤ 7 and any λ, can be obtained as a byproduct of computations explained in the first part of [98], which focused on L ²-cohomology. In fact by Proposition 4 these are given by the conceptually simple formula (32). The difficulty in evaluating this formula resides in computing certain coefficients, called masses, for which we gave an algorithm in [98]. The number $e({{{\mathcal A}}_{4}})$ was missing in [70] to complete the proof of Theorem 14.

Next we recall from [98] that the automorphic representations for $\operatorname {Sp}_{2g}$ contributing to $IH^{\bullet }({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}},{\mathbb {V}}_{\lambda })$ can be reconstructed from certain sets of automorphic representations of general linear groups, which we shall introduce in Definition 4. Thanks to Arthur’s endoscopic classification [9] specialized to level one and the identification by Arancibia et al. [4] of certain Arthur-Langlands packets with the concrete packets previously constructed by Adams and Johnson [1] in the case of the symplectic groups, combined with analogous computations for certain special orthogonal groups, we have computed the cardinalities of these “building blocks”. Again, this is explicit for g ≤ 7 and arbitrary λ. For g ≤ 11 and “small” λ, the classification by Chenevier and Lannes in [22] of level one algebraic automorphic representations of general linear groups over ${\mathbb {Q}}$ having “motivic weight” ≤ 22 (see Theorem 31 below) gives another method to compute these sets. Using either method, we deduce $IH^6({{{{\mathcal A}}_{3}^{\operatorname {Sat}}}}, {\mathbb {V}}_{1,1,0}) = 0$ in Corollary 3, which was a missing ingredient to complete the computation in [55] of $IH^{\bullet }({{{{\mathcal A}}_{4}^{\operatorname {Sat}}}}, {\mathbb {Q}})$ (case g = 4 in Theorem 17). In fact using the computation by Vogan and Zuckerman [102] of the $(\mathfrak {g}, K)$-cohomology of Adams-Johnson representations, including the trivial representation of $\operatorname {Sp}_{2g}({\mathbb {R}})$, we can prove that the intersection cohomology of ${{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}$ is isomorphic to the tautological ring R _g for all g ≤ 5 (see Theorem 17), again by either method.

One could deduce from [102] an algorithm to compute intersection cohomology also in the cases where there are non-trivial representations of $\operatorname {Sp}_{2g}({\mathbb {R}})$ contributing to $IH^{\bullet }({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}, {\mathbb {V}}_{\lambda })$, e.g. for all g ≥ 6 and $V_{\lambda } = {\mathbb {Q}}$. Instead of pursuing this, in Sect. 8 we make explicit the beautiful description by Langlands and Arthur of L ²-cohomology in terms of the Archimedean Arthur-Langlands parameters involved, i.e. Adams-Johnson parameters. In fact the correct way to state this description would be to use the endoscopic classification of automorphic representations for $\operatorname {GSp}_{2g}$. Although this classification is not yet known, we can give an unconditional recipe in the case of level one automorphic representations. We conclude the appendix with the example of the computation of $IH^{\bullet }({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}, {\mathbb {Q}})$ for g = 6, 7, and relatively simple formulas to compute the polynomials

$$\displaystyle \begin{aligned} \sum_k T^k \dim IH^k({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}, {\mathbb{V}}_{\lambda}) \end{aligned}$$

for all values of (g, λ) such that the corresponding set of substitutes for Arthur-Langlands parameters of conductor on is known (currently g ≤ 7 and arbitrary λ and all pairs (g, λ) with g + λ ₁ ≤ 11).

Let us recall that for n = 3, 4, 5mod8 there is a (unique by [52, Proposition 2.1]) reductive group G over ${\mathbb {Z}}$ such that $G_{{\mathbb {R}}} \simeq \operatorname {SO}(n-2,2)$. Such G is a special orthogonal group of a lattice, for example E ₈ ⊕ H ^⊕2 where H is a hyperbolic lattice. If K is a maximal compact subgroup of $G({\mathbb {R}})$, we can also consider the hermitian locally symmetric space

$$\displaystyle \begin{aligned} G({\mathbb{Z}}) \backslash G({\mathbb{R}}) / K^0 = G({\mathbb{Z}})^0 \backslash G({\mathbb{R}})^0 / K^0 \end{aligned}$$

where $G({\mathbb {R}})^0$ (resp. K ⁰) is the identity component of $G({\mathbb {R}})$ (resp. K) and $G({\mathbb {Z}})^0 = G({\mathbb {Z}}) \cap G({\mathbb {R}})^0$. Then everything explained in this appendix also applies to this situation, except for the simplification in Proposition 4 which can only be applied to the simply connected cover of G. Using [23, §4.3] one can see that this amounts to considering $(\mathfrak {so}_n, SO(2) \times SO(n-2))$-cohomology instead of $(\mathfrak {so}_n, S(O(2) \times O(n-2)))$-cohomology as in [98], and this simply multiplies Euler characteristics by 2. In fact the analogue of Sect. 8 is much simpler for special orthogonal groups, since they do occur in Shimura data (of abelian type).

To be complete we mention that Arthur’s endoscopic classification in [9] is conditional on several announced results which, to the best of our knowledge, are not yet available (see [98, §1.3]).

1.1 Evaluation of a Trace Formula

Our first goal is to prove the following result.

Proposition 3

We have $e({{{\mathcal A}}_{4}}) = 9$.

This number is a byproduct of the explicit computation in [98] of

$$\displaystyle \begin{aligned} e_{(2)}({{{\mathcal A}}_{g}}, {\mathbb{V}}_{\lambda}) := \sum_i (-1)^i \dim_{{\mathbb{R}}} H^i_{(2)}({{{\mathcal A}}_{g}}, {\mathbb{V}}_{\lambda}) \end{aligned} $$

(31)

for arbitrary irreducible algebraic representations V _λ of $\operatorname {Sp}_{2g}$. Each representation V _λ is defined over ${\mathbb {Q}}$, and L ²-cohomology is defined with respect to an admissible inner product on ${\mathbb {R}} \otimes _{{\mathbb {Q}}} V_{\lambda }$. In particular $H^{\bullet }_{(2)}({{{\mathcal A}}_{g}}, {\mathbb {V}}_{\lambda })$ is a graded real vector space (for arbitrary arithmetic symmetric spaces the representation V _λ may not be defined over ${\mathbb {Q}}$ and so in general L ²-cohomology is only naturally defined over ${{\mathbb {C}}}$). Recall that by Zucker’s conjecture (31) is also equal to

$$\displaystyle \begin{aligned} \sum_i (-1)^i \dim IH^i({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}, {\mathbb{V}}_{\lambda}). \end{aligned}$$

To evaluate the Euler characteristic (31) we use Arthur’s L ²-Lefschetz trace formula [6]. This is a special case since this is the alternating trace of the unit in the unramified Hecke algebra on these cohomology groups. Arthur obtained this formula by specializing his more general invariant trace formula. In general Arthur’s invariant trace formula yields transcendental values, but for particular functions at the real place (stable sums of pseudo-coefficients of discrete series representations) Arthur obtained a simplified expression all of whose terms can be seen to be rational. Goresky et al. [46, 49] gave a different proof of this formula, by a topological method. In fact they obtained more generally a trace formula for weighted cohomology [48], the case of a lower middle or upper middle weight profile on a hermitian locally symmetric space recovering intersection cohomology of the Baily-Borel compactification. We shall also use split reductive groups over ${\mathbb {Q}}$ other than $\operatorname {Sp}_{2g}$ below, which will be of equal rank at the real place but do not give rise to hermitian symmetric spaces. For this reason it is reassuring that Nair [84] proved that in general weighted cohomology groups coincide with Franke’s weighted L ² cohomology groups [36] defined in terms of automorphic forms. The case of usual L ² cohomology corresponds to the lower and upper middle weight profiles in [48]. In particular Nair’s result implies that [49] is a generalization of [6].

In our situation the trace formula can be written

$$\displaystyle \begin{aligned} e_{(2)}({{{\mathcal A}}_{g}}, {\mathbb{V}}_{\lambda}) = \sum_M T(\operatorname{Sp}_{2g}, M, \lambda) \end{aligned}$$

where the sum is over conjugacy classes of Levi subgroups M of $\operatorname {Sp}_{2g}$ which are ${\mathbb {R}}$-cuspidal, i.e. isomorphic to $\operatorname {GL}_1^a \times \operatorname {GL}_2^c \times \operatorname {Sp}_{2d}$ with a + 2c + d = g. The right hand side is traditionally called the geometric side, although this terminology is confusing in the present context. The most interesting term in the sum is the elliptic part $T_{\operatorname {ell}}(\operatorname {Sp}_{2g}, \lambda ) := T(\operatorname {Sp}_{2g}, \operatorname {Sp}_{2g}, \lambda )$ which is defined as

$$\displaystyle \begin{aligned} T_{\operatorname{ell}}(\operatorname{Sp}_{2g}, \lambda) = \sum_{c \in C(\operatorname{Sp}_{2g})} m_c \mathrm{tr} \left( c \,\middle|\, V_{\lambda} \right). \end{aligned} $$

(32)

Here $C(\operatorname {Sp}_{2g})$ is the finite set of torsion ${\mathbb {R}}$-elliptic elements in $\operatorname {Sp}_{2g}({\mathbb {Q}})$ up to conjugation in $\operatorname {Sp}_{2g}(\overline {{\mathbb {Q}}})$. These can be simply described by certain products of degree 2n of cyclotomic polynomials. The rational numbers m _c are “masses” (in the sense of the mass formula, so it would be more correct to call them “weights”) computed adelically, essentially as products of local orbital integrals (at all prime numbers) and global terms (involving Tamagawa numbers and values of certain Artin L-functions at negative integers). We refer the reader to [98] for details. Let us simply mention that for $c = \pm 1 \in C(\operatorname {Sp}_{2g})$, the local orbital integrals are all equal to 1, and m _c is the familiar product ζ(−1)ζ(−3)…ζ(1 − 2g). The appearance of other terms in $T_{\operatorname {ell}}(\operatorname {Sp}_{2g}, \lambda )$ corresponding to non-central elements in $C(\operatorname {Sp}_{2g})$ is explained by the fact that the action of $\operatorname {Sp}_{2g}({\mathbb {Z}}) / \{ \pm 1 \}$ on ${\mathbb {H}}_g$ is not free.

To evaluate $T_{\operatorname {ell}}(\operatorname {Sp}_{2g}, \lambda )$ explicitly, the main difficulty consists in computing the local orbital integrals. An algorithm was given in [98, §3.2]. In practice these are computable (by a computer) at least for g ≤ 7. For g = 2 they were essentially computed by Tsushima in [101]. For g = 3 they could also be computed by a (dedicated) human being. See [98, Table 9] for g = 3, and [100] for higher g. The following table contains the number of masses in each rank, taking into account that m _−c = m _c.

In general the elliptic part of the geometric side of the L ²-Lefschetz trace formula does not seem to have any spectral or cohomological meaning, but for unit Hecke operators and simply connected groups, such as $\operatorname {Sp}_{2g}$, it turns out that it does.

Proposition 4

Let G be a simply connected reductive group over ${\mathbb {Q}}$ . Assume that $G_{{\mathbb {R}}}$ has equal rank, i.e. $G_{{\mathbb {R}}}$ admits a maximal torus (defined over ${\mathbb {R}}$ ) which is anisotropic, and that $G_{{\mathbb {R}}}$ is not anisotropic, i.e. $G({\mathbb {R}})$ is not compact. Let K _f be a compact open subgroup of $G({{\mathbb {A}}}_f)$ and let $\varGamma = G({\mathbb {Q}}) \cap K_f$ . Let K _∞ be a maximal compact subgroup of $G({\mathbb {R}})$ (which is connected). Then for any irreducible algebraic representation V _λ of $G({{\mathbb {C}}})$ , in Arthur’s L ² -Lefschetz trace formula

$$\displaystyle \begin{aligned} e_{(2)}( \varGamma \backslash G({\mathbb{R}}) / K_{\infty}, {\mathbb{V}}_{\lambda}) = \sum_M T(G, K_f, M, \lambda) \end{aligned}$$

where the sum is over $G({\mathbb {Q}})$ -conjugacy classes of cuspidal Levi subgroups of G, the elliptic term

$$\displaystyle \begin{aligned} T_{\operatorname{ell}}(G, K_f, \lambda) := T(G, K_f, G, \lambda) \end{aligned}$$

is equal to

$$\displaystyle \begin{aligned} e_c( \varGamma \backslash G({\mathbb{R}}) / K_{\infty}, {\mathbb{V}}_{\lambda}) = \sum_i (-1)^i \dim H^i_c( \varGamma \backslash G({\mathbb{R}}) / K_{\infty}, {\mathbb{V}}_{\lambda}). \end{aligned}$$

Proof

Note that by strong approximation (using that $G({\mathbb {R}})$ is not compact), the natural inclusion $\varGamma \backslash G({\mathbb {R}}) / K_{\infty } \rightarrow G({\mathbb {Q}}) \backslash G({{\mathbb {A}}}_f) / K_{\infty } K_f$ is an isomorphism between orbifolds.

We claim that in the formula [49, §7.17] for $e_c( \varGamma \backslash G({\mathbb {R}}) / K_{\infty }, {\mathbb {V}}_{\lambda })$, every term corresponding to M ≠ G vanishes. Note that in [49] the right action of Hecke operators is considered: see [49, §7.19]. Thus to recover the trace of the usual left action of Hecke operators one has to exchange E (our V _λ) and E ^∗ in [49, §7.17]. Since we are only considering a unit Hecke operator, the orbital integrals at finite places denoted $O_{\gamma }(f_M^{\infty })$ in [49, Theorem 7.14.B] vanish unless $\gamma \in M({\mathbb {Q}})$ is power-bounded in $M({\mathbb {Q}}_p)$ for every prime p. Recall that γ is also required to be elliptic in $M({\mathbb {R}})$, so that by the adelic product formula this condition on γ at all finite places implies that γ is also power-bounded in $M({\mathbb {R}})$. Thus it is enough to show that for any cuspidal Levi subgroup M of $G_{{\mathbb {R}}}$ distinct from $G_{{\mathbb {R}}}$, and any power-bounded $\gamma \in M({\mathbb {R}})$, we have Φ _M(γ, V _λ) = 0 (where Φ _M is defined on p. 498 of [49]). Choose an elliptic maximal torus T in $M_{{\mathbb {R}}}$ such that $\gamma \in T({\mathbb {R}})$. Since the character of V _λ is already a continuous function on $M({\mathbb {R}})$, it is enough to show that there is a root α of T in G not in M such that α(γ) = 1. All roots of T in M are imaginary, so it is enough to show that there exists a real root α of T in G such that α(γ) = 1. Since γ is power-bounded in $M({\mathbb {R}})$, for any real root of T in G we have α(γ) ∈{±1}. Thus it is enough to show that there is a real root α such that α(γ) > 0. This follows (for $M \neq G_{{\mathbb {R}}}$) from the argument at the bottom of p. 499 in [49] (this is were the assumption that G is simply connected is used). □

The proposition is a generalization of [60] to the orbifold case (Γ not neat) with non-trivial coefficients, but note that Harder’s formula is used in [49].

In particular for any g ≥ 1 and dominant weight λ we simply have $T_{\operatorname {ell}}(\operatorname {Sp}_{2g}, \lambda ) = e_c({{{\mathcal A}}_{g}}, {\mathbb {V}}_{\lambda }) = e({{{\mathcal A}}_{g}}, {\mathbb {V}}_{\lambda })$ by Poincaré duality and self-duality of V _λ. As a special case we have the simple formula $e({{{\mathcal A}}_{g}}) = \sum _{c \in C(\operatorname {Sp}_{2g})} m_c$ and Proposition 3 follows (with the help of a computer). See [100] for tables of masses m _c, where the source code computing these masses (using [96]) can also be found.

In the following table we record the value of $e({{{\mathcal A}}_{g}})$ for small g.

The Euler characteristic of L ²-cohomology can also be evaluated explicitly. Theorem 3.3.4 in [98] expresses $e_{(2)}({{{\mathcal A}}_{g}}, {\mathbb {V}}_{\lambda })$ in a (relatively) simple manner from $T_{\operatorname {ell}}(\operatorname {Sp}_{2g'}, \lambda ')$ for g′≤ g and dominant weights λ′. Hence $e_{(2)}({{{\mathcal A}}_{g}}, {\mathbb {V}}_{\lambda })$ can be derived from tables of masses, for any λ. Of course this does not directly yield dimensions of individual cohomology groups. Fortunately, Arthur’s endoscopic classification of automorphic representations for $\operatorname {Sp}_{2g}$ allows us to write this Euler characteristic as a sum of two contributions: “old” contributions coming from automorphic representations for groups of lower dimension, and new contributions which only contribute to middle degree. Thus it is natural to try to compute old contributions by induction. As we shall see below, they can be described combinatorially from certain self-dual level one automorphic cuspidal representations of general linear groups over ${\mathbb {Q}}$.

1.2 Arthur’s Endoscopic Classification in Level One

We will explain the decomposition of $H^{\bullet }_{(2)}({{{\mathcal A}}_{g}}, {\mathbb {V}}_{\lambda })$ that can be deduced from Arthur’s endoscopic classification. These real graded vector spaces are naturally endowed with a real Hodge structure, a Lefschetz operator and a compatible action of a commutative Hecke algebra. This last action will make the decomposition canonical. For this reason we will recall what is known about this Hecke action. We denote ${{\mathbb {A}}}$ for the adele ring of ${\mathbb {Q}}$.

Definition 3

Let G be a reductive group over ${\mathbb {Z}}$ (in the sense of [92, Exposé XIX, Définition 2.7]).

• (i) If p is a prime, let ${\mathcal H}_p^{\operatorname {unr}}(G)$ be the commutative convolution algebra (“Hecke algebra”) of functions $G({\mathbb {Z}}_p) \backslash G({\mathbb {Q}}_p) / G({\mathbb {Z}}_p) \rightarrow {\mathbb {Q}}$ having finite support.

• (ii) Let ${\mathcal H}^{\operatorname {unr}}_f(G)$ be the commutative algebra of functions

  $$\displaystyle \begin{aligned} G(\widehat{{\mathbb{Z}}}) \backslash G({{\mathbb{A}}}) / G(\widehat{{\mathbb{Z}}}) \rightarrow {\mathbb{Q}} \end{aligned}$$

  having finite support, so that ${\mathcal H}^{\operatorname {unr}}_f(G)$ is the restricted tensor product $\bigotimes _p^{\prime } {\mathcal H}_p^{\operatorname {unr}}(G)$.

Recall the Langlands dual group $\widehat {G}$ of a reductive group G, which we consider as a split reductive group over ${\mathbb {Q}}$. We will mainly consider the following cases.

Recall from [53, 91] the Satake isomorphism: for F an algebraically closed field of characteristic zero and G a reductive group over ${\mathbb {Z}}$, if we choose a square root of p in F then ${\mathbb {Q}}$-algebra morphisms ${\mathcal H}^{\operatorname {unr}}_p(G) \rightarrow F$ correspond naturally and bijectively to semisimple conjugacy classes in $\widehat {G}(F)$.

If π is an automorphic cuspidal representation of $\operatorname {GL}_N({{\mathbb {A}}})$, then it admits a decomposition as a restricted tensor product $\pi = \pi _{\infty } \otimes \bigotimes ^{\prime }_p \pi _p$, where the last restricted tensor product is over all prime numbers p. Assume moreover that π has level one, i.e. that $\pi _p^{\operatorname {GL}_N({\mathbb {Z}}_p)} \neq 0$ for any prime p. Then π has the following invariants:

• (i) the infinitesimal character $\operatorname {ic}(\pi _{\infty })$, which is a semisimple conjugacy class in $\mathfrak {gl}_N({{\mathbb {C}}}) = M_N({{\mathbb {C}}})$ obtained using the Harish-Chandra isomorphism [62],

• (ii) for each prime number p, the Satake parameter c(π _p) of the unramified representation π _p of $\operatorname {GL}_N({\mathbb {Q}}_p)$, which is a semisimple conjugacy class in $\operatorname {GL}_N({{\mathbb {C}}})$ corresponding to the character by which ${\mathcal H}_p^{\operatorname {unr}}(\operatorname {GL}_N)$ acts on the complex line $\pi _p^{\operatorname {GL}_N({\mathbb {Z}}_p)}$.

We now introduce three families of automorphic cuspidal representations for general linear groups that will be exactly those contributing to intersection cohomology of ${{{\mathcal A}}_{g}}$’s.

Definition 4

• (i) For g ≥ 0 and integers w ₁ > ⋯ > w _g > 0, let O _o(w ₁, …, w _g) be the set of self-dual level one automorphic cuspidal representations π = π _∞⊗ π _f of $\operatorname {GL}_{2g+1}({{\mathbb {A}}})$ such that $\operatorname {ic}(\pi _{\infty })$ has eigenvalues w ₁ > ⋯ > w _g > 0 > −w _g > ⋯ > −w ₁. For π ∈ O _o(w ₁, …, w _g) we let $\widehat {G_{\pi }} = \operatorname {SO}_{2g+1}({{\mathbb {C}}})$.

• (ii) For g ≥ 1 and integers w ₁ > ⋯ > w _2g > 0, let O _e(w ₁, …, w _2g) be the set of self-dual level one automorphic cuspidal representations π = π _∞⊗ π _f of $\operatorname {GL}_{4g}({{\mathbb {A}}})$ such that $\operatorname {ic}(\pi _{\infty })$ has eigenvalues w ₁ > ⋯ > w _2g > −w _2g > ⋯ > −w ₁. For π ∈ O _e(w ₁, …, w _2g) we let $\widehat {G_{\pi }} = \operatorname {SO}_{4g}({{\mathbb {C}}})$.

• (iii) For g ≥ 1 and w ₁ > ⋯ > w _g > 0 with $w_i \in 1/2 + {\mathbb {Z}}$, let S(w ₁, …, w _g) be the set of self-dual level one automorphic cuspidal representations π = π _∞⊗ π _f of $\operatorname {GL}_{2g}({{\mathbb {A}}})$ such that $\operatorname {ic}(\pi _{\infty })$ has eigenvalues w ₁ > ⋯ > w _g > −w _g > ⋯ > −w ₁. For π ∈ S(w ₁, …, w _g) we let $\widehat {G_{\pi }} = \operatorname {Sp}_{2g}({{\mathbb {C}}})$.

These sets are all finite by [63, Theorem 1], and O _o (resp. O _e, S) is short for “odd orthogonal” (resp. “even orthogonal”, “symplectic”). A fact related to vanishing of cohomology with coefficients in ${\mathbb {V}}_{\lambda }$ for w(λ) odd is that O _o(w ₁, …, w _g) = ∅ if w ₁ + ⋯ + w _g ≠ g(g + 1)∕2mod2 and O _e(w ₁, …, w _2g) = ∅ if w ₁ + ⋯ + w _2g ≠ gmod2. See [98, Remark 4.1.6] or [23, Proposition 1.8].

Remark 4

For small g the sets in Definition 4 are completely described in terms of level one (elliptic) eigenforms, due to accidental isomorphisms between classical groups in small rank (see [98, §6] for details).

• (i) For k > 0 the set $S(k-\frac {1}{2})$ is naturally in bijection with the set of normalized eigenforms of weight 2k for $\operatorname {SL}_2({\mathbb {Z}})$, and $O_o(2k-1) \simeq S(k-\frac {1}{2})$.

• (ii) For integers w ₁ > w ₂ > 0 such that w ₁ + w ₂ is odd, $O_e(w_1, w_2) \simeq S(\frac {w_1+w_2}{2}) \times S(\frac {w_1-w_2}{2})$.

Let us recall some properties of the representations appearing in Definition 4.

Let ${\mathbb {Q}}^{\operatorname {real}}$ be the maximal totally real algebraic extension of ${\mathbb {Q}}$ in ${{\mathbb {C}}}$. Then ${\mathbb {Q}}^{\operatorname {real}}$ is an infinite Galois extension of ${\mathbb {Q}}$ which contains $\sqrt {p} > 0$ for any prime p.

Theorem 28

For π as in Definition 4 there exists a finite subextension E of ${\mathbb {Q}}^{\operatorname {real}} / {\mathbb {Q}}$ such that for any prime number p, the characteristic polynomial of $p^{w_1} c(\pi _p)$ has coefficients in E. Moreover c(π _p) is compact (i.e. power-bounded).

Proof

That there exists a finite extension E of ${\mathbb {Q}}$ satisfying this condition is a special case of [24, Théorème 3.13]. The fact that it can be taken totally real follows from unitarity and self-duality of π.

The last statement is a consequence of [95] and [25]. □

Let E(π) be the smallest such extension. Then π _f is defined over E(π), and this structure is unique up to ${{\mathbb {C}}}^{\times } / E(\pi )^{\times }$. There is an action of the Galois group $\operatorname {Gal}({\mathbb {Q}}^{\operatorname {real}} / {\mathbb {Q}})$ on O _o(w ₁, … ) (resp. O _e(w ₁, … ), S(w ₁, … )): if π = π _∞⊗ π _f, σ(π) := π _∞⊗ σ(π _f) belongs to the same set. Dually we have $c(\sigma (\pi )_p) = p^{-w_1} \sigma ( p^{w_1} c(\pi _p))$. For π ∈ O _o(w ₁, … ) or O _e(w ₁, … ) the power of p is not necessary since $w_1 \in {\mathbb {Z}}$.

In all three cases c(π _p) can be lifted to a semisimple conjugacy class in $\widehat {G_{\pi }}({{\mathbb {C}}})$, uniquely except in the second case, where it is unique only up to conjugation in $\operatorname {O}_{4g}({{\mathbb {C}}})$. Identifying semisimple conjugacy classes to Weyl group orbits in maximal tori, this is elementary. We abusively denote this conjugacy class by c(π _p).

We now consider the Archimedean place of ${\mathbb {Q}}$, which will be of particular importance for real Hodge structures. Recall that the Weil group of ${\mathbb {R}}$ is defined as the non-trivial extension of $\operatorname {Gal}({{\mathbb {C}}} / {\mathbb {R}})$ by ${{\mathbb {C}}}^{\times }$. If H is a complex reductive group and $\varphi : {{\mathbb {C}}}^{\times } \rightarrow H({{\mathbb {C}}})$ is a continuous semisimple morphism, there is a maximal torus T of H such that φ factors through $T({{\mathbb {C}}})$ and takes the form $z \mapsto z^{\tau _1} \bar {z}^{\tau _2}$ for uniquely determined $\tau _1, \tau _2 \in X_*(T) \otimes _{{\mathbb {Z}}} {{\mathbb {C}}}$ such that τ ₁ − τ ₂ ∈ X _∗(T). Here X _∗(T) is the group of cocharacters of T and $z^{\tau _1} \bar {z}^{\tau _2}$ is defined as $(z/|z|)^{\tau _1-\tau _2} |z|{ }^{\tau _1+\tau _2}$. We call the $H({{\mathbb {C}}})$-conjugacy class of τ ₁ in $\mathfrak {h} = \operatorname {Lie}(H)$ (complex analytic Lie algebra) the infinitesimal character of φ, denoted $\operatorname {ic}(\varphi )$. If $\varphi : W_{{\mathbb {R}}} \rightarrow H({{\mathbb {C}}})$ is continuous semisimple we let $\operatorname {ic}(\varphi ) = \operatorname {ic}(\varphi |{ }_{{{\mathbb {C}}}^{\times }})$.

In all three cases in Definition 4 there is a continuous semisimple morphism $\varphi _{\pi _{\infty }} : W_{{\mathbb {R}}} \rightarrow \widehat {G_{\pi }}({{\mathbb {C}}})$ such that for any $z \in {{\mathbb {C}}}^{\times }$, $\varphi _{\pi _{\infty }}(z)$ has eigenvalues

$$\displaystyle \begin{aligned} \begin{cases} (z/\bar{z})^{\pm w_1}, \dots, (z/\bar{z})^{\pm w_g}, 1 & \text{ if } \pi \in O_o(w_1, \dots, w_g), \\ (z/\bar{z})^{\pm w_1}, \dots, (z/\bar{z})^{\pm w_{2g}} & \text{ if } \pi \in O_e(w_1, \dots, w_{2g}), \\ (z/|z|)^{\pm 2 w_1}, \dots, (z/|z|)^{\pm 2 w_g} & \text{ if } \pi \in S(w_1, \dots, w_g), \end{cases} \end{aligned}$$

in the standard representation of $\widehat {G_{\pi }}$. The parameter $\varphi _{\pi _{\infty }}$ is characterized up to conjugacy by this property except in the second case where it is only characterized up to $\operatorname {O}_{4g}({{\mathbb {C}}})$-conjugacy. To rigidify the situation we choose a semisimple conjugacy class τ _π in the Lie algebra of $\widehat {G_{\pi }}$ whose image via the standard representation has eigenvalues ± w ₁, …, ±w _g, 0 (resp. ± w ₁, …, ±w _2g, resp. ± w ₁, …, ±w _g). Then the pair $(\widehat {G_{\pi }}, \tau _{\pi })$ is well-defined up to isomorphism unique up to conjugation by $\widehat {G_{\pi }}$, since in the even orthogonal case τ _π is not fixed by an outer automorphism of $\widehat {G_{\pi }}$. Up to conjugation by $\widehat {G_{\pi }}$ there is a unique $\varphi _{\pi _{\infty }} : W_{{\mathbb {R}}} \rightarrow \widehat {G_{\pi }}({{\mathbb {C}}})$ as above and such that $\operatorname {ic}(\varphi _{\pi _{\infty }}) = \tau _{\pi }$. In all cases the centralizer of $\varphi _{\pi _{\infty }}(W_{{\mathbb {R}}})$ in $\widehat {G_{\pi }}({{\mathbb {C}}})$ is finite.

Let us now indicate how the general definition of substitutes for Arthur-Langlands parameters in [9] specializes to the case at hand.

Definition 5

Let V _λ be an irreducible algebraic representation of $\operatorname {Sp}_{2g}$, given by the dominant weight λ = (λ ₁ ≥⋯ ≥ λ _g ≥ 0). Let ρ be half the sum of the positive roots for $\operatorname {Sp}_{2g}$, and τ = λ + ρ = (w ₁ > ⋯ > w _g > 0) where $w_i = \lambda _i + n+1-i \in {\mathbb {Z}}$, which we can see as the regular semisimple conjugacy class in $\mathfrak {so}_{2g+1}({{\mathbb {C}}})$. Let $\varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$ be the set of pairs (ψ ₀, {ψ ₁, …, ψ _r}) with r ≥ 0 and such that

• (i) ψ ₀ = (π ₀, d ₀) where $\pi _0 \in O_o(w_1^{(0)}, \dots , w_{g_0}^{(0)})$ and d ₀ ≥ 1 is an odd integer,

• (ii) The ψ _i’s, for i ∈{1, …, r}, are distinct pairs (π _i, d _i) where d _i ≥ 1 is an integer and $\pi _i \in O_e(w_1^{(i)}, \dots , w_{g_i}^{(i)})$ with g _i even (resp. $\pi _i \in S(w_1^{(i)}, \dots , w_{g_i}^{(i)})$) if d _i is odd (resp. even).

• (iii) $2g+1 = (2g_0+1) d_0 + \sum _{i=1}^r 2 g_i d_i$,

• (iv) The sets

  • a. $\{ \frac {d_0-1}{2}, \frac {d_0-3}{2}, \dots , 1 \}$,

  • b. $\{ w_1^{(0)} + \frac {d_0-1}{2} - j, \dots , w^{(0)}_{g_0} + \frac {d_0-1}{2} -j \}$ for j ∈{0, …, d ₀ − 1},

  • c. $\{ w_1^{(i)} + \frac {d_i-1}{2} - j, \dots , w_{g_i}^{(i)} + \frac {d_i-1}{2} -j \}$ for i ∈{1, …, r} and j ∈{0, …, d _i − 1}

  are disjoint and their union equals {w ₁, …, w _g}.

We will write more simply

This could be defined as an “isobaric sum”, but for the purpose of this appendix we can simply consider this expression as a formal unordered sum. If π ₀ is the trivial representation of $\operatorname {GL}_1({{\mathbb {A}}})$, we write [d ₀] for 1[d ₀], and when d = 1 we simply write π for π[1].

Example 1

• (i) For any g ≥ 1, $[2g+1] \in \varPsi _{\operatorname {disc}}^{\operatorname {unr}, \rho }(\operatorname {Sp}_{2g})$.

• (ii) $[9] \boxplus \varDelta _{11}[2] \in \varPsi _{\operatorname {disc}}^{\operatorname {unr}, \rho }(\operatorname {Sp}_{12})$ where $\varDelta _{11} \in S(\frac {11}{2})$ is the automorphic representation of $\operatorname {GL}_2({{\mathbb {A}}})$ corresponding to the Ramanujan Δ function.

• (iii) For g ≥ 1, τ = (w ₁ > ⋯ > w _g > 0), for any π ∈ O _o(w ₁, …, w _g) we have $\pi = \pi [1] \in \varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$.

Definition 6

• (i) For $\psi = \pi _0[d_0] \boxplus \dots \boxplus \pi _r[d_r] \in \varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$, let ${\mathcal L}_{\psi } = \prod _{i=0}^r \widehat {G_{\pi _i}}({{\mathbb {C}}})$. Let $\dot {\psi } : {\mathcal L}_{\psi } \times \operatorname {SL}_2({{\mathbb {C}}}) \rightarrow \operatorname {SO}_{2n+1}({{\mathbb {C}}})$ be a morphism such that composing with the standard representation gives $\bigoplus _{0 \leq i \leq r} \operatorname {Std}_{\widehat {G_{\pi _i}}} \otimes \nu _{d_i}$ where $\operatorname {Std}_{\widehat {G_{\pi _i}}}$ is the standard representation of $\widehat {G_{\pi _i}}$ and $\nu _{d_i}$ is the irreducible representation of $\operatorname {SL}_2({{\mathbb {C}}})$ of dimension d _i. Then $\dot {\psi }$ is well-defined up to conjugation by $\operatorname {SO}_{2n+1}({{\mathbb {C}}})$.

• (ii) Let ${\mathcal S}_{\psi }$ be the centralizer of $\dot {\psi }$ in $\operatorname {SO}_{2g+1}({{\mathbb {C}}})$, which is isomorphic to $({\mathbb {Z}}/2{\mathbb {Z}})^r$. A basis is given by (s _i)_1≤i≤r where s _i is the image by $\dot {\psi }$ of the non-trivial element in the center of $\widehat {G_{\pi _i}}$.

• (iii) Let ψ _∞ be the morphism $W_{{\mathbb {R}}} \times \operatorname {SL}_2({{\mathbb {C}}}) \rightarrow \operatorname {SO}_{2g+1}({{\mathbb {C}}})$ obtained by composing $\dot {\psi }$ with the morphisms $\varphi _{\pi _{i, \infty }} : W_{{\mathbb {R}}} \rightarrow \widehat {G_{\pi _i}}({{\mathbb {C}}})$. The centralizer ${\mathcal S}_{\psi _{\infty }}$ of ψ _∞ in $\operatorname {SO}_{2g+1}({{\mathbb {C}}})$ contains ${\mathcal S}_{\psi }$ and is isomorphic to $({\mathbb {Z}} / 2 {\mathbb {Z}} )^x$ where $x = \sum _{i=1}^r g_i$.

• (iv) For p a prime let c _p(ψ) be the image of $((c(\pi _{i,p}))_{0 \leq i \leq r}, \operatorname {diag}(p^{1/2}, p^{-1/2}))$ by $\dot {\psi }$, a well-defined semisimple conjugacy class in $\operatorname {SO}_{2g+1}({{\mathbb {C}}})$. Let $\chi _p(\psi ) : {\mathcal H}_p^{\operatorname {unr}}(\operatorname {Sp}_{2g}) \rightarrow {{\mathbb {C}}}$ be the associated character.

• (v) Let $\chi _f(\psi ) = \prod _p \chi _p(\psi ) : {\mathcal H}_f^{\operatorname {unr}}(\operatorname {Sp}_{2g}) \rightarrow {{\mathbb {C}}}$ be the product of the χ _p(ψ)’s. It takes values in the smallest subextension E(ψ) of ${\mathbb {Q}}^{\operatorname {real}}$ containing E(π ₀), …, E(π _r), which is also finite over ${\mathbb {Q}}$. For any prime p the characteristic polynomial of c _p(ψ) has coefficients in E(ψ). In particular we have a continuous action of $\operatorname {Gal}({\mathbb {Q}}^{\operatorname {real}} / {\mathbb {Q}})$ on $\varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$ which is compatible with χ _f.

The following theorem is a consequence of [72].

Theorem 29

For any g ≥ 1 the map $(\lambda , \psi \in \varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g}) ) \mapsto \chi _f(\psi )$ is injective.

The last condition in Definition 5 is explained by compatibility with infinitesimal characters, stated after the following definition.

Definition 7

Let

$$\displaystyle \begin{aligned} \delta_{\infty} : {{\mathbb{C}}}^{\times} & \longrightarrow {{\mathbb{C}}}^{\times} \times \operatorname{SL}_2({{\mathbb{C}}}) \\ z & \longmapsto (z, \operatorname{diag}(||z||{}^{1/2}, ||z||{}^{-1/2})).\end{aligned} $$

For a complex reductive group H and a morphism $\psi _{\infty } : {{\mathbb {C}}}^{\times } \times \operatorname {SL}_2({{\mathbb {C}}}) \rightarrow H({{\mathbb {C}}})$ which is continuous semisimple and algebraic on $\operatorname {SL}_2({{\mathbb {C}}})$, let $\operatorname {ic}(\psi _{\infty }) = \operatorname {ic}(\psi _{\infty } \circ \delta _{\infty })$. Similarly, if $\psi _{\infty } : W_{{\mathbb {R}}} \times \operatorname {SL}_2({{\mathbb {C}}}) \rightarrow H({{\mathbb {C}}})$, let $\operatorname {ic}(\psi _{\infty }) = \operatorname {ic}(\psi _{\infty }|{ }_{{{\mathbb {C}}}^{\times }})$.

For $\psi \in \varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$, we have $\operatorname {ic}(\psi _{\infty }) = \tau$ (equality between semisimple conjugacy classes in $\mathfrak {so}_{2g+1}({{\mathbb {C}}})$), and this explains the last condition in Definition 5.

For ψ as above Arthur constructed [9, Theorem 1.5.1] a finite set Π(ψ _∞) of irreducible unitary representations of $\operatorname {Sp}_{2g}({\mathbb {R}})$ and a map $\varPi (\psi _{\infty }) \rightarrow {\mathcal S}_{\psi _{\infty }}^{\vee }$, where $A^{\vee } = \operatorname {Hom}(A, {{\mathbb {C}}}^{\times })$. We simply denote this map by π _∞↦〈⋅, π _∞〉. Arthur also defined a character 𝜖 _ψ of ${\mathcal S}_{\psi }$, in terms of symplectic root numbers. We do not recall the definition, but note that for everywhere unramified parameters considered here, this character can be computed easily from the infinitesimal characters of the π _i’s (see [23, §3.9]).

We can now formulate the specialization of [9, Theorem 1.5.2] to level one and algebraic regular infinitesimal character, and its consequence for L ²-cohomology thanks to [17].

Theorem 30

Let g ≥ 1. Let V _λ be an irreducible algebraic representation of $\operatorname {Sp}_{2g}$ with dominant weight λ. Let τ = λ + ρ. The part of the discrete automorphic spectrum for $\operatorname {Sp}_{2g}$ having level $\operatorname {Sp}_{2g}(\widehat {{\mathbb {Z}}})$ and infinitesimal character τ decomposes as a completed orthogonal direct sum

$$\displaystyle \begin{aligned} L^2_{\operatorname{disc}}( \operatorname{Sp}_{2g}({\mathbb{Q}}) \backslash \operatorname{Sp}_{2g}({{\mathbb{A}}}) / \operatorname{Sp}_{2g}(\widehat{{\mathbb{Z}}}))^{\operatorname{ic} = \tau} \simeq \bigoplus_{\psi \in \varPsi_{\operatorname{disc}}^{\operatorname{unr}, \tau}(\operatorname{Sp}_{2g})} \bigoplus_{\substack{\pi_{\infty} \in \varPi(\psi_{\infty}) \\ \langle \cdot, \pi_{\infty} \rangle = \epsilon_{\psi}}} \pi_{\infty} \otimes \chi_f(\psi). \end{aligned}$$

Therefore

$$\displaystyle \begin{aligned} H^{\bullet}_{(2)}({{{\mathcal A}}_{g}}, {\mathbb{V}}_{\lambda}) \simeq \bigoplus_{\psi \in \varPsi_{\operatorname{disc}}^{\operatorname{unr}, \tau}(\operatorname{Sp}_{2g})} \bigoplus_{\substack{\pi_{\infty} \in \varPi(\psi_{\infty}) \\ \langle \cdot, \pi_{\infty} \rangle = \epsilon_{\psi}}} H^{\bullet}((\mathfrak{g}, K), \pi_{\infty} \otimes V_{\lambda}) \otimes \chi_f(\psi) \end{aligned}$$

where as before $\mathfrak {g} = \mathfrak {sp}_{2g}({{\mathbb {C}}})$ and K = U(g).

To be more precise the specialization of Arthur’s theorem relies on [98, Lemma 4.1.1] and its generalization giving the Satake parameters, considering traces of arbitrary elements of the unramified Hecke algebra.

Thanks to [4] for $\psi \in \varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$ the sets Π(ψ _∞) and characters 〈⋅, π _∞〉 for π _∞∈ Π(ψ _∞) are known to coincide with those constructed by Adams and Johnson in [1]. Furthermore, the cohomology groups

$$\displaystyle \begin{aligned} H^{\bullet}((\mathfrak{g}, K), \pi_{\infty} \otimes V_{\lambda}) \end{aligned} $$

(33)

for π _∞∈ Π(ψ _∞) were computed explicitly in [102, Proposition 6.19], including the real Hodge structure.

Thus in principle one can compute (algorithmically) the dimensions of the cohomology groups $H^i_{(2)}({{{\mathcal A}}_{g}}, {\mathbb {V}}_{\lambda })$ if the cardinalities of the sets O _o(… ), O _e(… ) and S(… ) are known. As explained in the previous section, for small g the Euler characteristic $e_{(2)}({{{\mathcal A}}_{g}}, {\mathbb {V}}_{\lambda })$ can be evaluated using the trace formula. For $\pi \in O_o(\tau ) \subset \varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$, the contribution of π to the Euler characteristic expanded using Theorem 30 is simply (−1)^g(g+1)∕22^g ≠ 0. The contributions of other elements of $\varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$ to $e_{(2)}({{{\mathcal A}}_{g}}, {\mathbb {V}}_{\lambda })$ can be evaluated inductively, using the trace formula and the analogue of Theorem 30 also for the groups $\operatorname {Sp}_{2g'}$ for g′ < g, $\operatorname {SO}_{4m}$ for m ≤ g∕2 and $\operatorname {SO}_{2m+1}$ for m ≤ g∕2. We refer to [98] for details, and simply emphasize that computing the contribution of some ψ to the Euler characteristic is much easier than computing all dimensions using [102]. For example for $\psi \in \varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$ we have

$$\displaystyle \begin{aligned} \sum_{\substack{\pi_{\infty} \in \varPi(\psi_{\infty}) \\ \langle \cdot, \pi_{\infty} \rangle|{}_{{\mathcal S}_{\psi}} = \epsilon_{\psi}}} e( (\mathfrak{sp}_{2g}, K), \pi_{\infty} \otimes V_{\lambda}) = \pm 2^{g-r} \end{aligned} $$

(34)

where the integer r is as in Definition 5 and the sign is more subtle but easily computable. To sum up, using the trace formula, we obtained tables of cardinalities for the three families of sets in Definition 4. See [100].

For small λ, more precisely for g + λ ₁ ≤ 11, there is another way to enumerate all elements of $\varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$, which still relies on some computer calculations, but much simpler ones and of a very different nature. The following is a consequence of [22, Théorème 3.3]. The proof uses the Riemann-Weil explicit formula for automorphic L-functions, and follows work of Stark, Odlyzko and Serre for zeta functions of number fields giving lower bound of their discriminants, and of Mestre, Fermigier and Miller for L-functions of automorphic representations. The striking contribution of [22, Théorème 3.3] is the fact that the rank of the general linear group is not bounded a priori, but for the purpose of the present appendix we impose regular infinitesimal characters.

Theorem 31

For w ₁ ≤ 11, the only non-empty O _o(w ₁, … ), O _e(w ₁, … ) or S(w ₁, … ) are the following.

• (i) For g = 0, O _o() = {1}.

• (ii) For 2w ₁ ∈{11, 15, 17, 19, 21}, $S(w_1) = \{ \varDelta _{2 w_1} \}$ where $\varDelta _{2 w_1}$ corresponds to the unique eigenform in $S_{2w_1+1}(\operatorname {SL}_2({{\mathbb {C}}})({\mathbb {Z}}))$.

• (iii) $O_o(11) = \{ \operatorname {Sym}^2 ( \varDelta _{11} ) \}$ ( $\operatorname {Sym}^2$ functoriality was constructed in [ 44 ]),

• (iv) For (2w ₁, 2w ₂) ∈{(19, 7), (21, 5), (21, 9), (21, 13)}, $S(w_1, w_2) = \{ \varDelta _{2w_1, 2w_2} \}$ . These correspond to certain Siegel eigenforms in genus two and level one.

Of course this is coherent with our tables.

As a result, for g + λ ₁ ≤ 11, i.e in

$$\displaystyle \begin{aligned} \sum_{g=1}^{11} \operatorname{card} \left\{ \lambda \,\middle|\, w(\lambda) \text{ even and } g+ \lambda_1 \leq 11 \right\} = 1055 \end{aligned}$$

non-trivial cases, $\varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$ can be described explicitly in terms of the 11 automorphic representations of general linear groups appearing in Theorem 31. In most of these cases $\varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$ is just empty, in fact

$$\displaystyle \begin{aligned} \sum_{g + \lambda_1 \leq 11} \operatorname{card} \varPsi_{\operatorname{disc}}^{\operatorname{unr}, \tau}(\operatorname{Sp}_{2g}) = 197 \end{aligned}$$

with 146 non-vanishing terms.

Corollary 3

For g = 3 and λ = (1, 1, 0) we have

$$\displaystyle \begin{aligned} IH^{\bullet}( {{{{\mathcal A}}_{3}^{\operatorname {Sat}}}}, {\mathbb{V}}_{\lambda}) = 0. \end{aligned}$$

Proof

Using Theorem 31 and Definition 5 we see that $\varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_6) = \emptyset$. □

Remark 5

Of course this result also follows from [98]. More precisely, without the a priori knowledge given by Theorem 31 we have for λ = (1, 1, 0) that $\varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_6)$ is the disjoint union of O _o(4, 3, 1) with the two sets

By Remark 4 and vanishing of $S_{2k}(\operatorname {SL}_2({\mathbb {Z}}))$ for 0 < k < 6 both sets are empty, so Corollary 3 follows from the computation of $e_{(2)}({{{\mathcal A}}_{3}}, {\mathbb {V}}_{\lambda }) = 0$. Note that computationally this result is easier than Proposition 3, since computing masses for $\operatorname {Sp}_8$ is much more work than for $\operatorname {Sp}_6$.

If we now focus on λ = 0, we have the following classification result.

Corollary 4

For 1 ≤ g ≤ 5 we have $\varPsi _{\operatorname {disc}}^{\operatorname {unr}, \rho }(\operatorname {Sp}_{2g}) = \{ [2g+1] \}$ . For 6 ≤ g ≤ 11, all elements of $\varPsi _{\operatorname {disc}}^{\operatorname {unr}, \rho }(\operatorname {Sp}_{2g}) \smallsetminus \{ [2g+1] \}$ are listed in the following tables.

In the next section we will recall and make explicit the description by Langlands and Arthur of L ²-cohomology in terms of ψ _∞, which is simpler than using [102] directly but imposes to work with the group $\operatorname {GSp}_{2g}$ (which occurs in a Shimura datum) instead of $\operatorname {Sp}_{2g}$.

Let us work out the simple case of $\psi = [2g+1] \in \varPsi _{\operatorname {disc}}^{\operatorname {unr}, \rho }(\operatorname {Sp}_{2g})$ directly using [102, Proposition 6.19], using their notation, in particular $\mathfrak {g} = \mathfrak {k} \oplus \mathfrak {p}$ (complexification of the Cartan decomposition of $\mathfrak {g}_0 = \operatorname {Lie}(\operatorname {Sp}_{2g}({\mathbb {R}}))$ and $\mathfrak {p} = \mathfrak {p}^+ \oplus \mathfrak {p}^-$ (decomposition of $\mathfrak {p}$ according to the action of the center of K, which is isomorphic to U(1)). In this case Π(ψ _∞) = {1}, $\mathfrak {u} = 0$ and $\mathfrak {l} = \mathfrak {sp}_{2g}$, so

$$\displaystyle \begin{aligned} H^{2k}((\mathfrak{g}, K), 1) = H^{k,k}((\mathfrak{g}, K), 1) \simeq \operatorname{Hom}_K \left(\bigwedge^{2k} \mathfrak{p}, {{\mathbb{C}}} \right)\end{aligned} $$

and the dimension of this space is the number of constituents in the multiplicity-free representation $\bigwedge ^k( \mathfrak {p}^+)$. One can show (as for any hermitian symmetric space) that this number equals the number of elements of length p in $W(\operatorname {Sp}_{2g}) / W(K) \simeq W(\operatorname {Sp}_{2g}) / W(\operatorname {GL}_g)$. A simple explicit computation that we omit shows that this number equals the number of partitions of k as a sum of distinct integers between 1 and n. In conclusion,

$$\displaystyle \begin{aligned} \sum_i T^i \dim H^i( (\mathfrak{sp}_{2n}, K), 1) = \prod_{k=1}^n (1+T^{2k}).\end{aligned} $$

Combining the first part of Corollary 4 and this computation we obtain Theorem 17.

Theorem 32

For 1 ≤ g ≤ 5 we have $IH^{\bullet }({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}, {\mathbb {Q}}) \simeq R_g$ as graded vector spaces over ${\mathbb {Q}}$.

We can sharpen Theorem 25 in the particular case of level one, although what we obtain is not a stabilization result (see the remark after the theorem).

Theorem 33

For g ≥ 2, λ a dominant weight for $\operatorname {Sp}_{2g}$ and k < 2g − 2,

$$\displaystyle \begin{aligned} IH^k({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}, {\mathbb{V}}_{\lambda}) = \begin{cases} R_g^k & \mathit{\text{ if }} \lambda = 0 \\ 0 & \mathit{\text{ otherwise }} \end{cases} \end{aligned}$$

where $R^k_g$ denotes the degree k part of R _g , u _i having degree 2i.

Proof

For $\psi \in \varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$ different from [2g + 1], one sees easily from the construction in [1] that the trivial representation of $\operatorname {Sp}_{2g}({\mathbb {R}})$ does not belong to Π(ψ _∞). So using Zucker’s conjecture, Borel-Casselman and Arthur’s multiplicity formula, we are left to show that for $\psi \in \varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g}) \smallsetminus \{ [2g+1] \}$, for any π _∞∈ Π(ψ _∞), $H^{\bullet }((\mathfrak {g}, K), \pi _{\infty } \otimes V_{\lambda })$ vanishes in degree less than 2g − 2. We can read this from [102, Proposition 6.19], and we use the notation from this paper. Let θ be the Cartan involution of $\operatorname {Sp}_{2g}({\mathbb {R}})$ corresponding to K, so that $\mathfrak {p} = \mathfrak {g}^{-\theta }$. The representation π _∞ is constructed from a θ-stable parabolic subalgebra $\mathfrak {q} = \mathfrak {l} \oplus \mathfrak {u}$ of $\mathfrak {g}$, where $\mathfrak {l}$ is also θ-stable. We will show that $\dim \mathfrak {u} \cap \mathfrak {p} \geq 2g-2$. The (complex) Lie algebra $\mathfrak {l}$ is isomorphic to $\prod _j \mathfrak {gl}(a_j+b_j) \times \mathfrak {sp}_{2c}$ with c +∑_j a _j + b _j = g. The action of the involution θ on the factor $\mathfrak {gl}(a_k+b_k)$ is such that the associated real Lie algebra is isomorphic to $\mathfrak {u}(a_k,b_k)$. Using notation of Definition 5, the integer c equals (d ₀ − 1)∕2. Since ψ ≠ [2g + 1] we have r ≥ 1 and this implies that c ≤ g − 2 (this is particular to level one, in arbitrary level one would simply get c < g). We have

$$\displaystyle \begin{aligned} 2 \dim \mathfrak{u} \cap \mathfrak{p} + \dim \mathfrak{l} \cap \mathfrak{p} = \dim \mathfrak{p} = g(g+1) \end{aligned}$$

since $\mathfrak {l}$, $\mathfrak {u}$ and its opposite Lie algebra with respect to $\mathfrak {l}$ are all stable under θ. We compute

$$\displaystyle \begin{aligned} \dim \mathfrak{l} \cap \mathfrak{k} = \sum_j (a_j^2+b_j^2) + c^2 \end{aligned}$$

and so

$$\displaystyle \begin{aligned} \dim \mathfrak{l} \cap \mathfrak{p} = \dim \mathfrak{l} - \dim \mathfrak{l} \cap \mathfrak{k} = 2 \sum_j a_jb_j + c(c+1) .\end{aligned}$$

We get

$$\displaystyle \begin{aligned} \dim \mathfrak{u} \cap \mathfrak{p} = \frac{g(g+1)}{2} - \frac{c(c+1)}{2} - \sum_j a_jb_j. \end{aligned}$$

We have

$$\displaystyle \begin{aligned} \sum_j a_jb_j \leq (\sum_j a_j)(\sum_j b_j) \leq \frac{(g-c)^2}{4} \end{aligned}$$

which implies

$$\displaystyle \begin{aligned} \dim \mathfrak{u} \cap \mathfrak{p} \geq \frac{g(g+1)}{2} - \frac{c(c+1)}{2} - \frac{(g-c)^2}{4}. \end{aligned} $$

The right hand side is a concave function of c, thus its maximal value for c ∈{0, …, g − 2} is $g(g+1)/2 - \min (g^2/4, (g-2)(g-1)/2+1)$. For integral g ≠ 3 one easily checks that this equals g(g + 1)∕2 − (g − 2)(g − 1)∕2 − 1 = 2g − 2. For g = 3 we get 2g − 2 − 1∕4, and ⌈2g − 2 − 1∕4⌉ = 2g − 2. □

Remark 6

• (i) For k ≥ g even the surjective map ${\mathbb {Q}}[\lambda _1, \lambda _3, \dots ]^k \rightarrow R_g^k$ has non-trivial kernel, so in the range g ≤ k < 2g − 2 we do not get stabilization.

• (ii) One can show that for the trivial system of coefficients (i.e. λ = 0) the bound in Theorem 33 is sharp for even g ≥ 6, is not sharp for odd g (i.e. $IH^{2g-2}({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}, {\mathbb {Q}}) = R_g^{2g-2}$) but that for odd g ≥ 9 we have $IH^{2g-1}({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}, {\mathbb {Q}}) \neq 0$. For even g ≥ 6 and odd g ≥ 9 this is due to ψ of the form $\pi [2] \boxplus [2g-3]$ where π ∈ S(g − 1∕2) corresponds to a weight 2g eigenform for $\operatorname {SL}_2({\mathbb {Z}})$.

• (iii) Of course for non-trivial λ one can improve on this result, e.g. for λ ₁ > ⋯ > λ _g > 0 we have vanishing in degree ≠ g(g + 1)∕2 (see [88, Theorem 5] and [75, Theorem 5.5] for a vanishing result for ordinary cohomology). If we only assume λ _g > 0, this forces c = 0 in the proof and we obtain vanishing in degree less than (g ² + 2g)∕4.

• (iv) An argument similar to the proof of Theorem 33 can be used to show the same result for k < g in arbitrary level. It seems likely that one could extend the sharper bound in Theorem 33 to certain deeper levels, e.g. Iwahori level at a finite number of primes.

There is also a striking consequence of [102, Proposition 6.19] (and [74, Lemma 9.1]) that was observed in [80], namely the fact that any ψ only contributes in degrees of a certain parity. This implies the dimension part in the following proposition which is a natural first step towards the complete description in the next section.

Proposition 5

There is a canonical decomposition

$$\displaystyle \begin{aligned} {\mathbb{Q}}^{\operatorname{real}} \otimes_{{\mathbb{Q}}} IH^{\bullet}({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}, {\mathbb{V}}_{\lambda}) = \bigoplus_{\psi \in \varPsi_{\operatorname{disc}}^{\operatorname{unr}, \tau}(\operatorname{Sp}_{2g})} {\mathbb{Q}}^{\operatorname{real}} \otimes_{E(\psi)} H_{\psi}^{\bullet} \end{aligned}$$

where $H_{\psi }^{\bullet }$ is a graded vector space of total dimension 2^n−r (r as in Definition 5) over the totally real number field E(ψ), endowed with

• (i) for any n ≥ 0, a pure Hodge structure of weight n on $H_{\psi }^n$ , inducing a bigrading ${{\mathbb {C}}} \otimes _{E(\psi )} H_{\psi }^n = \bigoplus _{p+q=n} H_{\psi }^{p,q}$ ,

• (ii) a linear operator $L : {\mathbb {R}} \otimes _{E(\psi )} H_{\psi }^{\bullet } \rightarrow {\mathbb {R}} \otimes _{E(\psi )} H_{\psi }^{\bullet }$ mapping $H_{\psi }^{p,q}$ to $H_{\psi }^{p+1,q+1}$ and such that for any 0 < n ≤ g(g + 1)∕2,

  $$\displaystyle \begin{aligned} L^n : {\mathbb{R}} \otimes_{E(\psi)} H_{\psi}^{g(g+1)/2-n} \rightarrow {\mathbb{R}} \otimes_{E(\psi)} H_{\psi}^{g(g+1)/2+n} \end{aligned}$$

  is an isomorphism.

This decomposition is ${\mathcal H}_f^{\operatorname {unr}}(\operatorname {Sp}_{2n})$ -equivariant, the action on H _ψ being by the character χ _f(ψ).

Proof

Recall that by Zucker’s conjecture [76, 78, 89] we have a Hecke-equivariant isomorphism

$$\displaystyle \begin{aligned} IH^{\bullet}({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}, {\mathbb{V}}_{\lambda}) \otimes_{{\mathbb{Q}}} {\mathbb{R}} \simeq H^{\bullet}_{(2)}({{{\mathcal A}}_{g}}, {\mathbb{V}}_{\lambda} \otimes_{{\mathbb{Q}}} {\mathbb{R}}). \end{aligned}$$

By Theorem 30 there are graded vector spaces H _ψ that can be defined over E _ψ such that the left hand side is isomorphic to the right hand side. By Theorem 29 each summand on the right hand side can be cut out using Hecke operators, so the decomposition is canonical and the E(ψ)-structure on H _ψ is canonical as well. We endow ${\mathbb {R}} \otimes _{{\mathbb {Q}}} IH^{\bullet }({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}, {\mathbb {V}}_{\lambda }) \simeq H^{\bullet }_{(2)}({{{\mathcal A}}_{g}}, {\mathbb {R}} \otimes _{{\mathbb {Q}}} {\mathbb {V}}_{\lambda })$ with the real Hodge structure given by Hodge theory on L ²-cohomology of the non-compact Kähler manifold ${{{\mathcal A}}_{g}}$. There is a natural Lefschetz operator L given by cup-product with the Kähler form. It commutes with Hecke operators and one can check that L is i times the operator X defined on p. 60 of [7]. The hard Lefschetz property of L is known both in L ²-cohomology and $(\mathfrak {g}, K)$-cohomology. It follows from [80, Theorem 1.5] that any ψ contributes in only one parity, so the claim about dim_E(ψ) H _ψ follows from (34). □

If o is any $\operatorname {Gal}({\mathbb {Q}}^{\operatorname {real}} / {\mathbb {Q}})$-orbit in $\varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$, ⊕_{ψ ∈ o} H _ψ is naturally defined over ${\mathbb {Q}}$ and endowed with an action of a quotient of ${\mathcal H}_f^{\operatorname {unr}}(\operatorname {Sp}_{2g})$ which is a finite totally real field extension E(o) of ${\mathbb {Q}}$, and elements of o correspond bijectively to ${\mathbb {Q}}$-embeddings $E(o) \rightarrow {\mathbb {Q}}^{\operatorname {real}}$.

Remark 7

There is also a rational Hodge structure defined on intersection cohomology groups thanks to Morihiko Saito’s theory of mixed Hodge modules, but unfortunately it is not known whether the induced real Hodge structure coincides with the one defined using L ² theory (see [64, §5]). Similarly, there is another natural Lefschetz operator acting on the cohomology $IH^{\bullet }({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}, {\mathbb {V}}_{\lambda })$ (using the first Chern class of an ample line bundle on ${{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}$), and it does not seem obvious that it coincides (up to a real scalar) with the Kähler operator L above, although it could perhaps be deduced from arguments as in [47, §16.6].

1.3 Description in Terms of Archimedean Arthur-Langlands Parameters

Langlands and Arthur [7, 8] gave a conceptually simpler point of view on the Hodge structure with Lefschetz operator on L ²-cohomology. This applies to Shimura varieties and so one would have to work with the reductive group $\operatorname {GSp}_{2g}$ instead of $\operatorname {Sp}_{2g}$, since only $\operatorname {GSp}_{2g}$ is part of a Shimura datum, $(\operatorname {GSp}_{2g}, {\mathbb {H}}_g \sqcup \overline {{\mathbb {H}}_g})$. Very roughly, the idea of this description for a Shimura datum (G, X) is that for K ₁ the stabilizer in $G({\mathbb {R}})$ of a point in X, representations of $G({\mathbb {R}})$ in an Adams-Johnson packet are parametrized by certain cosets in W(G, T)∕W(K ₁, T) for a maximal torus T of $G({\mathbb {R}})$ contained in K ₁, and W(K ₁, T) is also identified with the stabilizer of the cocharacter $\mu : \operatorname {GL}_1({{\mathbb {C}}}) \rightarrow G({{\mathbb {C}}})$ obtained from the Shimura datum. This cocharacter can be seen as an extremal weight for an irreducible algebraic representation r _μ of $\widehat {G}$ and r _μ is minuscule, i.e. its weights form a single orbit under the Weyl group of $\widehat {G}$, which is identified with W(G, T).

In the case of $\operatorname {GSp}_{2g}$ the Langlands dual group is $\widehat {\operatorname {GSp}_{2g}} = \operatorname {GSpin}_{2g+1}$ and r _μ is a spin representation. Morphisms taking values in a spin group cannot simply be described as self-dual linear representations. For this reason we do not have substitutes for Arthur-Langlands parameters for $\operatorname {GSp}_{2g}$ constructed using automorphic representations of general linear groups (that is the analogue of 5 for $\operatorname {GSp}_{2g}$ and arbitrary level), and no precise multiplicity formula yet. Bin Xu [103] obtained a multiplicity formula in many cases, but his work does not cover the case of non-tempered Arthur-Langlands parameters that is typical when λ = 0. For example all parameters appearing in Corollary 4 are non-tempered. Fortunately in level one it turns out that we can simply formulate the result in terms of $\operatorname {Sp}_{2g}$. This is in part due to the fact that, letting $K_1 = {\mathbb {R}}_{>0} \operatorname {Sp}_{2g}({\mathbb {R}}) \subset \operatorname {GSp}_{2g}({\mathbb {R}})$, the natural map

$$\displaystyle \begin{aligned} {{{\mathcal A}}_{g}} = \operatorname{Sp}_{2g}({\mathbb{Q}}) \backslash \operatorname{Sp}_{2g}({{\mathbb{A}}}) / K \operatorname{Sp}_{2g}(\widehat{{\mathbb{Z}}}) \rightarrow \operatorname{GSp}_{2g}({\mathbb{Q}}) \backslash \operatorname{GSp}_{2g}({{\mathbb{A}}}) / K_1 \operatorname{GSp}_{2g}(\widehat{{\mathbb{Z}}}) \end{aligned} $$

(35)

is an isomorphism. This is a special case of a more general principle in level one, see §4.3 and Appendix B in [23] for a conceptual explanation. Since the cohomology of the intermediate extension to ${{{{\mathcal A}}_{g}^{\operatorname {Sat}}}}$ of ${\mathbb {V}}_{\lambda }$ vanishes when the weight w(λ) := λ ₁ + ⋯ + λ _g is odd, we will be able to formulate the result using $\operatorname {Spin}_{2g+1}({{\mathbb {C}}})$ instead of $\operatorname {GSpin}_{2g+1}({{\mathbb {C}}})$.

Let g ≥ 1 and λ a dominant weight for $\operatorname {Sp}_{2g}$, as usual let τ = λ + ρ. Consider

as in Definition 5. First we recall how to equip ${{\mathbb {C}}} \otimes _{E(\psi )} H_{\psi }^{\bullet }$ with a continuous semisimple linear action ρ _ψ of ${{\mathbb {C}}}^{\times } \times \operatorname {SL}_2({{\mathbb {C}}})$. This action will be trivial on ${\mathbb {R}}_{>0} \subset {{\mathbb {C}}}^{\times }$ by construction.

• (i) We let $z \in {{\mathbb {C}}}^{\times }$ act on $H_{\psi }^{p,q}$ by multiplication by (z∕|z|)^q−p.

• (ii) There is a unique algebraic action of $\operatorname {SL}_2({{\mathbb {C}}})$ on ${{\mathbb {C}}} \otimes _{E(\psi )} H_{\psi }^{\bullet }$ such that the action of $\begin {pmatrix} 0 & 1 \\ 0 & 0 \end {pmatrix} \in \mathfrak {sl}_2$ is given by the Lefschetz operator L and the diagonal torus in $\operatorname {SL}_2({{\mathbb {C}}})$ preserves the grading on ${{\mathbb {C}}} \otimes _{E(\psi )} H_{\psi }^{\bullet }$. Explicitly, by hard Lefschetz we have that for $t \in \operatorname {GL}_1$, $\operatorname {diag}(t, t^{-1}) \in \operatorname {SL}_2$ acts on $H_{\psi }^i$ by multiplication by t ^{g(g+1)∕2−i}. This algebraic action is defined over ${\mathbb {R}}$, and if we knew that L is rational it would even be defined over E(ψ).

• (iii) These actions commute and we obtain

  $$\displaystyle \begin{aligned} \rho_{\psi} : {{\mathbb{C}}}^{\times} \times \operatorname{SL}_2({{\mathbb{C}}}) \longrightarrow \operatorname{GL}({{\mathbb{C}}} \otimes_{E(\psi)} H_{\psi}). \end{aligned}$$

The dimension g(g + 1)∕2 being fixed, we see that the isomorphism class of the real Hodge structure with Lefschetz operator $({\mathbb {R}} \otimes _{E(\psi )} H_{\psi }, L)$ determines and is determined by the isomorphism class of ρ _ψ. In fact they are both determined by the Hodge diamond of ${\mathbb {R}} \otimes _{E(\psi )} H_{\psi }^{\bullet }$.

To state the description of these isomorphism classes in terms of Arthur-Langlands parameters we need a few more definitions. For i ∈{0, …, r} let m _i be the product of d _i with the dimension of the standard representation of $\widehat {G_{\pi _i}}$, so that $2g+1 = \sum _{i=0}^r m_i$. Let ${\mathcal M}_{\psi _0} = \operatorname {SO}_{m_0}({{\mathbb {C}}})$. For 1 ≤ i ≤ r let $({\mathcal M}_{\psi _i}, \tau _{\psi _i})$ be a pair such that ${\mathcal M}_{\psi _i} \simeq \operatorname {SO}_{m_i}({{\mathbb {C}}})$ and $\tau _{\psi _i}$ is a semisimple element in the Lie algebra of ${\mathcal M}_{\psi _i}$ whose image via the standard representation has eigenvalues

$$\displaystyle \begin{aligned} \pm \left(w_1^{(i)} + \frac{d_i-1}{2} - j\right) , \dots, \pm \left( w_{g_i}^{(i)} + \frac{d_i-1}{2} -j\right) \text{ for } 0 \leq j \leq d_i-1. \end{aligned}$$

As in Definition 6 the point of this definition is that the group of automorphisms of $({\mathcal M}_{\psi _i}, \tau _{\psi _i})$ is the adjoint group of ${\mathcal M}_{\psi _i}$, because $\tau _{\psi _i}$ is not invariant under the outer automorphism of ${\mathcal M}_{\psi _i}$. Note that ${\mathcal M}_{\psi _i}$ is semisimple since m _i ≠ 2.

Let ${\mathcal M}_{\psi } = \prod _{0 \leq i \leq r} {\mathcal M}_{\psi _i}$. There is a natural embedding $\iota _{\psi } : {\mathcal M}_{\psi } \rightarrow \operatorname {SO}_{2g+1}({{\mathbb {C}}})$. Up to conjugation by ${\mathcal M}_{\psi }$ there is a unique morphism $f_{\psi } : {\mathcal L}_{\psi } \times \operatorname {SL}_2({{\mathbb {C}}}) \rightarrow {\mathcal M}_{\psi }$ such that

• (i) ι _ψ ∘ f _ψ is conjugated to $\dot {\psi }$, which implies that f _ψ is an algebraic morphism,

• (ii) the differential of f _ψ maps $( (\tau _{\pi _i})_{1 \leq i \leq r}, \operatorname {diag}(\frac {1}{2}, - \frac {1}{2}))$ to $\tau _{\psi } := (\tau _{\psi _i})_{0 \leq i \leq r}$.

We can conjugate $\dot {\psi }$ in $\operatorname {SO}_{2g+1}({{\mathbb {C}}})$ so that $\iota _{\psi } \circ f_{\psi } = \dot {\psi }$, so we assume this equality from now on. The centralizer of ι _ψ in $\operatorname {SO}_{2g+1}({{\mathbb {C}}})$ coincides with ${\mathcal S}_{\psi }$. Let

$$\displaystyle \begin{aligned} f_{\psi, \infty} = f_{\psi} \circ \left((\varphi_{\pi_{i, \infty}})_{0 \leq i \leq r}, \operatorname{Id}_{\operatorname{SL}_2({{\mathbb{C}}})} \right) : W_{{\mathbb{R}}} \times \operatorname{SL}_2({{\mathbb{C}}}) \rightarrow {\mathcal M}_{\psi}. \end{aligned}$$

Condition (ii) above is equivalent to $\operatorname {ic}(f_{\psi , \infty }) = \tau _{\psi }$. We have ψ _∞ = ι _ψ ∘ f _ψ,∞.

Let ${\mathcal M}_{\psi , \operatorname {sc}} = \prod _{i=0}^r {\mathcal M}_{\psi _i, \operatorname {sc}} \simeq \prod _{i=0}^r \operatorname {Spin}_{m_i}({{\mathbb {C}}})$ be the simply connected cover of ${\mathcal M}_{\psi }$. Let $\operatorname {spin}_{\psi _0}$ be the spin representation of ${\mathcal M}_{\psi _0, \operatorname {sc}}$, of dimension $2^{(m_0-1)/2}$. The group ${\mathcal M}_{\psi _i, \operatorname {sc}}$ has two half-spin representations $\operatorname {spin}^{\pm }_{\psi _i}$, distinguished by the fact that the largest eigenvalue of $\operatorname {spin}^+_{\psi _i}(\tau _{\psi _i})$ is greater than that of $\operatorname {spin}^-_{\psi _i}(\tau _{\psi _i})$. They both have dimension $2^{m_i/2-1}$. Let ${\mathcal L}_{\psi , \operatorname {sc}} = \prod _{i=0}^r (\widehat {G_{\pi _i}})_{\operatorname {sc}}$ be the simply connected cover of ${\mathcal L}_{\psi }$, a product of spin and symplectic groups. There is a unique algebraic lift $\widetilde {f_{\psi }} : {\mathcal L}_{\psi , \operatorname {sc}} \times \operatorname {SL}_2({{\mathbb {C}}}) \rightarrow {\mathcal M}_{\psi , \operatorname {sc}}$ of f _ψ. There is a unique algebraic lift $\widetilde {\iota _{\psi }} : {\mathcal M}_{\psi , \operatorname {sc}} \rightarrow \operatorname {Spin}_{2g+1}({{\mathbb {C}}})$ of ι _ψ and it has finite kernel. The pullback of the spin representation of $\operatorname {Spin}_{2g+1}({{\mathbb {C}}})$ via $\widetilde {\iota _{\psi }}$ decomposes as

$$\displaystyle \begin{aligned} \operatorname{spin}_{\psi_0} \otimes \bigoplus_{(\epsilon_i)_i \in \{ \pm \}^r} \operatorname{spin}^{\epsilon_1}_{\psi_1} \otimes \dots \otimes \operatorname{spin}^{\epsilon_r}_{\psi_r}. \end{aligned} $$

(36)

Let us be more specific. It turns out that the preimage ${\mathcal S}_{\psi , \operatorname {sc}} \simeq ({\mathbb {Z}}/2{\mathbb {Z}})^{r+1}$ of ${\mathcal S}_{\psi }$ in $\operatorname {Spin}_{2g+1}({{\mathbb {C}}})$ commutes with $\widetilde {\iota _{\psi }}({\mathcal M}_{\psi , \operatorname {sc}})$. This is specific to conductor one, i.e. level $\operatorname {Sp}_{2g}(\widehat {{\mathbb {Z}}})$. Thus the spin representation of $\operatorname {Spin}_{2g+1}({{\mathbb {C}}})$ restricts to a representation of ${\mathcal M}_{\psi , \operatorname {sc}} \times {\mathcal S}_{\psi , \operatorname {sc}}$, and (36) is realized by decomposing into isotypical components for ${\mathcal S}_{\psi , \operatorname {sc}}$. More precisely, the non-trivial element of the center of $\operatorname {Spin}_{2g+1}({{\mathbb {C}}})$ is mapped to − 1 in the spin representation, and there is a natural basis (s _i)_1≤i≤r of ${\mathcal S}_{\psi }$ over $({\mathbb {Z}}/2{\mathbb {Z}})^r$ and lifts $(\tilde {s}_i)_{1 \leq i \leq r}$ in ${\mathcal S}_{\psi , \operatorname {sc}}$ such that in each factor of (36) $\tilde {s}_i$ acts by 𝜖 _i.

For 0 ≤ i ≤ r there is a unique continuous lift $\widetilde {\varphi _{\pi _{i, \infty }}} : {{\mathbb {C}}}^{\times } \rightarrow (\widehat {G_{\pi _i}})_{\operatorname {sc}}$ of the restriction $\varphi _{\pi _{i, \infty }}|{ }_{{{\mathbb {C}}}^{\times }}$. One could lift morphisms from $W_{{\mathbb {R}}}$ but the lift is not unique in general. Finally we can define

$$\displaystyle \begin{aligned} \widetilde{f_{\psi, \infty}} = \widetilde{f_{\psi}} \circ \left((\widetilde{\varphi_{\pi_{i, \infty}}})_{0 \leq i \leq r}, \operatorname{Id}_{\operatorname{SL}_2({{\mathbb{C}}})} \right) : {{\mathbb{C}}}^{\times} \times \operatorname{SL}_2({{\mathbb{C}}}) \rightarrow {\mathcal M}_{\psi}. \end{aligned}$$

Theorem 34

For g ≥ 1, λ a dominant weight for $\operatorname {Sp}_{2g}$ and

we have an isomorphism of continuous semisimple representations of ${{\mathbb {C}}}^{\times } \times \operatorname {SL}_2({{\mathbb {C}}})$ :

$$\displaystyle \begin{aligned} \rho_{\psi} \simeq \left( \operatorname{spin}_{\psi_0} \otimes \operatorname{spin}^{u_1}_{\psi_1} \otimes \dots \otimes \operatorname{spin}^{u_r}_{\psi_r} \right) \circ \widetilde{f_{\psi, \infty}} \end{aligned}$$

where u ₁, …, u _r ∈{+, −} can be determined explicitly (see [ 99 ] for details).

Proof

This is essentially a consequence of [7, Proposition 9.1] and Arthur’s multiplicity formula, but we need to argue that in level one the argument goes through with the multiplicity formula for $\operatorname {Sp}_{2g}$ (Theorem 30) instead. This is due to two simple facts. Firstly, the group K ₁ in [7, §9] is simply ${\mathbb {R}}_{>0} \times K$, and this implies that for π _∞ a unitary irreducible representation of $\operatorname {GSp}_{2g}({\mathbb {R}}) / {\mathbb {R}}_{>0}$,

$$\displaystyle \begin{aligned} H^{\bullet}((\mathfrak{gsp}_{2g}, K_1), \pi_{\infty} \otimes V_{\lambda}) = H^{\bullet}((\mathfrak{sp}_{2g}, K), \pi_{\infty}|{}_{\operatorname{Sp}_{2g}({\mathbb{R}})} \otimes V_{\lambda} ) \end{aligned}$$

where on the left hand side V _λ is seen as an algebraic representation of $\operatorname {PGSp}_{2g}$ (since we can assume that w(λ) even). Secondly, as we observed above the preimage ${\mathcal S}_{\psi , \operatorname {sc}}$ of ${\mathcal S}_{\psi }$ in $\operatorname {Spin}_{2g+1}({{\mathbb {C}}})$ still commutes with $\widetilde {\iota _{\psi }}({\mathcal M}_{\psi , \operatorname {sc}})$, making the representation σ _ψ of [7, §9] well-defined. This second fact is particular to the level one case. □

To conclude, if we know $\varPsi _{\operatorname {disc}}^{\operatorname {unr}, \tau }(\operatorname {Sp}_{2g})$, making the decomposition in Proposition 5 completely explicit boils down to computing signs (u _i)_1≤i≤r and branching in the following cases:

• (i) for the morphism $\operatorname {Spin}_{2a+1} \times \operatorname {SL}_2 \rightarrow \operatorname {Spin}_{(2a+1)(2b+1)}$ lifting the representation

  $$\displaystyle \begin{aligned} \operatorname{Std}_{\text{SO}_{2a+1}} \otimes \operatorname{Sym}^{2b}(\operatorname{Std}_{\operatorname{SL}_2}) : \operatorname{SO}_{2a+1} \times \operatorname{SL}_2 \rightarrow \operatorname{SO}_{(2a+1)(2b+1)} \end{aligned}$$

  and the spin representation of $\operatorname {Spin}_{(2a+1)(2b+1)}$,

• (ii) for $\operatorname {Spin}_{4a} \times \operatorname {SL}_2 \rightarrow \operatorname {Spin}_{4a(2b+1)}$ and both half-spin representations,

• (iii) for $\operatorname {Sp}_{2a} \times \operatorname {SL}_2 \rightarrow \operatorname {Spin}_{4ab}$ and both half-spin representations.

For example one can using Corollary 4, Proposition 5 and Theorem 34 one can explicitly compute $IH^{\bullet }({{{\mathcal A}}_{g}})$ for all g ≤ 11.

Example 2

• (i) For any g ≥ 1 and ψ = [2g + 1], the group ${\mathcal L}_{\psi }$ is trivial and up to a shift we recover the graded vector space R _g as the composition of the spin representation of $\operatorname {Spin}_{2g+1}$ composed with the principal morphism $\operatorname {SL}_2 \rightarrow \operatorname {Spin}_{2g+1}$, graded by weights of a maximal torus of $\operatorname {SL}_2$.

• (ii) Consider g = 6 and $\psi = \varDelta _{11}[2] \boxplus [9]$. For ψ ₀ we have $\operatorname {spin}_{\psi _0} \circ \widetilde {f_{\psi _0}} \simeq \nu _{11} \oplus \nu _5$, where as before ν _d denotes the irreducible d-dimensional representation of $\operatorname {SL}_2$. For ψ ₁ = Δ ₁₁[2] we have ${\mathcal L}_{\psi _1} = \operatorname {Sp}_2({{\mathbb {C}}})$, ${\mathcal M}_{\psi _1} = \operatorname {SO}_4({{\mathbb {C}}})$, $\operatorname {spin}^+_{\psi _1} \circ \widetilde {f_{\psi _1}} \simeq \operatorname {Std}_{\operatorname {Sp}_2} \otimes 1_{\operatorname {SL}_2}$ and $\operatorname {spin}^-_{\psi _1} \circ \widetilde {f_{\psi _1}} \simeq 1_{\operatorname {Sp}_2} \otimes \nu _2$. It turns out that u ₁ = −, so $\rho _{\psi }|{ }_{{{\mathbb {C}}}^{\times }}$ is trivial and

  $$\displaystyle \begin{aligned} \rho_{\psi}|{}_{\operatorname{SL}_2} \simeq (\nu_{11} \oplus \nu_5 ) \otimes \nu_2 \simeq \nu_{12} \oplus \nu_{10} \oplus \nu_6 \oplus \nu_4. \end{aligned}$$

  Thus $H_{\psi }^{\bullet }$ has primitive cohomology classes in degrees 10, 12, 16, 18 (a factor ν _d contributes a primitive cohomology class in degree g(g + 1)∕2 − d + 1). Surprisingly, these classes are all Hodge, i.e. they belong to $H_{\psi }^{2k} \cap H_{\psi }^{k,k}$, despite the fact that the parameter ψ is explained by a non-trivial motive over ${\mathbb {Q}}$ (attached to Δ ₁₁).

• (iii) Consider g = 7 and $\psi = \varDelta _{11}[4] \boxplus [7]$. Again ${\mathcal L}_{\psi , \operatorname {sc}} \simeq \operatorname {Sp}_2({{\mathbb {C}}})$. For ψ ₀ = [7] we have $\operatorname {spin}_{\psi _0} \circ \widetilde {f_{\psi _0}} \simeq \nu _7 \oplus 1$. For ψ ₁ = Δ ₁₁[4] we have ${\mathcal L}_{\psi _1} = \operatorname {Sp}_2({{\mathbb {C}}})$, ${\mathcal M}_{\psi _1} = \operatorname {SO}_8({{\mathbb {C}}})$, $\operatorname {spin}^+ \simeq \operatorname {Sym}^2(\operatorname {Std}_{\operatorname {Sp}_2}) \oplus \nu _5$ and $\operatorname {spin}^- \simeq \operatorname {Std}_{\operatorname {Sp}_2} \otimes \nu _4$. Here u ₁ = +, and we conclude

  $$\displaystyle \begin{aligned} \rho_{\psi} & \simeq (\nu_7 \oplus 1) \otimes (\operatorname{Sym}^2(\operatorname{Std}_{\operatorname{Sp}_2}) \oplus \nu_5) \\ & \simeq \operatorname{Sym}^2(\operatorname{Std}_{\operatorname{Sp}_2}) \otimes (\nu_7 \oplus 1) \oplus \nu_{11} \oplus \nu_9 \oplus \nu_7 \oplus \nu_5^{\oplus 2} \oplus \nu_3. \end{aligned} $$

  In this example, as in general, we would love to know that the above formula is valid for the rational Hodge structure $H_{\psi }^{\bullet }$, replacing $\operatorname {Sym}^2(\operatorname {Std}_{\operatorname {Sp}_2})$ by $\operatorname {Sym}^2(M)(11)$ where M is the motivic Hodge structure associated to Δ ₁₁. In [99] this (and generalizations) is proved at the level of ℓ-adic Galois representations.

Forgetting the Hodge structure, the graded vector space $H_{\psi }^{\bullet }$ is completely described by the restriction of ρ _ψ to $\operatorname {SL}_2({{\mathbb {C}}})$. The Laurent polynomial

$$\displaystyle \begin{aligned}T^{-g(g+1)/2} \sum_{k=0}^{g(g+1)} T^k \dim H_{\psi}^k\end{aligned}$$

can easily be computed by taking the product over 0 ≤ i ≤ r of the following Laurent polynomials (with choice of signs as in Theorem 34). Denote $x = (1, \operatorname {diag}(T,T^{-1})) \in {{\mathbb {C}}}^{\times } \times \operatorname {SL}_2({{\mathbb {C}}})$.

• (i) For ψ ₀ = π ₀[2d + 1] with π ₀ ∈ O _o(w ₁, …, w _m), we have

  $$\displaystyle \begin{aligned} \left( \operatorname{spin}_{\psi_0} \circ \widetilde{f_{\psi, \infty}} \right)(x) = 2^m \prod_{j=1}^d (T^{-j} + T^j)^{2m+1}. \end{aligned}$$

• (ii) For ψ _i = π _i[2d + 1] with π _i ∈ O _e(w ₁, …, w _2m) we have

  $$\displaystyle \begin{aligned} \left( \operatorname{spin}^{\pm}_{\psi_i} \circ \widetilde{f_{\psi, \infty}} \right)(x) = 2^{2m-1} \prod_{j=1}^d (T^{-j} + T^j)^{4m}. \end{aligned}$$

• (iii) For ψ _i = π _i[2d] with π _i ∈ S(w ₁, …, w _m) we have

  $$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \left( \operatorname{spin}^{\pm}_{\psi_i} \circ \widetilde{f_{\psi, \infty}} \right)(x) = \\ &\displaystyle &\displaystyle \qquad \qquad \quad \frac{1}{2} \left( \prod_{j=1}^d (2+T^{2j-1}+T^{1-2j})^m \pm \prod_{j=1}^d (2-T^{2j-1}-T^{1-2j})^m \right). \end{array} \end{aligned} $$

Acknowledgements Klaus Hulek presented his joint work with Sam Grushevsky [55] at the Oberwolfach workshop “Moduli spaces and Modular forms” in April 2016. During this workshop Dan Petersen pointed out that $IH^{\bullet }({{{{\mathcal A}}_{g}^{\operatorname {Sat}}}})$ can also be computed using [98]. I thank Dan Petersen, the organizers of this workshop (Jan Hendrik Bruinier, Gerard van der Geer and Valery Gritsenko) and the Mathematisches Institut Oberwolfach. I also thank Eduard Looijenga for kindly answering questions related to Zucker’s conjecture.