Tempered D-modules and Borel-Moore homology vanishing

We characterize the tempered part of the automorphic Langlands category D-mod(Bun_G) using the geometry of the big cell in the affine Grassmannian. We deduce that, for $G$ non-abelian, tempered D-modules have no de Rham cohomology with compact supports. The latter fact boils down to a concrete statement, which we prove using the Ran space and some explicit t-structure estimates: for $G$ non-abelian and $\Sigma$ a smooth affine curve, the Borel-Moore homology of the indscheme $Maps(\Sigma,G)$ vanishes.


1.1.
Overview. The present paper is devoted to the study of the tempered condition appearing on the automorphic side of the geometric Langlands conjecture. In this overview, we recall the statement of the geometric Langlands conjecture, review how the tempered condition comes about and explain why this condition is important. We will then state the goals of the paper.
1.1.1. Let G be a connected reductive group and X a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Denote by Bun G := Bun G (X) the stack of G-bundles on X and by D(Bun G ) the differential graded (DG, from now on) category of D-modules on it. This is the DG category appearing on the automorphic side of the geometric Langlands correspondence.
For details on the geometry of Bun G and on the DG category of D-modules on a stack, we recommend the papers [16], [17] and [21].
1.1.2. The geometric Langlands conjecture calls for an equivalence between D(Bun G ) and a different-looking DG category whose definition explicitly involvesǦ, the Langlands dual group of G. At first approximation, the candidate is QCoh(LSǦ), the DG category of quasi-coherent sheaves on the stack LSǦ := LSǦ(X) of de RhamǦ-local systems on X. This is the so-called spectral side of the conjecture. For details on the definition of LSǦ, and in particular for its derived nature, we refer to [6,Section 2.11] and [1, Section 10].
1.1.4. A viable candidate for the enlarged spectral DG category was introduced in [1]; to define it, one needs the theory of ind-coherent sheaves (developed in [22]) and the theory of singular support for coherent sheaves on quasi-smooth stacks (see [1,). We will briefly discuss this material in Section 2.3.8; for now, we just need to know that: • IndCoh(LSǦ), the DG category of ind-coherent sheaves on LSǦ, contains QCoh(LSǦ) as a full subcategory; • ind-coherent sheaves on LSǦ can be classified according to their singular support, with possible singular supports being closed subsets of the space of geometric Arthur parameters (1.2) ArthǦ := {(σ, A)|σ ∈ LSǦ, A ∈ H 0 (X dR ,ǧ σ )}. After [1], the above is the official form of the conjecture. It should come with a series of compatibilities (with the Hecke action, with the Whittaker normalization, with Eisenstein series) that we do not discuss here.
1.1.7. The alternative way to correct (1.1) is to shrink the automorphic side. This form of the conjecture, to be called the tempered geometric Langlands conjecture, was also introduced in [1]. It calls for an equivalence where temp D(Bun G ) is the full-subcategory of D(Bun G ) consisting of tempered D-modules. We will recall the definition of the tempered condition in Section 1.3; meanwhile, let us explain why the latter form of the conjecture is more fundamental than the official one. This has to do with the gluing statements appearing on the two sides of the conjecture.
1.1.8. In [2] and [12], it is proven that IndCohŇglob(LSǦ) can be reconstructed using QCoh(LSǦ) as well as similar DG categories for smaller Levi subgroups ofǦ. The details of those two papers are rather technical, but luckily we do not need them here; very informally 1 , we have whereM runs through the poset of standard Levi subgroups ofǦ, while the symbol Glue means "glue the DG categories QCoh(LSM ) in a certain precise and explicit way that we do not explain in this paper". This is the content of the spectral gluing theorem.
1.1.9. The official version of the geometric Langlands conjecture then predicts that a similar decomposition must occur on the automorphic side; in other words, we expect the following automorphic gluing statement: (1.6) D(Bun G ) ≃ Glue M⊆G temp D(Bun M ) .
1 Warning: the usage of the symbol Glue here does not match with the one adopted in [2].
For more details on the latter, we refer to [11,Section 1.11] and [12,Section 1.4]. While (1.5) is settled, the equivalence (1.6) is still in progress. Namely, it is possible to properly define the glued DG category and it is easy to write down a functor from D(Bun G ) to that glued DG category. The delicate part is proving that such a functor is an equivalence: this requires a good understanding of the tempered condition. The present paper (in particular, Theorem C below) is the first step towards our ongoing proof of (1.6).
rank equal to one. If this is true, then Theorem A ′ provides a very pleasant characterization of tempered D-modules in semisimple rank 1, and thus a very pleasant correction of the best hope conjecture (1.1) in that case.
1.2.3. We will prove Theorem A by establishing two other main results, Theorems B and C, which are possibly more interesting than Theorem A itself. The first of these results is the following concrete statement.
Remark 1.2.4 (Related results). The ordinary homology of G[Σ] was computed by C. Teleman in [41]. The homology of G[Σ] gen , the space of rational (alias: generic) maps from Σ to G, was computed by D. Gaitsgory in [21].

Let us look at G[Σ]
and at its Borel-Moore homology more closely. First off, note that A N [Σ] is an indscheme isomorphic to A ∞ := colim m≥0 A m . Indeed, where A N [Σ] ≤d is the finite dimensional vector space of maps whose poles at the points at infinity of Σ have order bounded by d; by Riemann-Roch, the dimension of A N [Σ] ≤d increases to ∞ with d.
Next, recall that any affine scheme Y of finite type (and in particular G) can be realized as a closed subscheme of A N . Since the induced map is an ind-affine indscheme of ind-finite type.
1.2.6. The Borel-Moore homology of a scheme Y of finite type can be defined using D-modules: we set where ω Y ∈ D(Y ) is the dualizing D-module and (p Y ) * ,dR the functor of de Rham cohomology. It follows formally that H BM is covariant with respect to proper maps, hence it is well-defined on indschemes (of ind-finite type). For example, for A ∞ we have [2n] ≃ 0.
1.2.7. The proof of Theorem B will be discussed in Section 1.5. Meanwhile, let us explain how to deduce Theorem A from Theorem B. For this, we must first digress and recall the definition of the tempered condition.
1.3. Tempered objects. The phenomenon of temperedness (and non-temperedness) was first observed in [1, Sections 1.1.10 and 12.1]. It arises as a consequence of three facts: the Hecke action on D(Bun G ), the derived Satake theorem, the discrepancy between ind-coherent sheaves and quasi-coherent sheaves on a quasi-smooth stack. Let us review these facts in order. To fix the conventions, we regard D(Bun G ) as acted upon by Sph G from the left. Similarly, for Gr G := G(O)\G(K) the affine Grassmannian, we regard D(Gr G ) as acted on by Sph G from the left (and compatibly by G(K) from the right). We denote by ½ Sph G the unit object of Sph G , described explicitly in Section 1.4.1 below.
1.3.2. The next ingredient is the derived Satake theorem (see [14] and [1,Section 12]), that is, the description of Sph G in Langlands dual terms. To appreciate this theorem, a certain familiarity with ind-coherent sheaves on quasi-smooth stacks and with the theory of singular support is desirable: we refer to Section 2.3.8 for the main tenets of these theories and to [1,Section 12] for the full treatment. Remark 1.3.4. We should stress the fact that the core of the proof of the geometric Satake is the second equivalence of [14,Theorem 5]. As explained in [1,Section 12.5], this equivalence is related to (1.7) by renormalization and Koszul duality.
1.3.5. In the above formula, Ωǧ denotes the self-intersection of the origin of the vector spaceǧ, that is, the derived scheme pt ×ǧ pt ≃ Spec Sym(ǧ * [1]). It is equipped with aǦ-action induced by the usual (co)adjoint action. The quotient stack Ωǧ/Ǧ is quasi-smooth with space of singularities equal toǧ * /Ǧ. Hence, we can consider ind-coherent sheaves on Ωǧ/Ǧ with singular support contained in any chosen closed conical G-invariant subset ofǧ * . In particular, the choice of the nilpotent coneŇ ⊆ǧ * yields the DG category appearing on the LHS of Theorem 1.3.3. On the other hand, the choice of 0 ∈ǧ * yields the DG category QCoh(Ωǧ/Ǧ). These two DG categories are related by a natural colocalization (that is, an adjunction with fully faithful left adjoint) QCoh(Ωǧ/Ǧ) IndCohŇ(Ωǧ/Ǧ).
Define temp Sph G to be the full subcategory of Sph G corresponding to QCoh(Ωǧ/Ǧ) under derived Satake. By construction, there is a colocalization where, abusing notation, we have denoted the two adjoint functors with the same symbols as above.
1.3.7. For C a DG category with a left action of Sph G , we set As above, and abusing notation again, there is a colocalization We always regard temp C as a full subcategory of C via the functor Ξ 0 Ň. Definition 1.3.8. We say that an object of C is tempered if it belongs to temp C. We say that an object of C is anti-tempered iff it is annihilated by the projection Ψ 0 Ň : C ։ temp C. Equivalently, by adjunction, c ∈ C is anti-tempered iff Hom C (t, c) ≃ 0 for all t ∈ temp C. Remark 1.3.9. The endofunctor Ξ 0 Ň • Ψ 0 Ň : C C will be often called the temperization functor, since it is the projector onto the tempered subcategory.
1.3.10. The above construction, applied to the Hecke action of Sph G on D(Bun G ) at a chosen point x ∈ X, yields the DG category temp D(Bun G ) we are interested in.
In principle, a different choice of x ∈ X might yield a different DG category. Thus, to be precise, we should write x -temp D(Bun G ) in place of temp D(Bun G ). However, [1,Conjecture 12.8.5] states that x -temp D(Bun G ) ought to be independent of the choice of the point x ∈ X. See [11,Section 1.4.2] for a sketch of the proof of this statement. Regardless of this conjecture and of its solution, our proof of Theorem A will show that ω Bun G is right-orthogonal to x -temp D(Bun G ) for any x.

Denote by ½ temp
Sph G the temperization of the unit ½ Sph G , that is, the object ½ temp We emphasize that this object is not very explicit, since the functors Ξ 0 Ň and Ψ 0 Ň are defined using the derived Satake equivalence. (Our theorem below will make it explicit.) For C a DG category endowed with a Sph G -action, indicate by the symbol ⋆ the action of Sph G on C. By construction, the temperization functor coincides with the functor ½ temp Hence, we immediately deduce that: • an object c ∈ C is tempered iff it is isomorphic to ½ temp Sph G ⋆ c; • an object d ∈ C is anti-tempered iff ½ temp 1.3.12. In view of the second item above, the idea of the proof of Theorem A is clear: as a first step, we should describe ½ temp Sph G explicitly (that is, only in terms of Sph G , without appealing to geometric Satake at all) and then, as a second step, we should prove that ½ temp The first step is exactly the content of Theorem C below, while the second one will turn out to be a quick consequence of Theorem B in the special case of Σ = A 1 .
It turns out that (f ! ) R agrees with the renormalized pushforward f * ,ren along f . The latter notion will be discussed in Section 5.1; in any case, the functor (f ! ) R can be described really explicitly as follows.
Recall, [18], that the map is an open embedding, the inclusion of the "big cell" of the affine Grassmannian. Hence f is the composition of an open embedding with a quotient by a pro-unipotent group (the first congruence subgroup of G(O)): .
Here we have used the notation of [8] for group actions on DG categories. This notation is reviewed in Section 2.5: in short, oblv G G(O) is the functor that forgets the G(O)-invariance while retaining only the residual G-invariance; (the * -averaging functor) is its continuous right adjoint.
Example 1.4.4. Suppose for a moment that G = T is a torus. In this case, the nilpotent cone equals the origin: this implies that every object of Sph T is tempered. In particular, ½ temp Sph T must agree with ½ Sph T .
Let us verify this fact using the formula of Theorem C. The key observation is that, at the reduced level, T (R) ≃ T . It follows that D(T \T (R)/T ) ≃ D(pt/T ) and that j coincides with the closed embedding i induced by the unit point of Gr T . We obtain that ½ temp which is indeed the unit ½ Sph T .
When Y is a scheme of finite type, one easily proves that D(Y ) ≤−∞ ≃ 0. On the other hand, the theorems below will exhibit several indschemes Y for which D(Y) ≤−∞ is nontrivial. 1.5.5. Given an affine scheme Y , one might ask what conditions on Y ensure that ω Y [Σ] is infinitely connective. The following result, whose proof uses only Riemann-Roch and an elementary t-structure estimate, gives a sufficient (but certainly not necessary) condition.
Theorem E. Let Y ⊆ A N be a closed subscheme defined as the zero locus of k polynomials of degrees n 1 , . . . , n k . If i n i < N , then ω Y [Σ] is infinitely connective.
1.5.6. Obviously, Theorem E settles Theorem D in the cases G = GL n and G = SL n . For more general groups, we take a completely different route, which uses the Ran space and a bit of representation theory.
The proof is outlined in Section 7.1.
1.6. The structure of the paper.
1.6.1. In Section 2, we collect some basic notions that we will need throughout.
1.6.2. In Section 3, we discuss geometric Langlands for X = P 1 , outline the proof of Theorem C and compute the Serre functor of the DG category D(Bun G (P 1 )). 1.6.3. In Section 4, we complete the proof of Theorem C by computing the Serre functor of three DG categories related to the nilpotent cone. This section and the previous one are the only ones that require some derived algebraic geometry.
1.6.4. In Section 5, we use Theorem C to characterize tempered D-modules on Bun G . We also show that Theorem A follows from the vanishing of H BM (Gr • G ).
1.6.5. In Section 6, we show that Theorem B is a simple corollary of Theorem D. We then prove Theorem E, which settles Theorem D for GL n and SL n .
1.6.6. Finally, in Section 7, we prove Theorem D for all reductive groups.
1.7. The main techniques and ideas. For the reader's convenience, let us highlight the eight most important notions and ideas employed in this paper. The first four are rather technical, while the second four form the geometric core of the paper. We refer to Section 2 for any undefined notation and terminology, as well as for the appropriate references.
1.7.1. Singular support. The notion of singular support for ind-coherent sheaves is unavoidable, as it is at the heart of the notion of temperedness. In Section 4, we will perform several singular support computations using Koszul duality and the shearing operation. These two devices allow to transform singular support for ind-coherent sheaves on a space into ordinary set-theoretic support for quasi-coherent sheaves on a different space.
1.7.2. Ind-coherent sheaves and formal geometry. In turn, quasi-coherent sheaves on a space Y with support on a closed subset Z ⊆ Y can be understood as quasi-coherent sheaves on Y ∧ Z , the formal completion of Y along Z. Contrarily to the case of ind-coherent sheaves, the functoriality of quasi-coherent sheaves is not welladapted to working with formal completions. For this reason, a number of passages between quasi-coherent sheaves and ind-coherent sheaves will occur. 1.7.3. Group actions on DG categories. The categories appearing in the geometric Langlands program are often DG categories of D-modules on (double) quotients: for instance, consider Sph G , D(Bun G (P 1 )), D(Gr G ). In particular, we often take quotients by infinite dimensional groups like G(O). The theory of loop group actions on DG categories is very convenient when dealing with such situations, and in particular when dealing with the Hecke action. It will be used in Section 5. 1.7.5. Serre functor calculations. While the definition of the Serre functor is abstract nonsense, the computation of the Serre functor in a given geometric situation is very much not abstract.
In Section 3.3, we use Drinfeld's miraculous duality to compute the Serre functor on the DG category D(Bun G (P 1 )). In Section 4, we compute the Serre functor of the DG category QCoh(N/G) and of some related DG categories. The calculations hinge on the fact that H * (N − {0}, O) is nonzero only in two degrees.
We believe that a systematic study of the behaviour of Serre functors of DG categories of quasi-coherent sheaves on quotient stacks could be really fruitful. 1.7.6. Sph G and D(Bun G (P 1 )). In Section 3, we crucially use the equivalence (due to V. Lafforgue) between the spherical DG category Sph G and the automorphic Langlands DG category D(Bun G (P 1 )). This equivalence is the reason for the appearance of G(R) in our Theorem C.
As in the previous point, we believe that a deeper study of this relation will yield interesting results. In general, it would be worthwhile to find several examples of indschemes (beyond the ones of Theorems D and E) whose dualizing sheaf is infinitely connective. Research supported by ERC-2016-ADG-741501.

Preliminaries and basic notations
In this section, we collect the notations, the basic notions, and the basic results that we use. We advise the reader to skip this material and return here only when it is necessary.
2.1. Representation theory and algebraic geometry. We follow the conventions of [1] and [2]. Let us recall the most relevant ones.
2.1.1. By the term "space", we mean a space of algebraic geometry: for instance, a (derived) scheme, an indscheme, a stack or a prestack. We fix a ground field , algebraically closed and of characteristic zero, and set pt := Spec( ). Every space Y appearing in this paper is defined over and the structure map Y pt will be denoted by p Y . The Langlands dual group of G, defined using the duality of root data, is denoted byǦ. It comes with a maximal torusŤ and a Borel subgroupB.
2.1.3. Let c G := g * / /G := Spec((Sym g) G ). By Chevalley's restriction theorem and the theory of exponents, , and a G is a -vector space generated by r-polynomials of degrees d 1 , . . . , d r (with d i equal one plus the i-th exponent of G).
2.1.4. We let N be the nilpotent cone of G (accordingly,Ň the nilpotent cone ofǦ). By definition, N is the closed subscheme of g * defined by Since the map g * c G is known to be flat, the fiber product defining N can be understood either in classical or in derived algebraic geometry. We usually wish to regard N as a closed subscheme of g: we do this by choosing a G-equivariant identification g ≃ g * .
2.1.5. We denote by Λ the lattice of (co)weights of G. Precisely, Λ means "coweights" in Sections 3 and 7, while it means "weights" in Section 4. Accordingly, the cone of dominant (co)weights is denoted by Λ dom . This changing notation is the price to pay to avoid usingλ in formulas. For two (co)weights µ and λ, the notation µ ≤ λ means that λ − µ is a sum of positive (co)roots.
2.1.6. Given an affine scheme Y and a smooth curve Σ, we denote by Y [Σ] the indscheme parametrizing maps from Σ to Y . As a functor of points, Y [Σ] sends a test affine scheme S to the set Y (S × Σ). We will often use the shortcut Σ S := S × Σ.
The complement of Σ inside its compactification is a finite set of "points at infinity", which we will denote by D ∞ . Let h be the cardinality of D ∞ : this is the number of "holes" that Σ has.
2.1.7. In Sections 3 and 4, we will need some formal and derived algebraic geometry. 4 The conventions for derived algebraic geometry follow [26]. In particular, fiber products of schemes in those sections are always derived. So, for example, the self-intersection of the origin in a finite dimensional vector space V is the derived affine scheme This derived scheme has appeared before with V =ǧ, and it will appear later with V = c G . 2.2.1. Denote by DGCat the ∞-category whose objects are ( -linear) cocomplete DG categories and whose 1-arrows are continuous (i.e., colimit preserving) functors. By default, when we say that C is a DG category, we mean that C ∈ DGCat, that is, we assume that C is cocomplete. When C is not cocomplete, we say so explicitly. Similarly, a functor between DG categories is assume to be continuous unless otherwise stated.

If
C is a DG category (cocomplete or not) with two objects c, c ′ , we denote by Hom C (c, c ′ ) the DG vector space of morphisms c c ′ .

2.2.3.
For C ∈ DGCat, we let C cpt be its non-cocomplete full subcategory of compact objects. We assume familiarity with the notions of dualizability, compact generation and ind-completion. When a DG category C is dualizable, we denote by C ∨ its dual. 4 The guiding principle is that derived algebraic geometry is required anytime we are dealing with quasi-coherent sheaves or ind-coherent sheaves. On the other hand, Sections 5-7 only deal with D-modules, so derived algebraic geometry is not needed there.

By
Vect, we denote the DG category of complexes of -vector spaces. We use cohomological indexing conventions throughout. Note that Vect cpt consists of those complexes with finite dimensional total cohomology. We usually say that V ∈ Vect is finite dimensional if it belongs to Vect cpt .
2.2.5. Let C be a compactly generated DG category. Following [28], we say that C is proper if When C is (compactly generated and) proper, we consider its Serre functor Serre C : C C. This is the continuous functor uniquely characterized by: Here, (−) * denotes the dual of a complex of vector spaces. If C is clear from the context, we sometimes write Serre instead of the more precise Serre C .
Remark 2.2.6. Observe that, in the defining formula for Serre C , the objects c, c ′ are required to be compact.
A simple colimit computation shows that in (2.2) we might just as well require c compact and c ′ arbitrary.
2.2.7. Our DG categories are sometimes equipped with t-structures. We use cohomological indexing, which means that, at the level of the underlying triangulated category, C ≤0 is left orthogonal to C ≥1 . A (continuous) colimits. This is a consequence of the Barr-Beck-Lurie theorem.
2.3.1. We denote by QCoh(Y) and IndCoh(Y) the DG categories of quasi-coherent and ind-coherent sheaves on a derived prestack Y. While QCoh(Y) is defined for arbitrary Y, some conditions are required for IndCoh(Y) to make sense. We will only consider ind-coherent sheaves on algebraic stacks (such as N/G, Ωg/G, LSǦ) and of formal completions of maps of algebraic stacks. Pushforwards, pullbacks and tensor products of sheaves are understood in the derived sense, unless otherwise stated. There is a natural action of QCoh(Y), equipped with its natural symmetric monoidal structure, on IndCoh(Y).

2.3.
3. When f is nice (for instance: inf-schematic), there is a well-defined pushforward f IndCoh * for indcoherent shaves. When f is furthermore inf-proper, f IndCoh * is left adjoint to f ! . 5 Warning: the D-module pullback and the D-module dualizing sheaf are denoted in the same way. We hope that the context will make it clear which one we are referring to.
However, we will use these notions only in the following situation. Let be a commutative diagram of algebraic stacks. Then the obvious map ξ : X ∧ W Z ∧ Y is inf-schematic (respectively: inf-proper, and inf-closed embedding) as soon as W Y is schematic (respectively: proper, a closed embedding). If W dR ≃ Y dR , we say that ξ is a nil-isomorphism; in this case ξ ! is conservative.
QCoh(Y), which is t-exact for the natural t-structures on both sides. When Y is quasi-smooth (and much more generally when Y is eventually coconnective), we have that: 2.3.5. The object Ψ X (ω X ) is a shifted line bundle more generally when X is quasi-smooth (and even more generally when X is Gorenstein), see [22,Section 7.3]. The following two computations will be useful.
Lemma 2.3.6. Let N ⊂ g be the nilpotent cone, as introduced above, and N/G the quotient stack given by the coajoint action. Then Proof. Using (2.3) applied to Y = BG, we quickly obtain that Ψ BG (ω BG ) ≃ O BG [dim(BG)]. Next we claim that Ψ g/G (ω g/G ) ≃ O g/G : this can be proven directly, or by using [26, Vol. 2, Chapter 9, Proposition 7.3.4], which is a relative version of (2.3). Now, recalling the notation of Section 2.1.4, we see that N/G ≃ g/G × cG 0. In particular, the inclusion N/G ֒ g/G is a regular embedding of relative codimension equal to dim(c G ). Then the assertion follows from Grothendieck's formula, see [26, Vol. 2, Chapter 9, Section 7]. Lemma 2.3.7. Recall the derived scheme ΩV = Spec Sym(V * [1]) that appeared in (2.1). We have: In particular (but we will not need this), Proof. The derived scheme ΩV is proper (indeed, by definition, properness is checked at the level of classical truncations, and the classical truncation of ΩV is pt). Hence, by adjunction and by the fully faithfulness of Υ ΩV , we obtain: The assertion follows.
2.3.8. Let Y be a quasi-smooth derived stack. We regard IndCoh(Y) as an enlargement of QCoh(Y) by means of the functor Ξ Y . Let us recall the main tenets of the theory of singular support of ind-coherent sheaves on Y.
• The (classical) stack of singularities of Y is defined as It admits a G m -action defined by λ · (y, ξ) = (y, λξ).
2.3.9. For f : X Y a map of algebraic stacks, denote by Y ∧ X its formal completion and let be the canonical factorization of f . Since ′ f is a nil-isomorphism, we have an adjunction the associated functor, which is easily seen to be monoidal. Since i is a nil-isomorphism, it follows that i IndCoh * generates the target under colimits.
Now, let us turn to IndCohŇ(Ωǧ/Ǧ). Its monoidal structure is the unique one making ΨŇ ǧ * monoidal. To make sense of this, we need the following observation: Proof. Since IndCoh(Ωǧ/Ǧ) is generated under colimits by the essential image of i IndCoh * , it suffices to check the statement for G = i IndCoh * (V ), with V arbitrary. By base-change, we have that where π : Ωǧ/Ǧ pt/Ǧ is the projection and act ⊗ denotes the action of QCoh on IndCoh. The assertion follows from the fact that such action commutes with Ψ functors in general.

D-modules.
With the exception of Section 5 and Section 2.5, we will only encounter D-modules on indschemes and algebraic stacks of ind-finite type. Let us recall our main conventions in this case. Our main references are [27] and [17].
As mentioned earlier in Corollary 1.5.3, this t-structure is Zariski local: this means that the (co)connectivity of objects can be tested on Zariski open covers. A proof is given in [24,Lemma 7.8.7]).
We will use this result several times.
Let Y Z be a map of schemes of finite type. As explained in [12, Section 2.3.2], we can write QCoh(Z ∧ Y ) and IndCoh(Z ∧ Y ) as tensor products of DG categories, using the action of D-modules on quasicoherent and ind-coherent sheaves: If Z is smooth, then the functors Ξ Z , Ψ Z : QCoh(Z) IndCoh(Z) are equivalences, hence the same holds true for the functors   G m -rep weak , which we denote by C C ⇒ . We use the same notation for morphisms in G m -rep weak : namely, whenever φ : C D is a G m -equivariant functor, we denote by φ ⇒ : C ⇒ D ⇒ the associated one. We refer to [12, Section 2.1], which is a section dedicated to this topic.

If
A is a graded DG algebra, then Amod acquires a natural G m -action and (Amod) ⇒ ≃ A ⇒mod.
Here A ⇒ is defined as follows. Let us view objects of Rep(G m ) as graded complexes (M i,k , d) of vector spaces, where i refers to the cohomological grading, j (called weight) refers to the grading given by the G m -action, and d is a horizontal differential. Then A ⇒ is the algebra whose underlying graded complex is (A i+2k,k , d). Here is the main example to keep in mind: for V a finite dimensional vector space, regard The following fact will be used in the sequel. Proof. The key is to show that φ is conservative if and only if so is the induced functor φ Gm : C Gm D Gm of (weak) invariant DG categories. One direction is obvious: forgetting G m -invariance C Gm C is conservative, so the conservativity of φ implies that of φ Gm . The opposite implication follows from Gaitsgory's 1-affineness theorem, [25], which states that C can be recovered as C ≃ C Gm ⊗

Rep(Gm)
Vect. Now, to conclude the proof of the lemma, it suffices to observe that, by construction, there is a natural isomorphism 7 (φ ⇒ ) Gm ≃ φ Gm of DG functors.

The tempered unit of the spherical category
In this section, we start the proof of Theorem C. Our strategy is explained in Section 3.2: it relies on a certain explicit equivalence Sph G ≃ − D(Bun G (P 1 )), reviewed in Section 3.1, and on the computation of two Serre functors. The latter computations will be performed partly in Section 3.3 and partly in Section 4.
3.1. Geometric Langlands for P 1 . Recall that Bun G (X) denotes the stack of G-bundles on the curve X.
In this section, we focus solely on the case X = P 1 . Let us fix a point x ∈ X and choose a local coordinate t at x. The data of X, x and t are regarded as fixed until the end of Section 3.2. Our goal is to construct an equivalence The present section does not contain any new mathematics: the functor γ is closely related to the same named functor introduced by V. Lafforgue in [32] and we limit ourselves to fill in some details. In passing, we will review, again following [32], the role of γ in the proof of the geometric Langlands equivalence for X = P 1 .
] the loop group and the arc group of G at the point x. For any λ ∈ Λ, we let t λ be the corresponding -point of be the affine Grassmannian at x, equipped with the obvious right action of G(K). By [3], we have: In particular, the open embedding which is the inclusion of the locus of trivializable G-bundles. Consider the object j ! (ω BG ) ∈ D(Bun G (P 1 )).
3.1.3. The above expression of Bun G (P 1 ) as a quotient equips D(Bun G (P 1 )) with a left action 8 of Sph G . We denote this action by ⋆ and set Our current goal is to prove that γ is an equivalence and to provide a formula for its inverse. To get there, we need some preliminary results.
3.1.4. We will use a few notions from the theory of strong group actions on DG categories, see [8] and Section 2.5. Since G(K) acts on Gr G from the right, it also acts on the DG category D(Gr G ), again from the right. Our two DG categories of interest are invariant categories for the action of G(O) and G(R) on By forgetting invariance along G ֒ G(O) and G ֒ G(R), we obtain the two functors: 3.1.5. Consider the partially defined left adjoint to oblv G G(R) , to be denoted by Av . Such pushforward is well-defined on holonomic D-modules 9 , but potentially not on all D-modules. Nevertheless, we have: Proof. Clearly, the functor in question is well-defined on any holonomic object of Sph G . So, it suffices Using this, we can concoct an indscheme presentation Gr G ≃ colim n≥0 Z n , where each Z n is a finite dimensional scheme such that: • G(O) acts on Z n with finitely many orbits and via a finite-dimensional quotient group H n ; • the kernel of the quotient G(O) ։ H n is pro-unipotent.
Hn is generated under colimits by holonomic D-modules. Since H n acts on Z n with finitely many orbits, the assertion easily follows by devissage along the H n -orbit stratification.
), well-defined by the above lemma, is canonically isomorphic to γ.
Proof. By its very construction, γ is Sph G -linear for the natural Sph G -actions on both sides. Let us first show that Av It follows that Av , being the left adjoint of a Sph G -linear functor, is colax linear. This means that, for any S ∈ Sph G and any F ∈ D(Gr G ) G , there are natural arrows 10 that are possibly not isomorphisms. We will now use derived Satake, and its relation with the usual (underived) geometric Satake, to show that this colax module structure is actually a genuine Sph G -module 9 more precisely, by "holonomic D-module on Y" we mean a an object of D(Y) that is smooth-locally represented as a complex of D-modules with holonomic cohomology sheaves 10 together with the usual higher coherences present anytime one is dealing with higher categories; we will not dwell on them here structure. Referring to the discussion of Section 2.3.10, observe that the natural functor is monoidal and generates the target under colimits (indeed, the right functor is essentially surjective, while the left one is left adjoint to a conservative functor). Combined with derived Satake, we obtain a monoidal functor Rep(Ǧ) Sph G that generates the target under colimits. Thus, to prove the strictness of the colax Sph-linear structure of Av , it suffices to prove that the induced colax Rep(Ǧ)-linear structure is strict. The latter is clear: Rep(Ǧ) is a rigid monoidal DG category, so we can apply [26, Vol. 1, Chapter 1, Lemma 9.3.6].
We have established that γ and Av To see they are isomorphic, it suffices to show that they agree when evaluated on the unit ½ Sph G . A straightforward base-change yields . Then the right adjoint to γ can be expressed as the continuous functor By abstract nonsense, any functor with a continuous right adjoint preserves compact objects. Proof. In the first three steps, we show that γ is fully faithful; in the remaining ones, we show that it is essentially surjective.
Step 1. Denote by φ : Rep(Ǧ) Sph G the monoidal functor introduced in the proof above. As we know, φ generates the target under colimits. Hence, to show that γ is fully faithful, it suffices to prove that for all F ∈ Sph G and all V ∈ Rep(G) cpt . Note that φ(V ) is dualizable (indeed, compact objects of Rep (G) are dualizable and φ is monoidal); thus, up to replacing F with φ(V * ) ⋆ F, we may restrict to the case where V is the trivial G-representation. In other words, it suffices to prove that for arbitrary F ∈ Sph G .
Step 2. We would like to make sure that it is enough to verify the above claim in the special case where F ∈ Sph G is compact. This is not immediate, as ½ Sph G is not compact. 11 To get around this issue, recall from Section 2.4.4 that D(pt/G) is generated by a single compact object q ! ( ). Setting M := ι(q ! ( )) ∈ Sph G , we 11 To see that ½ Sph G is indeed non-compact, notice that it is the image of the non-compact object ω pt/G ∈ D(pt/G) under the (compact preserving) fully faithful functor see that M is compact and that ½ Sph G can be written as a colimit of copies of M. Hence, it suffices to prove that (3.4) for arbitrary F ∈ Sph G . Since γ preserves compact objects and M is compact, it is enough to prove (3.4) under the assumption that F is compact.
Next, to return to ½ Sph G , we use the fact that ω pt/G is a (non-compact) generator of D(pt/G), so that M can be expressed as a colimit of copies of ½ Sph G . Consequently, it is enough to prove that Step 3. The latter isomorphism for F compact is proven in [32,, by means of the contraction principle. Actually, V. Lafforgue proves that isomorphism for all F locally compact, see [1,Section 12.2.3] for the definition. It is easy to see that compact objects are in particular locally compact. This concludes the proof that γ is fully faithful.
Step 4. Let us now proceed to the essential the surjectivity of γ. In view of the already established fully faithfulness, it suffices to show that γ generates the target under colimits.
Step 5. Let us use again the fact that, when L is a connected algebraic group, D(pt/L) is generated under colimits by a single object. This observation, together with the Birkhoff decomposition, implies that D(Bun G (P 1 )) is generated under colimits by the collection where: • E λ is the G-bundle on P 1 corresponding to t λ ; • Aut(E λ ) := G(O) ∩ Ad t λ (G(R)) is its automorphism group; • j λ is the locally closed embedding • G λ is a generator of D(pt/Aut(E λ )).
We will prove, by induction on the height |λ|, that for each λ ∈ Λ dom the cocompletion of the essential image of γ contains one M λ as above. The base case is obvious: Step 6. Now recall that each λ ∈ Λ dom yields a quasi-compact open substack Bun Step 7. We first prove that is an object !-extended from the open substack Bun (≤λ) G . Using the expression Av is the natural action map. We need to show that m λ factors through Bun Step 8. Next, let i λ : pt/ Aut(E λ ) ≃ Bun we see that the square is cartesian. We deduce that G λ ≃ (f λ ) ! (ω pt/P λ ) up to a cohomological shift. Then the assertion follows from Lemma 2.4.5.
In the sequel, we will only need the second and the third equivalences (that is, Sat G and γ), not the first one.

3.2.
Overview of the proof of Theorem C. Let us explain our strategy of the proof of Theorem C.

3.2.1.
Step 1: the setup. The construction of the previous section and derived Satake show that the two functors are equivalences.

3.2.3.
Step 3: properness. Recall the notion of properness for DG categories, see Section 2.2.5. We claim that the DG categories appearing in (3.7) are proper. We will give two different proofs of this fact. The quickest proof, explained in Section 3.3, shows that D(Bun G (P 1 )) is proper. Another proof, discussed in Section 4.3, shows the properness of IndCohŇ(Ωǧ/Ǧ).

3.2.4.
Step 4: the Serre functor on the spectral side. Thanks to properness, it makes sense to consider the Serre functors of the three DG categories of (3.7). On theǦ-side, we will prove that Serre functors are obviously intertwined by equivalences of DG categories: in our case, derived Satake implies that (3.8)

3.2.5.
Step 5: the Serre functor on the automorphic side. On the automorphic side, we will show that where j : BG ֒ Bun G (P 1 ) is the open embedding induced by the trivial G-bundle. More generally, we wil check that a certain explicit functor T Bun G (P 1 ) (see below for the definition) equals the Serre functor on D(Bun G (P 1 )).
3.3. The Serre functor on the automorphic side. In this short section, we prove the Serre functor formula that appeared in (3.9). For this, we need to review a few facts on the pseudo-identity functor Ps-Id Y,! , also called Drinfeld's miraculous duality. We return to the general case of X of arbitrary genus and we let, as is standard, Bun G := Bun G (X The dualizability of D(Y) implies that functors from the dual DG category D(Y) ∨ to D(Y) correspond precisely to objects of D(Y × Y). We will consider two particularly interesting functors, called "pseudo- The first one is defined by the kernel (

These functors were introduced and discussed in detail in [16, Section 4] and [24, Sections 6-7].
Here are some relevant facts that we need: • If Y is quasi-compact, then Ps-Id Y, * is an equivalence; • if Ps-Id Y,! is an equivalence, Y is said to be miraculous; • Bun G is miraculous, see [20] for the proof; • Bun G can be exhausted by a sequence of miraculous quasi-compact opens, see [16, Lemma 4.5.7].

When Y is miraculous, we can consider the functor
If Y is quasi-compact and miraculous, then T Y is an equivalence. On the other hand, T Bun G is not at all Lemma 3.3.5. In the above notation, we have a canonical isomorphism Proof. By [16,Lemma 4.4.12], we know that Hence, it remains to show that as functors D(U ) ∨ D(Bun G ). Each of these two functors is given by a kernel in D(U × Bun G ). Tautologically, these two kernels are repsectively To conclude, observe that these two objects match by base-change.
3.3.6. Now assume again that X = P 1 . In this case, T BunG(P 1 ) is quite special. Indeed: Lemma 3.3.7. The DG category D(Bun G (P 1 )) is proper and T Bun G (P 1 ) is its Serre functor.
Proof. We already know that D(Bun G (P 1 )) is compactly generated, and Section 3.3.4 describes how compact objects look like. In view of that description, the properness of D(Bun G (P 1 )) is an immediate consequence of the following claim: for any quasi-compact open U ⊂ D(Bun G (P 1 )), the DG category D(U ) is proper. The Birkhoff decomposition guarantees that U has finitely many isomorphism classes of -points. Then the claim follows from the first part of [28, Theorem 2.1.5].
Next, let us show that Serre D(BunG(P 1 )) ≃ T Bun G (P 1 ) . We need to provide, for any F ∈ D(Bun G (P 1 )) and any G ∈ D(Bun G (P 1 )) cpt , a natural isomorphism From this, we obtain that or, equivalently, It remains to invoke (3.10).

The Serre functor of the spectral spherical category
In this section, we compute the Serre functor of IndCohŇ(Ωǧ/Ǧ) and complete the proof of the fourth step of Section 3.2. Since Langlands duality does not appear here, we will formulate our results for G, keeping in mind that they have been applied toǦ in Section 3.2.
Our main result is that the Serre functor of IndCoh N (Ωg/G) equals the temperization functor up to a cohomological shift. The proof hinges on a preliminary result, which might be of independent interest: the computation of the Serre functor of the DG category QCoh(N/G).
4.1. The nilpotent cone. Let N be the nilpotent cone associated to the group G. We show that the DG category QCoh(N/G) is proper and compute its Serre functor explicitly. Proof. For λ ∈ Λ dom a dominant weight, let V λ be the irreducible G-representation of highest weight λ. Denote by π : N/G pt/G the obvious projection. Since π is affine, π * is conservative and consequently the essential image of π * : Rep(G) ≃ QCoh(pt/G) QCoh(N/G) generates QCoh(N/G) under colimits. More precisely, the perfect objects

form a collection of compact generators of QCoh(N/G).
Thus, it suffices to show that Hom QCoh(N/G) (A λ , A µ ) is finite dimensional 12 for all λ, µ ∈ Λ dom . By adjunction and the projection formula, we have: where w 0 is the longest element of the Weyl group, so that V −w0(ν) ≃ V * ν . It follows that Hom Rep(G) (V ν , R) ≃ (V −w0(ν) ) T , which is indeed finite dimensional. Proof. Let us retain the notation of the previous lemma.
Step 1. By the defining property of the Serre functor, see Section 2.2.5, we just need to exhibit natural isomorphisms Hom QCoh(N/G) A ν , A λ ⊗ S 0 ≃ Hom QCoh(N/G) A λ , A ν * 12 the locution "being finite dimensional" applied to a complex of vector spaces is a shortcut for "having finite dimensional total cohomology" for all λ, ν ∈ Λ dom . Since A λ is dualizable with dual A −w0(λ) , it is easy to see that we may assume λ = 0.
Reasoning as in the previous lemma (using adjunction and the projection formula), it suffices to establish a functorial isomorphism where we have set R := H 0 (N, O) and R × := H * (N × , O), both regarded as G-representations in the natural way.
Step 2. Consider now the Springer resolution µ : T * (G/B) N. In view of [30, Theorem A], the canonical arrow is an isomorphism. 13 In other words, the nilpotent cone has rational singularities. Pulling back µ along j, we obtain a map µ × : T * (G/B) × N × , where T * (G/B) × is the complement of the zero section. Thus, by base-change, the canonical map is an isomorphism, too. Upon taking global sections, we get: where τ ≥m is the usual truncation functor associated to the standard tstructure on complexes of vector spaces. Then (4.2) simplifies as To prove our result, it suffices to show that R × has higher cohomology only in degree (2 dim n − 1), and that such higher cohomology decomposes (as a G-representation) as Step 4. Now observe that T * (G/B) × ≃ G × B (n − 0). In view of the B-equivariant isomorphism we obtain that In the above formulas, we have used H >0 (−) as a shortcut for τ ≥1 (H * (−)).
Step 5. Comparing (4.5) with the formula (4.1) obtained from Kostant's theorem, we deduce that Thus, (4.4) simplifies as To conclude our proof, we need to prove the above formula, as well as the fact that the only higher cohomology of R × occurs in degree 2 dim(n) − 1. By looking at (4.6), it is clear that both claims boil down to proving that We prove this in the next two steps.
Step 6. By [28,Section 1.3.3], the Serre functor for Rep(B) equals the functor − ⊗ Λ dim n n[dim n]. Recall also that the defining property of Serre C requires the second argument to be compact, see Remark 2.2.6. In our case, we proceed as follows: Step 7. Thanks to (4.7), we know that Hom Rep(B) (V −w0(ν) , Sym n * ) is finite dimensional: in particular, we can replace the direct sum above with a direct product. Hence, where the last step used (4.7) again. This concludes the proof.
Proof. Denote by R ⇒ and R ×,⇒ the sheared versions of R and R × . Arguing as before, it suffices to construct, for each ν ∈ Λ dom , an isomorphism To determine both sides explicitly, we need to decompose both R and R × as (G × G m )-representation. In view of (4.5), the (G × G m )-decomposition of R is the tautological one coming from the grading of Sym n * : It follows that so that the RHS of (4.9) can be rewritten as Similarly, in view of (4.6), the ( with the expression in parentheses of weight (−m − dim n). Hence, From this, we see that the LHS of (4.9) equals Comparing this equation with (4.10), it remains to exhibit, for each ν and m, an isomorphism Such isomorphism is the one induced by the Serre functor of Rep(B).

4.3.
The main Serre computation. In this section, we finally show that the Serre functor on IndCoh N (Ωg/G) equals the temperization functor up to a cohomological shift: this is the content of Theorem 4.3.11. A key ingredient will be the following pair of Koszul duality equivalences.  where N is regarded as a subscheme of g * . Remark 4.3.2. As the proof below shows, on the RHS of (4.11) and (4.12) we could replace IndCoh by QCoh.
However, the functoriality of ind-coherent sheaves is more convenient when dealing with formal completions: the usage of Lemma 4.3.1 in the proof of Theorem 4.3.11 will make this clear (see also Section 2.3).
Proof. We will only prove the first assertion; for the second one, follow the exact same steps using the inclusion pt ֒ g * of the origin instead of the inclusion of the nilpotent cone. Recall that we have been abusing notation: the DG category IndCoh N (Ωg/G) should be more properly denoted by IndCoh N/G (Ωg/G), since possible singular supports of ind-coherent sheaves live inside g * /G, and in the case at hands we are looking at the subset N/G ⊂ g * /G. For clarity, in this proof, we will use the more precise notation. Since singular support can be computed smooth-locally ([1, Section 8]), we have: where the two maps in the fiber product are the natural pullback and the inclusion Ξ N֒ g * , respectively. Now, we use the Koszul duality equivalences of [1,Proposition 12.4.2]: The first of these two equivalences transforms singular support on the LHS into set-theoretic support on the RHS. This is proven in [1, Section 9.1, especially 9.1.6 and Corollary 9.1.7]; see also [12, Section 2.2] for a slightly different point of view. In particular, where(g * ) ∧ N denotes the formal completion of the closed embedding N ֒ g * . Since g * and g * /G are smooth, we can identify quasi-cohererent sheaves with ind-coherent sheaves on them, so that IndCoh(Ωg) ≃ IndCoh(g * ) ⇒ ;

All in all, Koszul duality yields
which is in turn equivalent to Let q : g * g * /G be the quotient and j : g * − N ֒ g * the open embedding complementary to the nilpotent cone. To conclude the proof, it remains to show that the G m -equivariant functor (4.13) IndCoh induced by the inclusion IndCoh((g * /G) ∧ N/G ) ֒ IndCoh(g * /G) is an equivalence. This is clear: both sides of (4.13) identify with the full subcategory of IndCoh(g * /G) spanned by those objects F such that j ! q ! F ≃ 0. 4.3.3. In the above discussion, N was naturally viewed as a subscheme of g * . However, to use the Serre computations of the previous sections, we prefer to realize N as a closed subscheme of g. Thus, we fix once and for all a G-equivariant identification g ≃ g * . Incorporating this into the above Koszul dualities, we obtain equivalences (4.14) IndCoh where now the shearing is such that (Sym g * ) ⇒ ≃ Sym(g * [−2]). To guide the reader through the shearings that will follow, it suffices to remember that dual Lie algebras (g * and n * ) have weight 1. In particular, we are in agreement with the shearing of Section 4.2.
4.3.4. Let ι 0 : (g/G) ∧ pt/G (g/G) ∧ N/G be the map induced by the inclusion of the origin 0 ∈ N. This map is an inf-closed embedding (see Section 2.3.3), and thus it yields the adjunction These two adjoint functors are both G m -equivariant with respect to the two natural G m -actions on both sides, hence they can be sheared.

4.3.5.
We have seen above that Koszul duality interchanges singular support and the usual set-theoretic support. Consequently, under the above equivalences, the comonad goes over to the comonad 4.3.6. Now recall that IndCoh N (Ωg/G) is proper: in Section 3 we showed that IndCoh N (Ωg/G) is equivalent to D(BunǦ(P 1 )), and in Section 3.2.3 we proved that the latter DG category is proper. We will nevertheless give a more direct proof: this will help us progress with the computation of the Serre functor of IndCoh N (Ωg/G). Using Koszul duality to identify IndCoh N (Ωg/G) ≃ IndCoh((g/G) ∧ N/G ) ⇒ , our plan will be to prove the properness of the latter DG category. The starting point is Lemma 4.3.8, in which we exhibit a convenient collection of compact generators.
form a collection of compact generators of IndCoh((g/G) ∧ N/G ) ⇒ .
Proof. As mentioned earlier, the objects (π * ) ⇒ (V λ ) compactly generate QCoh(N/G) ⇒ . Hence, it suffices to prove that the functor which is evidently continuous. Let us show conservativity. Observe first that the functor is an equivalence: this follows from Section 2.4.3 and descent (to take care of the stackyness). Thus, it suffices In general, Υ intertwines * -pullbacks of quasi-coherent sheaves with !-pullbacks of ind-coherent sheaves; in our case, this implies that The quasi-smoothness of N/G guarantees that Ψ N/G • Υ N/G is an equivalence: it is the functor of tensoring with a shifted line bundle, see [22,Section 7] or Lemma 2.3.6. It remains then to prove that is conservative. This can be seen in various ways. For instance, using the equivalence Υ (g/G) ∧ N/G again, the assertion is equivalent to the conservativity of The latter is clear: f ! is conservative as mentioned earlier, while Υ N/G is even fully faithful.  (4.16) with the following similar object: We claim that these two objects differ by a cohomological shift: precisely, To see this, observe that the last step being an application of Lemma 2.3.6. Proof. Since the F λ above form a collection of compact generators, it suffices to prove that is finite dimensional for any pair λ, µ ∈ Λ dom . Thanks to the fact that f is a nil-isomorphism, we obtain by adjunction that where U(T (N/G)/(g/G) ) is the universal envelope of the Lie algebroid T (N/G)/(g/G) T N/G . See [26, Volume II, Chapter 8] for these notions: in particular, U(T (N/G)/(g/G) ) is a monad acting on IndCoh(N/G), and relative tangent complexes are regarded as ind-coherent sheaves 15 .
To proceed, let us compute the relative tangent complex T (N/G)/(g/G) and then its universal envelope. We will use the isomorphism where c G ≃ Spec(Sym(g * ) G ) is isomorphic, after our G-equivariant identification g ≃ g * , to the Chevalley space of Section 2.1.3. Since the fiber product on the RHS is derived (as well as classical, in view of the flatness of g c G ), we can compute the relative tangent complex algorithmically: Consequently, the functor underlying the monad U(T (N/G)/(g * /G) ) is just the functor of tensoring with the graded vector space Sym(T cG,0 [−1]). Using the fully faithfulness of Ξ N/G and turning on the shearing, we conclude that Thanks to the already established properness of QCoh(N/G) ⇒ , it remains to check that Sym(T cG,0 [−1]) ⇒ is finite dimensional. For this, we need to recall the weight decomposition of T cG,0 (or, which is the same, of c G , since the latter is a vector space). We have where z G = Lie(Z G ) is in weight −1, each l di is a line in weight −d i (the negative of the i-th fundamental invariant of the group), and r G is the semisimple rank. It follows that This is an exterior algebra with finitely many generators, hence in particular finite dimensional. Proof. Consider again the natural map ι 0 : (g/G) ∧ pt/G (g/G) ∧ N/G . By Koszul duality, the statement of the theorem is equivalent to the fact that the Serre functor of the DG category IndCoh((g (G)]. This is what we will prove below, in several steps.
Step 1. For any pair (λ, µ) of dominant weights, we need to provide an isomorphism Using Remark 4.3.9, this is equivalent to providing an isomorphism The reason for replacing F ν with F ′ ν is that the primed expressions are more amenable to the base-change manipulations of Step 2 below.
We have already computed the RHS in the lemma above. Taking that calculation into account and setting it remains to prove that (4.19) Hom Step 2. Let us manipulate the LHS of (4.19). Base-change gives are the natural maps. Then where abusing notation we are considering V λ and V µ as objects of IndCoh(BG) by means on the equivlaence Υ BG . Consider now the fiber square Step 3. The convolution right action of Ω(c G ) := pt × cG pt on N/G = g/G × cG pt induces the isomorphism From this point of view, the functor (q 2 ) IndCoh * (q 1 ) ! is the functor of acting with ω ΩcG on objects of IndCoh((N/G) ∧ BG ). Next, notice that the action and the projection ( In other words, the group Ω(c G ) acts trivially on objects in the essential image of α ! . Hence, Step 4. The above formula needs to be applied in its sheared version: this amounts to apply ⇒ to the functors and to the graded vector space (p ΩcG ) IndCoh * (ω ΩcG ). By Lemma 2.3.7, we have: which is vector space W defined earlier. We obtain that Step 5. It is clear that where the last step used Corollary 4.2.3. Hence, the LHS of (4.19) equals Step 6. On the other hand, the RHS of (4.19) equals Hence, it remains to show that W * ≃ W [− dim G]. For this, recall that with each l di a line. Now, the classical formula immediately yields the claim.

Proof of Theorem A
In this section, we deduce Theorem A from the combination of Theorem B and Theorem C. In more detail: using the expression of ½ temp Sph G given by Theorem C, we obtain an explicit formula, see (5.6), for the Hecke action of ½ temp Sph G on D(Bun G ). In particular, we obtain an explicit formula for the object ½ temp Sph G ⋆ ω BunG . We then show that the simplest case of Theorem B implies that ½ temp 5.1.1. We claim that the DG category D(G\G(R)/G) is naturally monoidal under convolution. Naively, the convolution product is defined as the pull-push along the natural correspondence where p is the obvious projection and m is the arrow induced by the multiplication of G(R). However, since m is not schematic (but only ind-schematic), specifying the pushforward to be used requires some care.

5.1.2.
To address this complication, we first observe that D(G\G(R)/G) is tautologically comonoidal: this structure is induced by the very same correspondence as above, just read from right to left. In this case, the above issue about the pushforward does not arise: the functor p * ,dR is well-defined since p is schematic (it is even smooth). Then, to turn this comonoidal structure into a monoidal one, we need: invertible matrices whose entries are polynomials in t −1 of degrees ≤ d. It is easy to see that the adjoint action of G on G(R) 1 preserves each Y d , so that 5.1.5. The above self-duality allows us to dualize the comonoidal structure to obtain the monoidal structure we were looking for. Concretely, the convolution product is defined as the pull-push along (5.1), but the pushforward to be used is m * ,ren , the renormalized de Rham pushforward along m (see [17,Section 9.3]), which is by definition the dual of m ! .
5.1.6. In general, the renormalized de Rham pushforward along a map h : X = colim i∈I X i Y of indschemes (of ind-finite type) admits an explicit description. Indeed, unraveling the self-dualities as in the proof above, one easily checks that h * ,ren : Y is the natural (schematic) map. This functor is different from h ! in general; indeed, the latter is only partially defined and, when defined, it is given by the formula However, the two functors agree when all the h i are proper, that is, when the map h : X Y is ind-proper.
Remark 5.1.7. As a particularly simple example, consider the map p X : X pt, where X ≃ colim i∈I X i is as above. Then (p X ) * ,ren {F i } i∈I ≃ colim i∈I (p Xi ) * ,dR (F i ).
Thus, (p X ) * ,ren yields a quick definition of the Borel-Moore homology of X via the formula and its pullback functor f ! : Sph G D(G\G(R)/G). According to Theorem C, Using the self-duality of D(G\G(R)/G) and that of Sph G (see Remark 5.1.4), we can express ½ temp Sph G slightly differently as Indeed, we have: Lemma 5.1.9. With the above notation, f * ,ren : , while j ! is easily seen to be dual to j * ,dR .

5.2.
The Hecke action of the tempered unit. Let us fix x ∈ X throughout and consider the Hecke action of Sph G on D(Bun G ) at x. We remind the reader that here X is a smooth projective curve of arbitrary genus, and that Bun G = Bun G (X). In this section, we provide an explicit formula for the Hecke action of where the pushforward along act is the !-pushforward. As act is ind-proper, act ! agrees with the renormalized pushforward act * ,ren . 5.2.2. Now we claim that the DG category D(G\ Bun G ) admits an action of D(G\G(R)/G). This action is given by convolution, however we have the same issue as in Section 5.1.1 together with a new issue: G\ Bun G is of infinite type. To avoid these issue, and to place this action on the same footing as the Hecke action above, let us proceed using the language of categorical group actions.

5.2.
3. Let C be a DG category equipped with a left action of G(K). By formal nonsense, Sph G coacts on G(O) C from the left; we denote by the coaction functor. Now, we exploit the self-duality of Sph G to turn this coaction into an action. Explicitly, given S ∈ Sph G and c ∈ G(O) C, the formula for this action is where ·, · is the evaluation functor

5.2.4.
Let us now repeat the same construction with the pair (G(R), G) in place of (G(K), G(O)). We obtain that, given C acted upon by G(R), the comonoidal DG category D(G\G(R)/G) coacts on G C and this coaction can be turned into an action. We denote the coaction by coact G G(R),C and the action by * .
In formulas, for F ∈ D(G\G(R)/G) and d ∈ G C, we have where ·, · is the evaluation functor ½ temp Proof. We use (5.5) and the definition of the ⋆ action to write ½ temp By duality, , where, abusing notation, ·, · denotes the evaluation functor A straightforward diagram chase shows that Since oblv G G(O) is fully faithful (as a consequence of the pro-unipotence of ker(G(O) ։ G)), we obtain It follows that 5.2.6. Now let C = D ! ( Bun G ), where Bun G is defined as in Section 5.2.1 and D ! as in [38,8]. The left G(K)-action on Bun G yields a left G(K)-action on C. Unraveling the definition, the resulting Sph G -action whenever G has semisimple rank ≥ 1. In view of (5.6), this amounts to showing that In fact, we will prove that is already the zero object.

Consider the diagram
where α is the map induced by the G(R)-action on Bun G , and π the projection onto the second component. The functor is equivariant for the obvious "balanced" G-action on the source and the left G-action on the target. Abusing notation, we denote by π * ,ren the induced functor at the level of G-invariant categories. Unraveling the constructions, we have

5.3.3.
We need to show that the latter object is zero. It is enough to do so after applying the conservative functor oblv G : D(G\ Bun G ) D( Bun G ). But then we tautologically have oblv G • π * ,ren ω G\G(R)× G BunG ≃ (p G\G(R) ) * ,ren ω G\G(R) ⊗ ω BunG , so it suffices to prove that the vector space (p G\G(R) ) * ,ren ω G\G(R) vanishes.
imply that Tensoring with a nonzero vector space, in our case H BM (G) ≃ H * (G)[2 dim(G)], is conservative. So it remains to prove that H BM (G(R)) ≃ 0: this vanishing statement is exactly the content of Theorem B for the affine curve A 1 .

Proof of Theorem E
In the previous part of the paper, we have shown that Theorem A follows from the simplest case of Theorem B. In this section, we deduce Theorem B from Theorem D, and then prove the latter in some special cases. Proof. Let C be a DG category equipped with a t-structure. The t-structure is said to be left-complete if the functor is an equivalence. When the t-structure on C is left-complete, it is obvious that C ≤−∞ ≃ 0: in other words, there are no nontrivial infinitely connective objects. It is easy to see that the usual t-structure on Vect is left-complete, so that, in particular, Vect ≤−∞ ≃ 0. Hence, as by Remark 5.1.7, it suffices to show that (p Y ) * ,ren : D(Y) Vect is right t-exact. Let Y ≃ colim k∈K Y k be an indscheme presentation, with each Y k an affine scheme. As discussed in Sections 2.4.1 and 2.4.2, the natural t-structure on D(Y) is defined by requiring that D(Y) ≤0 be generated under colimits by the objects of the form (i k ) * ,dR (ind Y k (C)), for all k and all C ∈ Coh(Y k ) ≤0 .
Thus, it suffices to prove that for any such C. Let us simplify the composition Observe first that (p Y ) * ,ren •(i k ) * ,dR ≃ (p Y k ) * ,dR : indeed, the dual to (p Y ) * ,ren by the definition of the renormalized pushforward. It follows that the functor (6.2) is the ind-coherent pushforward along p Y k ; furthermore, as C is coherent, we obtain that where we have used the fact that Ψ intertwines quasi-coherent and ind-coherent pushforwards. Then the claim of (6.1) is evident: thanks to the affineness of Y k , the functor (p Y k ) * : Coh(Y k ) Vect is t-exact.
6.2. Proof of Theorem E. Let G = SL n . In this case, G is a closed subscheme of A n 2 determined by the vanishing of one equation of degree n. Thus, Theorem D for G = SL n is a simple case of the following general result, which went under the name of Theorem E in the introduction. Proof. Let X be the smooth compactification of Σ, obtained by adding h ≥ 1 points at infinity. Denote by D ∞ = X − Σ the union of such points and by g the genus of X.
Step 1. Consider the following indscheme presentation where we have set Denoting by i d : Y d Y [Σ] the tautological closed embeddings, we deduce that Since each pushforward (i d ) * ,dR is right t-exact by construction, it suffices to find a divergent sequence Step 2. Assume from now on that d > max(0, (2g − 2)/h). In this case, the scheme A N [Σ] ≤d is a vector space of dimension N (dh + 1 − g). Let us compute the number of equations needed to specify Step 3. Now let Since N − n > 0 and h > 0 by assumption, C d goes to infinity with d. On the other hand, the general lemma Proof. We proceed by induction on p. In case p = 0, we have For the sake of completeness, let us give a proof of this well-known statement. By [ . To see the latter fact, use Grothendieck duality for P n to prove that (p P n ) !,IndCoh ( ) ≃ K P n [n], and then further pullback along A n ֒ P n . Assume now that p > 0. We can write Z ≃ Z ′ × A 1 pt, for an affine scheme Z ′ of the form A m × A p−1 pt. Observe that i * ,dR (ω Z ) sits in the fiber sequence Moreover, j ! and j * ,dR are both t-exact (the latter because j is affine), so that Combined with the bound for ω Z ′ , this yields the assertion.

Proof of Theorem D in general
In this section we prove Theorem D: given a non-abelian connected reductive group G and a smooth affine curve Σ, the dualizing sheaf ω G[Σ] is infinitely connective for the natural t-structure on D(G[Σ]).  7.1.4. Before proceeding, it is convenient to introduce some more notation. Let fSet be the 1-category of nonempty finite sets and surjective maps between them. Given I ∈ fSet and given x ∈ Σ I (S) an I-tuple of maps from S to Σ, we denote by D x ⊂ Σ S the incidence divisor of x. be the indscheme whose S-points are those pairs x ∈ Σ I (S), φ : Σ S G with the following two properties: • the members of the I-tuple x have pairwise disjoint graphs in Σ S ; • φ sends Σ S − D x to G • .
In Section 7.4, we will show that the Theorem 7.1.3 is a consequence of the statement below. : its S-points are pairs (x ∈ Σ(S), φ : Σ S G) such that the restriction of φ to Σ S − D x factors through G • . Even more particularly, fix x ∈ Σ( ) and consider the indscheme 7.2.4. We will use a similar notation and terminology for elements of G • ((t)) λ . Namely, we represent them as triples φ = (φ − , φ T , φ + ) according to (7.2) and say that φ has poles bounded by n if so do all the Laurent series comprising φ ± . Lemma 7.2.5. For any λ ∈ Λ, there exists a number e(λ) ∈ N with the following property: if φ = (φ − , φ T , φ + ) ∈ G • ((t)) λ is contained in G • ((t)) λ × G((t)) G[ [t]], then φ ± has poles at x bounded by e(λ).
Proof. For G = GL 2 , this lemma is obvious. Indeed, letting R be a test ring, choose f, g ∈ R((t)) and m, n ∈ Z arbitrarily. If the element A similar computation, left to the reader, proves the lemma for G = GL d , with d ≥ 3. Now let G be arbitrary; we will reduce to the case of GL d as follows. Pick a faithful representation ρ : G ֒ GL(V ).
By choosing an appropriate ordered basis of V consisting of weight vectors, we can assume that ρ(N ) (respectively: ρ(N − ), ρ(T )) is contained in the subgroup of upper triangular (respectively: lower triangular, diagonal) matrices of GL(V ) ≃ GL dim(V ) . In particular, ρ sends the big cell of G to the big cell of GL dim(V ) . Now pick φ = (φ − , φ T , φ + ) ∈ G • ((t)) λ × G((t)) G[ [t]]. The case of GL d implies that ρ(φ ± ) both have poles at x bounded by some e ∈ N. It follows easily that the same is true, possibly with a different bound e ′ ∈ N, for the φ ± themselves. 7.2.9. Next, note that As usual, let X be the smooth compactification of Σ and D ∞ the divisor at infinity, of cardinality h ≥ 1.
We have: In the same way, T [Σ − x] λ is an indscheme of ind-finite type, again realized as a colimit over the poset of natural numbers. It remains to apply the following general result. Proof. We first show that ω A ∞ ×T is infinitely connective. By assumption, each T m is a scheme of finite type. It follows that there exists some N m ≥ 0 such that ω Tm ∈ D(T m ) ≤Nm : to prove this, recall that the t-structure is Zariski local, argue as in Lemma 6.2.2 and then use the quasi-compactness of T m . Up to replacing each N m with a larger number, let us assume that the sequence m N m is increasing and divergent. Then we have the indscheme presentation induced by exterior tensor product, which is valid because D(A 2Nm ) is dualizable. The t-structure on the LHS corresponds to the tensor product t-structure on the RHS (the latter is defined by declaring that connective objects of the tensor product are generated under colimits by tensor products of connective objects). Since the dualizing sheaf ω A 2Nm ×Tm corresponds to ω A 2Nm ⊠ ω Tm and ω A 2Nm ∈ D(A 2Nm ) ≤−2Nm , we obtain that Recall that each (i m ) * ,dR is right t-exact by construction; then, as N m goes to infinity with m, we have proven that ω A ∞ ×T is infinitely connective. Now let i : Y ֒ A ∞ × T be the obvious closed embedding. In view of ω Y ≃ i ! (ω A ∞ ×T ), it suffices to show that i ! is right t-exact up to a finite shift. This boils down to proving that, for any scheme Z of finite type mapping to Y , the !-pullback along Z × Y pt ֒ Z is right t-exact up to a shift that is independent of Z. This follows exactly as in Lemma 6.2.2: first by the Zariski-local nature of the t-structure, we can replace Y with an affine open Y ′ containing pt; then we write pt ∈ Y ′ as the zero locus of finitely many equations in Y ′ . 7.3. Moving points. Now we adapt the argument of Section 7.2 to prove Proposition 7.1.6. The idea is the same, but the notation heavier. that forgets the datum of y. In [21,Section 4.6.4], it is proven that such a map is ind-proper for any map J I of nonempty finite sets. Since our ξ [A I] is obtained from f source (J I) by setting J = A and base-changing, it is ind-proper too. The contractibility of the fibers of ξ follows exactly as in [21,Section 4.6]: it boils down to the contractibility of the Ran space of a smooth affine curve, which in turn is due to [5]. Proof. In view of the above formula, it suffices to verify that each functor (ξ [A I] ) ! is right t-exact up to a finite shift. We can write ξ [A I] as the composition of two obviously defined maps: It is proven [21, Section 4.6.4] that the leftmost map is an ind-closed embedding, hence it follows from the definition of the t-structure that the associated !-pushforward is right t-exact. On the other hand, the !-pushforward along the second map is right t-exact up to a shift by |I| = dim(X I ).
7.4.10. Let us come back to our group case. The big cell G • is a basic open subset of G: the explicit map realizing this is given in [29]. Thus, the above general results apply and, in particular, the following statement implies Theorem 7.1.3. where I ∈ fSet (in particular, I = ∅). This proves that Proposition 7.1.6 implies Proposition 7.4.11.