A New Approach

to the Geometric Satake Equivalence

Timo Richarz

Received: February 20, 2013

Communicated by Otmar Venjakob

Abstract. I give another proof of the geometric Satake equivalence from I. Mirković and K. Vilonen [16] over a separably closed field. Over a not necessarily separably closed field, I obtain a canonical construction of the Galois form of the full L-group.

2010 Mathematics Subject Classification: 14M15, 20G05

1 Introduction

Connected reductive groups over separably closed fields are classified by their root data. These come in pairs: to every root datum, there is associated its dual root datum and vice versa. Hence, to every connected reductive group G, there is associated its dual group Ĝ. Following Drinfeld’s geometric interpretation of Langlands’ philosophy, Mirković and Vilonen [16] show that the representation theory of Ĝ is encoded in the geometry of an ind-scheme canonically associated to G as follows. Let G be a connected reductive group over an arbitrary field F. The loop group LG is the fpqc-sheaf associated with group functor on the category of F -algebras

LG : R ⟼ G(R((t))).

The positive loop group L^{+}G is the fpqc-sheaf associated with the group functor

L^{+}G : R ⟼ G(R[[t]]).

Then L^{+}G ⊂ LG is a subgroup functor, and the fpqc-quotient Gr_{G} = LG/L^{+}G is called the affine Grassmannian. It is representable by an ind-projective indscheme (= inductive limit of projective schemes). Now fix a prime ℓ ≠ char(F), and consider the category P_{L^{+}G}(Gr_{G}) of L^{+}G-equivariant ℓ-adic perverse sheaves on Gr_{G}. This is a ℚ̄_{ℓ}-linear abelian category.

First assume that F is separably closed. Then the simple objects in P_{L^{+}G}(Gr_{G}) are as follows. Fix T ⊂ B ⊂ G a maximal torus contained in a Borel. For every cocharacter μ, denote by

\overline{O}_{μ} =^{def} \overline{L^{+}G · t^{μ}}

the reduced L^{+}G-orbit closure of t^{μ} ∈ T (F ((t))) inside Gr_{G}. Then \overline{O}_{μ} is a projective variety over F. Let IC_{μ} be the intersection complex of \overline{O}_{μ}. The simple objects of P_{L^{+}G}(Gr_{G}) are the IC_{μ}’s where μ ranges over the set of dominant cocharacters X_{+}^{∨}. Furthermore, the category P_{L^{+}G}(Gr_{G}) is equipped with an inner product: to every A_{1}, A_{2} ∈ P_{L^{+}G}(Gr_{G}), there is associated a perverse sheaf A_{1} ⋆ A_{2} ∈ P_{L^{+}G}(Gr_{G}) called the convolution product of A_{1} and A_{2} (cf. §3 below). Denote by

ω(-) =^{def} ⨁_{i∈ℤ} R^{i}Γ(Gr_{G}, -) : P_{L^{+}G}(Gr_{G}) ⟶ Vecℚ̄_{ℓ}

the global cohomology functor with values in the category of finite dimensional ℚ̄_{ℓ}-vector spaces. Fix a pinning of G, and let Ĝ be the Langlands dual group over ℚ̄_{ℓ}, i.e. the reductive group over ℚ̄_{ℓ} whose root datum is dual to the root datum of G. Let T̂ be the dual torus, i.e. the ℚ̄_{ℓ}-torus with X^{∗}(T̂) = X_{∗}(T).

Theorem 1.1. (i) The pair (P_{L^{+}G}(Gr_{G}), ⋆) admits a unique symmetric monoidal structure such that the functor ω is symmetric monoidal. (ii) The functor ω is a faithful exact tensor functor, and induces via the Tannakian formalism an equivalence of tensor categories

(P_{L^{+}G}(Gr_{G}), ⋆) ⟶^{≃} (Repℚ̄_{ℓ}(Ĝ), ⊗)
A ⟼ ω(A),

which is uniquely determined up to inner automorphisms by T̂ by the property that ω(IC_{μ}) is the irreducible representation of highest weight μ.

In the case F = ℂ, this reduces to the theorem of Mirković and Vilonen [16] for coefficient fields of characteristic 0. The drawback of our method is the restriction to ℚ̄_{ℓ}-coefficients. Mirkovic and Vilonen are able to establish a geometric Satake equivalence with coefficients in any Noetherian ring of finite global dimension (in the analytic topology). I give a proof of the theorem over any separably closed field F using ℓ-adic perverse sheaves. My proof is different from the one of Mirković and Vilonen. It proceeds in two main steps as follows. In the first step I show that the pair (P_{L^{+}G}(Gr_{G}), ⋆) is a symmetric monoidal category. This relies on the Beilinson-Drinfeld Grassmannians [2] and the comparison of the convolution product with the fusion product via Beilinson’s construction of the nearby cycles functor. Here the fact that the convolution of two perverse sheaves is perverse is deduced from the fact that nearby cycles preserve perversity. The method is based on ideas of Gaitsgory [7] which were extended by Reich [19]. The constructions in this first step are essentially known, my purpose was to give a coherent account of these results.

The second step is the identification of the group of tensor automorphisms \underline{Aut}^{⋆}(ω) with the reductive group Ĝ. I use a theorem of Kazhdan, Larsen and Varshavsky [10] which states that the root datum of a split reductive group can be reconstructed from the Grothendieck semiring of its algebraic representations. The reconstruction of the root datum relies on the PRVconjecture proven by Kumar [11]. I prove the following geometric analogue of the PRV-conjecture.

Theorem 1.2 (Geometric analogue of the PRV-Conjecture). Denote by W = W (G, T) the Weyl group. Let μ_{1}, …, μ_{n} ∈ X_{+}^{∨} be dominant coweights. Then, for every λ ∈ X_{+}^{∨} of the form λ = ν_{1} + … + ν_{k} with ν_{i} ∈ W μ_{i} for i = 1, …, k, the perverse sheaf IC_{λ} appears as a direct summand in the convolution product IC_{μ_{1}} ⋆ … ⋆ IC_{μ_{n}}.

Using this theorem and the method in [10], I show that the Grothendieck semirings of P_{L^{+}G}(Gr_{G}) and Repℚ̄_{ℓ}(Ĝ) are isomorphic. Hence, the root data of \underline{Aut}^{⋆}(ω) and Ĝ are the same. This shows that \underline{Aut}^{⋆}(ω) ≃ Ĝ uniquely up to inner automorphisms by T̂. If F is not neccessarily separably closed, we are able to apply Galois descent to reconstruct the full L-group. Fix a separable closure F̄ of F, and denote by Γ = Gal(F̄/F) the absolute Galois group. Let ^{L}G = Ĝ(ℚ̄_{ℓ}) ⋊ Γ be the Galois form of the full L-group with respect to some pinning.

Theorem 1.3. The functor A ↦ ω(AF̄) induces an equivalence of abelian tensor categories

(P_{L^{+}G}(Gr_{G}), ⋆) ≃ (Repc ; ℚ̄_{ℓ}(^{L}G), ⊗),

where Repc ; ℚ̄_{ℓ}(^{L}G) is the full subcategory of the category of finite dimensional continuous ℓ-adic representations of ^{L}G such that the restriction to Ĝ(ℚ̄_{ℓ}) is algebraic.

Theorem 1.3 may be seen as an extension of Theorem A.12 in my joint work with Zhu [20]. In [loc. cit.] we consider the category Repℚ̄_{ℓ}(^{L}G) of algebraic representations of ^{L}G regarded as a pro-algebraic group over ℚ̄_{ℓ}. Then Repℚ̄_{ℓ}(^{L}G) is a full subcategory of Repc ; ℚ̄_{ℓ}(^{L}G), and we identify the corresponding subcategory of P_{L^{+}G}(Gr_{G}) explicitly. My method of proof here is similiar to the method used in [20]. Besides some general Tannakian formalism, the key ingredient is the identification of the Γaction on Ĝ obtained via the geometric Satake equivalence over F̄. It differs from the usual action by a twist with the cyclotomic character, cf. Proposition 6.6 below. The structure of the paper is as follows. In §2 we introduce the Satake category P_{L^{+}G}(Gr_{G}). Appendix A supplements the definition of P_{L^{+}G}(Gr_{G}) and explains some basic facts on perverse sheaves on ind-schemes as used in the paper. In §3-§4 we clarify the tensor structure of the tuple (P_{L^{+}G}(Gr_{G}), ⋆), and show that it is neutralized Tannakian with fiber functor ω. Section 5 is devoted to the identification of the dual group. This section is supplemented by Appendix B on the reconstruction of root data from the Grothendieck semiring of algebraic representations. The reader who is just interested in the case of an algebraically closed ground field may assume F to be algebraically closed throughout §2-§5. The last section §6 is concerned with Galois descent and the reconstruction of the full L-group.

Acknowledgement 1. First of all I thank my advisor M. Rapoport for his steady encouragement and advice during the process of writing. I am grateful to the stimulating working atmosphere in Bonn and for the funding by the Max-Planck society.

2 The Satake Category

Let G a connected reductive group over any field F. The loop group LG is the fpqc-sheaf associated with the group functor on the category of F -algebras

LG : R ⟼ G(R((t))).

The positive loop group L^{+}G is the fpqc-sheaf associated with the group functor

L^{+}G : R ⟼ G(R[[t]]).

Then L^{+}G ⊂ LG is a subgroup functor, and the fpqc-quotient Gr_{G} = LG/L^{+}G is called the affine Grassmannian (associated to G over F).

Lemma 2.1. The affine Grassmannian Gr_{G} is representable by an ind-projective strict ind-scheme over F. It represents the functor which assigns to every F - algebra R the set of isomorphism classes of pairs (F, β), where F is a G-torsor over Spec(R[[t]]) and β a trivialization of F[\frac{1}{t}] over Spec(R((t))).

We postpone the proof of Lemma 2.1 to Section 3.1 below. For every i ≥ 0, let G_{i} denote i-th jet group, given for any F -algebra R by G_{i} : R ↦ G(R[t]/t^{i+1}). Then G_{i} is representable by a smooth connected affine group scheme over F and, as fpqc-sheaves,

L^{+}G ≃ \underleftarrow{lim}_{i} G_{i}.

In particular, if G is non trivial, then L^{+}G is not of finite type over F. The positive loop group L^{+}G operates on Gr_{G} and, for every orbit O, the L^{+}Gaction factors through G_{i} for some i. Let \overline{O} denote the reduced closure of O in Gr_{G}, a projective L^{+}G-stable subvariety. This presents the reduced locus as the direct limit of L^{+}G-stable subvarieties

(Gr_{G})_{red} = \underrightarrow{lim}_{O} \overline{O},

where the transition maps are closed immersions.

Fix a prime ℓ ≠ char(F), and denote by ℚ_{ℓ} the field of ℓ-adic numbers with algebraic closure ℚ̄_{ℓ}. For any separated scheme T of finite type over F, we consider the bounded derived category D_{c}^{b}(T, ℚ̄_{ℓ}) of constructible ℓ-adic complexes on T, and its abelian full subcategory P (T) of ℓ-adic perverse sheaves. If H is a connected smooth affine group scheme acting on T, then let P_{H}(T) be the abelian subcategory of P (T) of H-equivariant objects with H-equivariant morphisms. We refer to Appendix A for an explanation of these concepts. The category of ℓ-adic perverse sheaves P (Gr_{G}) on the affine Grassmannian is the direct limit

P (Gr_{G}) =^{def} \underrightarrow{lim}_{O} P (\overline{O}),

which is well-defined, since all transition maps are closed immersions, cf. Appendix A.

Definition 2.2. The Satake category is the category of L^{+}G-equivariant ℓ-adic perverse sheaves on the affine Grassmannian Gr_{G}

P_{L^{+}G}(Gr_{G}) =^{def} \underrightarrow{lim}_{O} P_{L^{+}G}(\overline{O}),

where O ranges over the L^{+}G-orbits.

The Satake category P_{L^{+}G}(Gr_{G}) is an abelian ℚ̄_{ℓ}-linear category, cf. Appendix A.

3 The Convolution Product

We are going to equip the category P_{L^{+}G}(Gr_{G}) with a tensor structure. Let

- ⋆ - : P (Gr_{G}) × P_{L^{+}G}(Gr_{G}) ⟶ D_{c}^{b}(Gr_{G}, ℚ̄_{ℓ})

be the convolution product with values in the derived category. We recall its definition [17, §2]. Consider the following diagram of ind-schemes

Gr_{G} × Gr_{G} ⟵^{p} LG × Gr_{G} ⟶^{q} LG ×^{L^{+}G} Gr_{G} ⟶^{m} Gr_{G}.  (3.1)

Here p (resp. q) is a right L^{+}G-torsor with respect to the L^{+}G-action on the left factor (resp. the diagonal action).The LG-action on Gr_{G} factors through q, giving rise to the morphism m. For perverse sheaves A_{1}, A_{2} on Gr_{G}, their box product A_{1} ⊠ A_{2} is a perverse sheaf on Gr_{G} × Gr_{G}. If A_{2} is L^{+}G-equivariant, then there is a unique perverse sheaf A_{1} ˜⊠A_{2} on LG ×^{L^{+}G} Gr_{G} such that there is an isomorphism equivariant for the diagonal L^{+}G-action^{1}

p^{∗}(A_{1} ⊠ A_{2}) ≃ q^{∗}(A_{1} ˜⊠A_{2}).

Then the convolution is defined as A_{1} ⋆ A_{2} =^{def} m_{∗}(A_{1} ˜⊠A_{2}).

1Though LG is not of ind-finite type, we use Lemma 3.20 below to define A_{1} ˜⊠A_{2}.

Documenta Mathematica 19 (2014) 209–246

Theorem 3.1. (i) For perverse sheaves A_{1}, A_{2} on Gr_{G} with A_{2} being L^{+}Gequivariant, their convolution A_{1} ⋆ A_{2} is a perverse sheaf. If A_{1} is also L^{+}Gequivariant, then A_{1} ⋆ A_{2} is L^{+}G-equivariant. (ii) Let F̄ be a separable closure of F. The convolution product is a bifunctor

- ⋆ - : P_{L^{+}G}(Gr_{G}) × P_{L^{+}G}(Gr_{G}) ⟶ P_{L^{+}G}(Gr_{G}),

and (P_{L^{+}G}(Gr_{G}), ⋆) has a unique structure of a symmetric monoidal category such that the cohomology functor with values in finite dimensional ℚ̄_{ℓ}-vector spaces

⨁_{i∈ℤ} R^{i}Γ(Gr_{G,F̄}, (-)F̄): P_{L^{+}G}(Gr_{G}) ⟶ Vecℚ̄_{ℓ}

is symmetric monoidal.

Part (i) is due to Lusztig [12] and Gaitsgory [7]. Part (ii) is based on methods due to Reich [19]. Both parts of Theorem 3.1 are proved simultaneously in Subsection 3.3 below using universally locally acyclic perverse sheaves (cf. Subsection 3.2 below) and a global version of diagram (3.1) which we introduce in the next subsection.

3.1 Beilinson-Drinfeld Grassmannians

Let X a smooth geometrically connected curve over F. For any F -algebra R, let X_{R} = X × Spec(R). Denote by Σ the moduli space of relative effective Cartier divisors on X, i.e. the fppf-sheaf associated with the functor on the category of F -algebras

R ⟼ {D ⊂ X_{R} relative effective Cartier divisor}.

Lemma 3.2. The fppf-sheaf Σ is represented by the disjoint union of fppfquotients ∐_{n≥1} X^{n}/S_{n}, where the symmetric group S_{n} acts on X^{n} by permuting its coordinates.

□

Definition 3.3. The Beilinson-Drinfeld Grassmannian (associated to G and X) is the functor Gr = Gr_{G,X} on the category of F -algebras which assings to every R the set of isomorphism classes of triples (D, F, β) with

D ∈ Σ(R) a relative effective Cartier divisor; F a G-torsor on X_{R};

β : F|_{X_{R}\D} →^{≃} F_{0}|_{X_{R}\D} a trivialisation,

where F_{0} denotes the trivial G-torsor. The projection Gr → Σ, (D, F, β) ↦ D is a morphism of functors.

Lemma 3.4. The Beilinson-Drinfeld Grassmannian Gr = Gr_{G,X} associated to a reductive group G and a smooth curve X is representable by an ind-proper strict ind-scheme over Σ.

Proof. This is proven in [7, Appendix A.5.]. We sketch the argument. If G = GL_{n}, consider the functor Gr_{(m)} parametrizing

J ⊂ O_{X_{R}}^{n}(−m · D)/O_{X_{R}}^{n}(m · D),

where J is a coherent O_{X_{R}}-submodule such that O_{X_{R}}(−m·D)/J is flat over R. By the theory of Hilbert schemes, the functor Gr_{(m)} is representable by a proper scheme over Σ. For m_{1} < m_{2}, there are closed immersions Gr_{(m_{1})} ↪ Gr_{(m_{2})}. Then as fpqc-sheaves

\underrightarrow{lim}_{m} Gr_{(m)} ⟶^{≃} Gr.

For general reductive G, choose an embedding G ↪ GL_{n}. Then the fppfquotient GL_{n}/G is affine, and the natural morphism Gr_{G} → Gr_{GL_{n}} is a closed immersion. The ind-scheme structure of Gr_{G} does not depend on the choosen embedding G ↪ GL_{n}. This proves the lemma.

Now we define a global version of the loop group. For every D ∈ Σ(R), the formal completion of X_{R} along D is a formal affine scheme. We denote by Ô_{X,D} its underlying R-algebra. Let D̂ = Spec(Ô_{X,D}) be the associated affine scheme over R. Then D is a closed subscheme of D̂, and we set D̂^{o} = D̂\D. The global loop group is the fpqc-sheaf associated with the group functor on the category of F -algebras

LG : R ↦ {(s, D) | D ∈ Σ(R), s ∈ G(D̂^{o})}.

The global positive loop group is the fpqc-sheaf associated with the group functor

L^{+}G : R ↦ {(s, D) | D ∈ Σ(R), s ∈ G(D̂)}.

Then L^{+}G ⊂ LG is a subgroup functor over Σ.

Lemma 3.5. (i) The global loop group LG is representable by an ind-group scheme over Σ. It represents the functor on the category of F -algebras which assigns to every R the set of isomorphism classes of quadruples (D, F, β, σ), where D ∈ Σ(R), F is a G-torsor on X_{R}, β : F →^{≃} F_{0} is a trivialisation over X_{R}\D and σ : F_{0} →^{≃} F|D̂ is a trivialisation over D̂. (ii) The global positive loop group L^{+}G is representable by an affine group scheme over Σ with geometrically connected fibers. (iii) The projection LG → Gr_{G}, (D, F, β, σ) → (D, F, β) is a right L^{+}G-torsor, and induces an isomorphism of fpqc-sheaves over Σ

LG/L^{+}G ⟶^{≃} Gr_{G}.

Proof. We reduce to the case that X is affine. Note that fppf-locally on R every D ∈ Σ(R) is of the form V (f). Then the moduli description in (i) follows from the descent lemma of Beauville-Laszlo [1] (cf. [14, Proposition 3.8]). The ind-representability follows from part (ii) and (iii). This proves (i). For any D ∈ Σ(R) denote by D^{(i)} its i-th infinitesimal neighbourhood in X_{R}. Then D^{(i)} is finite over R, and the Weil restriction Res_{D^{(i)}/R}(G) is representable by a smooth affine group scheme with geometrically connected fibers. For i ≤ j, there are affine transition maps Res_{D^{(j)}/R}(G) → Res_{D^{(i)}/R}(G) with geometrically connected fibers. Hence, \underleftarrow{lim}_{i} Res_{D^{(i)}/R}(G) is an affine scheme, and the canonical map

L^{+}G ×_{Σ,D} Spec(R) ⟶ \underleftarrow{lim}_{i} Res_{D^{(i)}/R}(G)

is an isomorphism of fpqc-sheaves. This proves (ii). To prove (iii), the crucial point is that after a faithfully flat extension R → R′ a G-torsor F on D̂ admits a global section. Indeed, F admits a R′-section which extends to D̂_{R′} by smoothness and Grothendieck’s algebraization theorem. This finishes (iii).

Remark 3.6. The connection with the affine Grassmannian Gr_{G} is as follows. Lemma 3.2 identifies X with a connected component of Σ. Choose a point x ∈ X(F) considered as an element D_{x} ∈ Σ(F). Then D̂_{x} ≃ Spec(F [[t]]), where t is a local parameter of X in x. Under this identification, there are isomorphisms of fpqc-sheaves

LG_{x} ≃ LG
L^{+}G_{x} ≃ L^{+}G
Gr_{G,x} ≃ Gr_{G}.

Using the theory of Hilbert schemes, the proof of Lemma 3.4 also implies that Gr_{GL_{n}}, and hence Gr_{G} is ind-projective. This proves Lemma 2.1 above.

By Lemma 3.5 (iii), the global positive loop groop L^{+}G acts on Gr from the left. For D ∈ Σ(R) and (D, F, β) ∈ Gr_{G}(R), denote the action by

((g, D), (F, β, D)) ⟼ (gF, gβ, D).

Corollary 3.7. The L^{+}G-orbits on Gr are of finite type and smooth over Σ.

Proof. Let D ∈ Σ(R). It is enough to prove that the action of

L^{+}G ×_{Σ,D} Spec(R) ≃ \underleftarrow{lim}_{i} Res_{D^{(i)}/R}(G)

on Gr ×_{Σ,D} Spec(R) factors over Res_{D^{(i)}/R}(G) for some i >> 0. Choose a faithful representation ρ : G → GL_{n}, and consider the corresponding closed immersion Gr_{G} → Gr_{GL_{n}}. This reduces us to the case G = GL_{n}. In this case, the Gr_{(m)}’s (cf. proof of Lemma 3.4) are L^{+}GL_{n} stable, and it is easy to see that the action on Gr_{(m)} factors through Res_{D^{(2m)}/R}(GL_{n}). This proves the corollary.

Now we globalize the convolution morphism m from diagram (3.1) above. The moduli space Σ of relative effective Cartier divisors has a natural monoid structure

- ∪ - : Σ × Σ ⟶ Σ
(D_{1}, D_{2}) ⟼ D_{1} ∪ D_{2}.

The key definition is the following.

Definition 3.8. For k ≥ 1, the k-fold convolution Grassmannian G̃r_{k} is the functor on the category of F -algebras which associates to every R the set of isomorphism classes of tuples ((D_{i}, F_{i}, β_{i})_{i=1,...,k}) with

D_{i} ∈ Σ(R) relative effective Cartier divisors, i = 1, …, k; F_{i} are G-torsors on X_{R};

β_{i} : F_{i}|_{X_{R}\D_{i}} →^{≃} F_{i−1}|_{X_{R}\D_{i}} isomorphisms, i = 1, …, k,

where F_{0} is the trivial G-torsor. The projection G̃r_{k} → Σ^{k}, ((D_{i}, F_{i}, β_{i})_{i=1,...,k}) ↦ ((D_{i})_{i=1,...,k}) is a morphism of functors.

Lemma 3.9. For k ≥ 1, the k-fold convolution Grassmannian G̃r_{k} is representable by a strict ind-scheme which is ind-proper over Σ^{k}.

Proof. The lemma follows by induction on k. If k = 1, then G̃r_{k} = Gr. For k > 1, consider the projection

p : G̃r_{k} ⟶ G̃r_{k−1} × Σ
((D_{i}, F_{i}, β_{i})_{i=1,...,k}) ⟼ ((D_{i}, F_{i}, β_{i})_{i=1,...,k−1}, D_{k}).

Then the fiber over a R-point ((D_{i}, F_{i}, β_{i})_{i=1,...,k−1}, D_{k}) is

p^{−1}(((D_{i}, F_{i}, β_{i})_{i=1,...,k−1}, D_{k})) ≃ F_{k−1} ×^{G} (Gr ×_{X_{R}} D_{k}),

which is ind-proper. This proves the lemma.

For k ≥ 1, there is the k-fold global convolution morphism

m_{k} : G̃r_{k} ⟶ Gr
((D_{i}, F_{i}, β_{i})_{i=1,...,k}) ⟼ (D, F_{k}, β_{1}|_{X_{R}\D} ∘ … ∘ β_{k}|_{X_{R}\D}),

where D = D_{1} ∪ … ∪ D_{k}. This yields a commutative diagram of ind-schemes

\begin{matrix}G̃r_{k} & Gr \\ Σ^{k} & Σ, \end{matrix} ; m_{k} ; ∪

i.e., regarding G̃r_{k} as a Σ-scheme via Σ^{k} → Σ, (D_{i})_{i} ↦ ∪_{i}D_{i}, the morphism m_{k} is a morphism of Σ-ind-schemes. The global positive loop group L^{+}G acts on G̃r_{k} over Σ as follows: let (D_{i}, F_{i}, β_{i})_{i} ∈ G̃r_{k}(R) and g ∈ G(D̂) with D = ∪_{i}D_{i}. Then the action is defined as

((g, D), (D_{i}, F_{i}, β_{i})_{i}) ⟼ (D_{i}, gF_{i}, gβ_{i}g^{−1})_{i}.

Corollary 3.10. The morphism m_{k} : G̃r_{k} → Gr is a L^{+}G-equivariant morphism of ind-proper strict ind-schemes over Σ.

Proof. The L^{+}G-equivariance is immediate from the definition of the action. Note that Σ^{k} →^{∪} Σ is finite, and hence G̃r_{k} is an ind-proper strict ind-scheme over Σ. This proves the corollary.

Now we explain the global analogue of the L^{+}G-torsors p and q from (3.1). For k ≥ 1, let L̃G_{k} be the functor on the category of F -algebras which associates to every R the set of isomorphism classes of tuples ((D_{i}, F_{i}, β_{i})_{i=1,...,k}, (σ_{i})_{i=2,...,k}) with

D_{i} ∈ Σ(R), i = 1, …, k; F_{i} are G-torsors on X_{R};

β_{i} : F_{i}|_{X_{R}\D_{i}} →^{≃} F_{0}|_{X_{R}\D_{i}} trivialisations, i = 1, …, k;

σ_{i} : F_{0}|D̂_{i} →^{≃} F_{i−1}|D̂_{i}, i = 2, …, k,

where F_{0} is the trivial G-torsor. There are two natural projections over Σ^{k}. Let

L^{+}G_{Σ}^{k−1} = Σ^{k} ×_{Σ^{k−1}} L^{+}G^{k−1}.

The first projection is given by

p_{k} : L̃G_{k} ⟶ Gr^{k}
((D_{i}, F_{i}, β_{i})_{i=1,...,k}, (σ_{i})_{i=2,...,k}) ⟼ ((D_{i}, F_{i}, β_{i})_{i=1,...,k}).

Then p_{k} is a right L^{+}G_{Σ}^{k−1}-torsor for the action on the σ_{i}’s. The second projection is given by

q_{k} : L̃G_{k} ⟶ G̃r_{k}
((D_{i}, F_{i}, β_{i})_{i=1,...,k}, (σ_{i})_{i=2,...,k}) ⟼ ((D_{i}, F′_{i}, β′_{i})_{i=1,...,k}),

where F′_{1} = F_{1} and for i ≥ 2, the G-torsor F′_{i} is defined successively by gluing F_{i}|_{X_{R}\D_{i}} to F′_{i−1}|D̂_{i} along σ_{i}|D̂_{i}^{o} ∘ β_{i}|D̂_{i}^{o}. Then q_{k} is a right L^{+}G_{Σ}^{k−1}-torsor for the action given by

(((D_{i}, F_{i}, β_{i})_{i≥1}, (σ_{i})_{i≥2}),(D_{1}, (D_{i}, g_{i})_{i≥2})) ⟼
((D_{1}, F_{1}, β_{1}), (D_{i}, g_{i}^{−1}F_{i}, g_{i}^{−1}β_{i})_{i≥2}, (σ_{i}g_{i})_{i≥2}).

In the following, we consider ind-schemes over Σ^{k} as ind-schemes over Σ via Σ^{k} → Σ.

Definition 3.11. For every k ≥ 1, the k-fold global convolution diagram is the diagram of ind-schemes over Σ

Gr^{k} ⟵^{p_{k}} L̃G_{k} ⟶^{q_{k}} G̃r_{k} ⟶^{m_{k}} Gr.

Remark 3.12. Fix x ∈ X(F), and choose a local coordinate t at x. Taking the fiber over diag({x}) ∈ X^{k}(F) in the k-fold global convolution diagram, then

Gr_{G}^{k} ⟵ LG^{k−1} × Gr_{G} ⟶ \underbrace{LG ×^{L^{+}G} … ×^{L^{+}G} Gr_{G}}_{k-times} ⟶ Gr_{G}.

For k = 2, we recover diagram (3.1).

3.2 Universal Local Acyclicity

The notion of universal local acyclicity (ULA) is used in Reich’s thesis [19], cf. also the paper [3] by Braverman and Gaitsgory. We recall the definition. Let S be a smooth geometrically connected scheme over F, and f : T → S a separated morphism of finite type. For complexes A_{T} ∈ D_{c}^{b}(T, ℚ̄_{ℓ}), A_{S} ∈ D_{c}^{b}(S, ℚ̄_{ℓ}), there is a natural morphism

A_{T} ⊗ f^{∗}A_{S} ⟶ (A_{T} ⊗^{!} f^{!}A_{S})[2 dim(S)],  (3.2)

where A ⊗^{!} B =^{def} 𝔻(𝔻A ⊗ 𝔻B) for A, B ∈ D_{c}^{b}(T, ℚ̄_{ℓ}). The morphism (3.2) is constructed as follows. Let Γ_{f} : T → T × S be the graph of f. The projection formula gives a map

Γ_{f,!}(Γ_{f}^{∗}(A_{T} ⊠ A_{S}) ⊗ Γ_{f}^{!} ℚ̄_{ℓ}) ≃ (A_{T} ⊠ A_{S}) ⊗ Γ_{f,!}Γ_{f}^{!} ℚ̄_{ℓ} ⟶ A_{T} ⊠ A_{S},

and by adjunction a map Γ_{f}^{∗}(A_{T} ⊠ A_{S}) ⊗ Γ_{f}^{!} ℚ̄_{ℓ} → Γ_{f}^{!}(A_{T} ⊠ A_{S}). Note that

Γ_{f}^{∗}(A_{T} ⊠ A_{S}) ≃ A_{T} ⊗ f^{∗}A_{S}  and  Γ_{f}^{!}(A_{T} ⊠ A_{S}) ≃ A_{T} ⊗^{!} f^{!}A_{S},

using 𝔻(A_{T} ⊠A_{S}) ≃ 𝔻A_{T} ⊠𝔻A_{S}. Since S is smooth, Γ_{f} is a regular embedding, and thus Γ_{f}^{!} ℚ̄_{ℓ} ≃ ℚ̄_{ℓ}[−2 dim(S)]. This gives after shifting by [2 dim(S)] the map (3.2).

Definition 3.13. (i) A complex A_{T} ∈ D_{c}^{b}(T, ℚ̄_{ℓ}) is called locally acyclic with respect to f (f-LA) if (3.2) is an isomorphism for all A_{S} ∈ D_{c}^{b}(S, ℚ̄_{ℓ}). (ii) A complex A_{T} ∈ D_{c}^{b}(T, ℚ̄_{ℓ}) is called universally locally acyclic with respect to f (f-ULA) if f_{S′}^{∗}A_{T} is f_{S′}-LA for all f_{S′} = f ×_{S} S′ with S′ → S smooth, S′ geometrically connected.

Remark 3.14. (i) If f is smooth, then the trivial complex A_{T} = ℚ̄_{ℓ} is f-ULA. (ii) If S = Spec(F) is a point, then every complex A_{T} ∈ D_{c}^{b}(T, ℚ̄_{ℓ}) is f-ULA. (iii) The ULA property is local in the smooth topology on T.

Lemma 3.15. Let g : T → T′ be a proper morphism of S-schemes of finite type. For every ULA complex A_{T} ∈ D_{c}^{b}(T, ℚ̄_{ℓ}), the push forward g_{∗}A_{T} is ULA.

Proof. For any morphism of finite type g : T → T′ and any two complexes A_{T}, A_{T′}, we have the projection formulas

g_{!}(A_{T} ⊗g^{∗}A_{T′}) ≃ g_{!}A_{T} ⊗A_{T′}  and  g_{∗}(A_{T} ⊗^{!} g^{!}A_{T′}) ≃ g_{∗}A_{T} ⊗^{!} A_{T′}.

If g is proper, then g_{∗} = g_{!}, and the lemma follows from an application of the projection formulas and proper base change.

Theorem 3.16 ([19]). Let D ⊂ S be a smooth Cartier divisor, and consider a cartesian diagram of morphisms of finite type

\begin{matrix}E & T & U \\ D & S & S\D. \end{matrix} i f j

Let A be a f-ULA complex on T such that A|_{U} is perverse. Then:

(i) There is a functorial isomorphism

i^{∗}[−1]A ≃ i^{!}[1]A,

and both complexes i^{∗}[−1]A, i^{!}[1]A are perverse. Furthermore, the complex A is perverse and is the middle perverse extension A ≃ j_{!∗}(A|_{U}). (ii) The complex i^{∗}[−1]A is f|_{E}-ULA.

□

Remark 3.17. The proof of Theorem 3.16 uses Beilinson’s construction of the unipotent part of the tame nearby cycles as follows. Suppose the Cartier divisor D is principal, this gives a morphism φ : S → 𝔸_{F}^{1} such that φ^{−1}({0}) = S\D. Let g = φ ∘ f be the composition. Fix a separable closure F̄ of F. In SGA VII, Deligne constructs the nearby cycles functor ψ = ψ_{g} : P (U) → P (EF̄). Let ψ_{tame} be the tame nearby cycles, i.e. the invariants under the pro-p-part of π_{1}(𝔾_{m,F̄}, 1). Fix a topological generator T of the maximal prime-p-quotient of π_{1}(𝔾_{m,F̄}, 1). Then T acts on ψ_{tame}, and there is an exact triangle

ψ_{tame} ⟶^{T −1} ψ_{tame} ⟶ i^{∗}j_{∗} ⟶^{+1}

Under the action of T − 1 the nearby cycles decompose as ψ_{tame} ≃ ψ_{tame}^{u} ⊕ ψ_{tame}^{nu}, where T − 1 acts nilpotently on ψ_{tame}^{u} and invertibly on ψ_{tame}^{un}. Let N : ψ_{tame} → ψ_{tame}(−1) be the logarithm of T, i.e. the unique nilpotent operator N such that T = exp(T̄N) where T̄ is the image of T under π_{1}(𝔾_{m,F̄}, 1) ↠ ℤ_{ℓ}(1). Then for any a ≥ 0, Beilinson constructs a local system L_{a} on 𝔾_{m} together with a nilpotent operator N_{a} such that for A_{U} ∈ P (U) and a ≥ 0 with N^{a+1}(ψ_{tame}^{u}(A_{U})) = 0 there is an isomorphism

(ψ_{tame}^{u}(A_{U}), N) ≃ (i^{∗}[−1]j_{!∗}(A_{U} ⊗ g^{∗}L_{a})F̄, 1 ⊗ N_{a}).

Set Ψ_{g}^{u}(A_{U}) =^{def} lim_{a→∞} i^{∗}[−1]j_{!∗}(A_{U} ⊗ g^{∗}L_{a}). Then Ψ_{g}^{u} : P (U) → P (E) is a functor, and we obtain that N acts trivially on ψ_{tame}^{u}(A_{U}) if and only if Ψ_{g}^{u}(A_{U}) = i^{∗}[−1]j_{!∗}(A_{U}). In this case, Ψ_{g}^{u} is also defined for non-principal Cartier divisors by the formula Ψ_{g}^{u} = i^{∗}[−1] ∘ j_{!∗}. In the situation of Theorem 3.16 above Reich shows that the unipotent monodromy along E is trivial, and consequently

i^{∗}[−1]A ≃ Ψ_{g}^{u} ∘ j^{∗}(A) ≃ i^{!}[1]A.

.

Corollary 3.18 ([19]). Let A be a perverse sheaf on S whose support contains an open subset of S. Then the following are equivalent:

(i) The perverse sheaf A is ULA with respect to the identity id : S → S. (ii) The complex A[− dim(S)] is a locally constant system, i.e. a lisse sheaf.

□

We use the universal local acyclicity to show the perversity of certain complexes on the Beilinson-Drinfeld Grassmannian. For every finite index set I, there is the quotient map X^{I} → Σ onto a connected component of Σ. Set

Gr_{I} =^{def} Gr ×_{Σ} X^{I}.

If I = {∗} has cardinality 1, we write Gr_{X} = Gr_{I}.

Remark 3.19. Let X = 𝔸_{F}^{1} with global coordinate t. Then 𝔾_{a} acts on X via translations. We construct a 𝔾_{a}-action on Gr as follows. For every x ∈ 𝔾_{a}(R), let a_{x} be the associated automorphism of X_{R}. If D ∈ Σ(R), then we get an isomorphism a_{−x} : a_{x}D → D. Let (D, F, β) ∈ Gr_{G}(R). Then the 𝔾_{a}-action on Gr_{G} → Σ is given as

(D, F, β) ⟼ (a_{−x}^{∗}F, a_{−x}^{∗}β, a_{x}D).

Let 𝔾_{a} act diagonally on X^{I}, then the structure morphism Gr_{I} → X^{I} is 𝔾_{a}- equivariant. If |I| = 1, then by the transitivity of the 𝔾_{a}-action on X, we get Gr_{X} = Gr_{G} × X. Let p : Gr_{X} → Gr_{G} be the projection. Then for every perverse sheaf A on Gr_{G}, the complex p^{∗}[1]A is a ULA perverse sheaf on Gr_{X} by Remark 3.14 (ii) and the smoothness of p.

Now fix a finite index set I of cardinality k ≥ 1. Consider the base change along X^{I} → Σ of the k-fold convolution diagram from Definition 3.11,

∏_{i∈I} Gr_{X,i} ⟵^{p_{I}} L̃G_{I} ⟶^{q_{I}} G̃r_{I} ⟶^{m_{I}} Gr_{I}.  (3.3)

Now choose a total order I = {1, …, k}, and set I^{o} = I\{1}. Then p_{I} (resp. q_{I}) is a L^{+}G_{I}^{o}-torsor, where L^{+}G_{I}^{o} = X^{I} ×_{X^{I^{o}}} L^{+}G_{I^{o}}. Let L^{+}G_{X} = L^{+}G ×_{Σ} X, and denote by P_{L^{+}G_{X}} (Gr_{X})^{ULA} the category of L^{+}G_{X}-equivariant ULA perverse sheaves on Gr_{X}. For any i ∈ I, let A_{X,i} ∈ P (Gr_{X})^{ULA} such that A_{X,i} are L^{+}G_{X}-equivariant for i ≥ 2. We have the ∏_{i≥2} L^{+}G_{X,i}-equivariant ULA perverse sheaf ⊠_{i∈I}A_{X,i} on ∏_{i∈I} Gr_{X,i}.

Lemma 3.20. There is a unique ULA perverse sheaf ˜⊠_{i∈I}A_{X,i} on G̃r_{I} such that there is a q_{I}-equivariant isomorphism^{2}

q_{I}^{∗}(˜⊠_{i∈I}A_{X,i}) ≃ p_{I}^{∗}(⊠_{i∈I}A_{X,i}),

where q_{I}-equivariant means with respect to the action on the L^{+}G_{I}^{o}-torsor q_{I} : L̃G_{I} → G̃r_{I}. If A_{X,1} is also L^{+}G_{X}-equivariant, then ˜⊠_{i∈I}A_{X,i} is L^{+}G_{I}- equivariant

Remark 3.21. The ind-scheme L̃G_{I} is not of ind-finite type. We explain how the pullback functors p_{I}^{∗}, q_{I}^{∗} should be understood. Let Y_{1}, …, Y_{k} be L^{+}Gequivariant closed subschemes of Gr_{X} containing the supports of A_{1}, …, A_{k} . Choose N >> 0 such that the action of L^{+}G_{X} on each Y_{1}, …, Y_{k} factors over the smooth affine group scheme H_{N} = Res_{D^{(N)}/X}(G), where D^{(N)} is the N-th infinitesimal neighbourhoud of the universal Cartier divisor D over X. Let K_{N} = ker(L^{+}G_{X} → H_{N}), and Y = Y_{1} × … Y_{k}. Then the left K_{N}-action on each Y_{i} is trivial, and hence the restriction of the p_{I}-action resp. q_{I}-action on p_{I}^{−1}(Y) to ∏_{i≥2} K_{N} agree. Let h_{N} : p_{I}^{−1}(Y) → Y_{N} be the resulting ∏_{i≥2} K_{N}- torsor. By Lemma A.4 below, we get a factorization

\begin{aligned}& p_{I}^{−1}(Y) \\ Y & Y_{N} & q_{I}(p_{I}^{−1}(Y)), \end{aligned} ; \begin{matrix}p_{I} \\ h_{N} \\ q_{I} \\ p_{I,N} \\ q_{I,N} \end{matrix}

where p_{I,N}, q_{I,N} are ∏_{i≥2} H_{N}-torsors. In particular, Y_{N} is a separated scheme of finite type, and we can replace p_{I}^{∗} (resp. q_{I}^{∗}) by p_{I,N}^{∗} (resp. q_{I,N}^{∗}).

Proof of Lemma 3.20. We use the notation from Remark 3.21 above. The sheaf p_{I;N}^{∗}(⊠_{i∈I}A_{X,i}) is ∏_{i≥2} H_{N}-equivariant for the q_{I,N}-action. Using descent along smooth torsors (cf. Lemma A.2 below), we get the perverse sheaf ˜⊠_{i∈I}A_{X,i}, which is ULA. Indeed, p_{I;N}^{∗}(⊠_{i∈I}A_{X,i}) is ULA, and the ULA property is local in the smooth topology. Since the diagram (3.3) is L^{+}G_{I}-equivariant, the sheaf ˜⊠_{i∈I}A_{X,i} is L^{+}G_{I}-equivariant, if A_{X,1} is L^{+}G_{X}- equivariant. This proves the lemma.

2See Remark 3.21 below.

Documenta Mathematica 19 (2014) 209–246

Let U_{I} be the open locus of pairwise distinct coordinates in X^{I}. There is a cartesian diagram

\begin{matrix}Gr_{I} & (Gr_{X}^{I})|_{U_{I}} \\ X^{I} & U_{I}. \end{matrix} ; j_{I}

Proposition 3.22. The complex m_{I,∗}(˜⊠_{i∈I}A_{X,i}) is a ULA perverse sheaf on Gr_{I}, and there is a unique isomorphism of perverse sheaves

m_{I,∗}(˜⊠_{i∈I}A_{X,i}) ≃ j_{I,!∗}(⊠_{i∈I}A_{X,i}|_{U_{I}}),

which is L^{+}G_{I}-equivariant, if A_{X,1} is L^{+}G_{X}-equivariant.

Proof. The sheaf ˜⊠_{i∈I}A_{X,i} is by Lemma 3.20 a ULA perverse sheaf on G̃r_{I}. Now the restriction of the global convolution morphism m_{I} to the support of ˜⊠_{i∈I}A_{X,i} is a proper morphism, and hence m_{I,∗}(˜⊠_{i∈I}A_{X,i}) is a ULA complex by Lemma 3.15. Then m_{I,∗}(˜⊠_{i∈I}A_{X,i}) ≃ j_{!∗}((⊠_{i∈I}A_{X,i})|_{U_{I}}), as follows from Theorem 3.16 (i) and the formula u_{!∗} ∘ v_{!∗} ≃ (u ∘ v)_{!∗} for open immersions V ↪^{v} U ↪^{u} T, because m_{I}|_{U_{I}} is an isomorphism. In particular, m_{I,∗}(˜⊠_{i∈I}A_{X,i}) is perverse. Since m_{I} is L^{+}G_{I}-equivariant, it follows from proper base change that m_{I,∗}(˜⊠_{i∈I}A_{X,i}) is L^{+}G_{I}-equivariant, if A_{X,1} is L^{+}G_{X}-equivariant. This proves the proposition.

3.3 The Symmetric Monoidal Structure

First we equip P_{L^{+}G_{X}} (Gr_{X})^{ULA} with a symmetric monoidal structure ✶ which allows us later to define a symmetric monoidal structure with respect to the usual convolution (3.1) of L^{+}G-equivariant perverse sheaves on Gr_{G}. Fix I, and let U_{I} be the open locus of pairwise distinct coordinates in X^{I}. Then the diagram

\begin{matrix}Gr_{X} & Gr_{I} & (Gr_{X}^{I})|_{U_{I}} \\ X & X^{I} & U_{I}. \end{matrix} ; i_{I} ; diag ; j_{I} (3.4)

is cartesian.

Definition 3.23. Fix some total order on I. For every tuple (A_{X,i})_{i∈I} with A_{X,i} ∈ P (Gr_{X})^{ULA} for i ∈ I, the I-fold fusion product ✶_{i∈I}A_{X,i} is the complex

✶_{i∈I}A_{X,i} =^{def} i_{I}^{∗}[−k + 1]j_{I,!∗}((⊠_{i∈I}A_{X,i})|_{U_{I}}) ∈ D_{c}^{b}(Gr_{X}, ℚ̄_{ℓ}),

where k = |I|.

Now let π : I → J be a surjection of finite index sets. For j ∈ J, let I_{j} = π^{−1}(j), and denote by U_{π} the open locus in X^{I} such that the I_{j}-coordinates are pairwise distinct from the I_{j′}-coordinates for j ≠ j′. Then the diagram

\begin{matrix}Gr_{J} & Gr_{I} & (∏_{j} Gr_{I_{j}})|_{U_{π}} \\ X^{J} & X^{I} & U_{π}, \end{matrix} ; i_{π} ; j_{π} (3.5)

is cartesian. The following theorem combined with Proposition 3.22 is the key to the symmetric monoidal structure:

Theorem 3.24. Let I be a finite index set, and let A_{X,i} ∈ P_{L^{+}G_{X}} (Gr_{X})^{ULA} for i ∈ I. Let π : I → J be a surjection of finite index sets, and set k_{π} = |I| − |J|.

(i) As complexes

i_{π}^{∗}[−k_{π}]j_{I,!∗}((⊠_{i∈I}A_{X,i})|_{U_{I}}) ≃ i_{π}^{!}[k_{π}]j_{I,!∗}((⊠_{i∈I}A_{X,i})|_{U_{I}}),

and both are L^{+}G_{J}-equivariant ULA perverse sheaves on Gr_{J}. Hence, ✶_{i∈I}A_{X,i} ∈ P_{L^{+}G_{X}} (Gr_{X})^{ULA}. (ii) There is an associativity and a commutativity constraint for the fusion product such that there is a canonical isomorphism

✶_{i∈I}A_{X,i} ≃ ✶_{j∈J}(✶_{i∈I_{j}}A_{X,i}),

where I_{j} = π^{−1}(j) for j ∈ J. In particular, (P_{L^{+}G_{X}} (Gr_{X})^{ULA}, ✶) is symmetric monoidal.

Proof. Factor π as a chain of surjective maps I = I_{1} → I_{2} → … → I_{k_{π}} = J with |I_{i+1}| = |I_{i}| + 1, and consider the corresponding chain of smooth Cartier divisors

X^{J} = X^{I_{kπ}} ⟶ … ⟶ X^{I_{2}} ⟶ X^{I_{1}} = X^{I}.

By Proposition 3.22, the complex j_{I,!∗}((⊠_{i∈I}A_{X,i})|_{U_{I}}) is ULA. Then part (i) follows inductively from Theorem 3.16 (i) and (ii). This shows (i). Let τ : I → I be a bijection. Then τ acts on X^{I} by permutation of coordinates, and diagram (3.4) becomes equivariant for this action. Then

τ^{∗}j_{I,!∗}((⊠_{i∈I}A_{X,i})|_{U_{I}}) ≃ j_{I,!∗}((⊠_{i∈I}A_{X,τ^{−1}(i)})|_{U_{I}}).

Since the action on diag(X) ⊂ X^{I} is trivial, we obtain

i_{I}^{∗}j_{I,!∗}((⊠_{i∈I}A_{X,i})|_{U_{I}}) ≃ i_{I}^{∗}τ^{∗}j_{I,!∗}((⊠_{i∈I}A_{X,i})|_{U_{I}})
≃ i_{I}^{∗}j_{I,!∗}((⊠_{i∈I}A_{X,τ^{−1}(i)})|_{U_{I}}),

and hence ✶_{i∈I}A_{X,i} ≃ ✶_{i∈I}A_{X,τ^{−1}(i)}. It remains to give the isomorphism defining the symmetric monoidal structure. Since j_{I} = j_{π} ∘ ∏_{j} j_{I_{j}}, diagram (3.5) gives

(j_{I,!∗}((⊠_{i∈I}A_{X,i})|_{U_{I}}))|_{U_{π}} ≃ ⊠_{j∈J}j_{I_{j},!∗}((⊠_{i∈I_{j}}A_{X,i})|_{U_{I_{j}}}).

Applying (i_{π}|_{U_{π}})^{∗}[k_{π}] and using that U_{π} ∩ X^{J} = U_{J}, we obtain

(i_{π}^{∗}[k_{π}]j_{I,!∗}((⊠_{i∈I}A_{X,i})|_{U_{I}}))|_{U_{J}} ≃ ⊠_{j∈J}(✶_{i∈I_{j}}A_{X,i}).

But by (i), the perverse sheaf i_{π}^{∗}[k_{π}]j_{I,!∗}((⊠_{i∈I}A_{X,i})|_{U_{I}}) is ULA, thus

i_{π}^{∗}[k_{π}]j_{I,!∗}((⊠_{i∈I}A_{X,i})|_{U_{I}}) ≃ j_{J,!∗}((⊠_{j∈J}(✶_{i∈I_{j}}A_{X,i}))|_{U_{J}}),

and restriction along the diagonal in X^{J} gives the isomorphism ✶_{i∈I}A_{X,i} ≃ ✶_{j∈J}(✶_{i∈I_{j}}A_{X,i}). This proves (ii).

Example 3.25. Let G = {e} be the trivial group. Then Gr_{X} = X. Let Loc(X) be the category of ℓ-adic local systems on X. Using Corollary 3.18, we obtain an equivalence of symmetric monoidal categories

H^{0} ∘ [−1] : (P (X)^{ULA}, ✶) ⟶^{≃} (Loc(X), ⊗),

where Loc(X) is endowed with the usual symmetric monoidal structure with respect to the tensor product ⊗.

Corollary 3.26. Let D_{c}^{b}(X, ℚ̄_{ℓ})^{ULA} be the category of ULA complexes on X. Denote by f : Gr_{X} → X the structure morphism. Then the functor

f_{∗}[−1] : (P (Gr_{X})^{ULA}, ✶) ⟶ (D_{c}^{b}(X, ℚ̄_{ℓ}), ⊗)

is symmetric monoidal.

Proof. If A_{X} ∈ P (Gr_{X})^{ULA}, then f_{∗}A_{X} ∈ D_{c}^{b}(X, ℚ̄_{ℓ})^{ULA} by Lemma 3.15 and the ind-properness of f. Now apply f_{∗} to the isomorphism in Theorem 3.24 (ii) defining the symmetric monoidal structure on P (Gr_{X})^{ULA}. Then by proper base change and going backwards through the arguments in the proof of Theorem 3.24 (ii), we get that f_{∗}[−1] is symmetric monoidal.

Corollary 3.27. Let X = 𝔸_{F}^{1} . Let p : Gr_{X} → Gr_{G} be the projection, cf. Remark 3.19.

(i) The functor

p^{∗}[1] : P_{L^{+}G}(Gr_{G}) ⟶ P_{L^{+}G_{X}} (Gr_{X})^{ULA}

embeds P_{L^{+}G}(Gr_{G}) as a full subcategory and is an equivalence of categories with the subcategory of 𝔾_{a}-equivariant objects in P_{L^{+}G_{X}} (Gr_{X})^{ULA}. (ii) For every I and A_{i} ∈ P_{L^{+}G}(Gr_{G}), i ∈ I, there is a canonical L^{+}G_{X}- equivariant isomorphism

p^{∗}[1](⋆_{i∈I}A_{i}) ≃ ✶_{i∈I}(p^{∗}[1]A_{i}),

where the product is taken with respect to some total order on I.

Proof. Under the simply transitive action of 𝔾_{a} on X, the isomorphism Gr_{X} ≃ Gr_{G} × X is compatible with the action of L^{+}G under the zero section L^{+}G ↪ L^{+}G_{X}. By Lemma 3.19, the complex p^{∗}[1]A is a ULA perverse sheaf on Gr_{X}. It is obvious that the functor p^{∗}[1] is fully faithful. Denote by i_{0} : Gr_{G} → Gr_{X} the zero section. If A_{X} on Gr_{X} is 𝔾_{a}-equivariant, then A_{X} ≃ p^{∗}[1]i_{0}^{∗}[−1]A_{X}. This proves (i). By Remark 3.12, the fiber over diag({0}) ∈ X^{I}(F) of (3.3) is the usual convolution diagram (3.1). Hence, by proper base change,

i_{0}^{∗}[−1](✶_{i∈I}p^{∗}[1]A_{i}) ≃ ⋆_{i∈I}i_{0}^{∗}[−1]p^{∗}[1]A_{i} ≃ ⋆_{i∈I}A_{i}.

Since ✶_{i∈I}p^{∗}[1]A_{i} is 𝔾_{a}-equivariant, this proves (ii).

Now we are prepared for the proof of Theorem 3.1.

Proof of Theorem 3.1. Let X = 𝔸_{F}^{1} . For every A_{1}, A_{2} ∈ P (Gr_{G}) with A_{2} being L^{+}G-equivariant, we have to prove that A_{1} ⋆ A_{2} ∈ P (Gr_{G}). By Theorem 3.24 (i), the ✶-convolution is perverse. Then the perversity of A_{1} ⋆ A_{2} follows from Corollary 3.27 (ii). Again by Corollary 3.27 (ii), the convolution A_{1} ⋆ A_{2} is L^{+}G-equivariant, if A_{1} is L^{+}G-equivariant. This proves (i). We have to equip (P_{L^{+}G}(Gr_{G}), ⋆) with a symmetric monoidal structure. By Corollary 3.27, the tuple (P_{L^{+}G}(Gr_{G}), ⋆) is a full subcategory of (P_{L^{+}G_{X}} (Gr_{X})^{ULA}, ✶), and the latter is symmetric monoidal by Theorem 3.24 (ii), hence so is (P_{L^{+}G}(Gr_{G}), ⋆). Since taking cohomology is only graded commutative, we need to modify the commutativity constraint of (P_{L^{+}G}(Gr_{G}), ⋆) by a sign as follows. Let F̄ be a separable closure of F. The L^{+}GF̄-orbits in one connected component of Gr_{G,F̄} are all either even or odd dimensional. Because the Galois action on Gr_{G,F̄} commutes with the L^{+}GF̄-action, the connected components of Gr_{G} are divided into those of even or odd parity. Consider the corresponding ℤ/2-grading on P_{L^{+}G}(Gr_{G}) given by the parity of the connected components of Gr_{G}. Then we equip (P_{L^{+}G}(Gr_{G}), ⋆) with the super commutativity constraint with respect to this ℤ/2-grading, i.e. if A (resp. B) is an L^{+}G-equivariant perverse sheaf supported on a connected component X_{A} (resp. X_{B}) of Gr_{G}, then the modified commutativity constraint differs by the sign (−1)^{p(X_{A})p(X_{B})}, where p(X) ∈ ℤ/2 denotes the parity of a connected component X of Gr_{G}. Now consider the global cohomology functor

ω(-) = ⨁_{i∈ℤ} R^{i}Γ(Gr_{G,F̄}, (-)F̄): P_{L^{+}G}(Gr_{G}) ⟶ Vecℚ̄_{ℓ}.

Let f : Gr_{X} → X be the structure morphism. Then the diagram

P_{L^{+}G_{X,F̄}} (Gr_{X,F̄})^{ULA} D_{c}^{b}(XF̄, ℚ̄_{ℓ})

P_{L^{+}G}(Gr_{G})  Vecℚ̄_{ℓ} f_{∗}[−1] p^{∗}[1] ∘ (-)F̄ ω ⊕_{i∈ℤ}H^{i} ∘ i_{0}^{∗}

is commutative up to natural isomorphism. Now if A is a perverse sheaf supported on a connected component X of Gr_{G}, then by a theorem of Lusztig [12, Theorem 11c],

R^{i}Γ(Gr_{G,F̄}, AF̄) = 0, i ≢ p(X) (mod 2),

where p(X) ∈ ℤ/2 denotes the parity of X. Hence, Corollary 3.26 shows that ω is symmetric monoidal with respect to the super commutativity constraint on P_{L^{+}G}(Gr_{G}). To prove uniqueness of the symmetric monoidal structure, it is enough to prove that ω is faithful, which follows from Lemma 4.4 below. This proves (ii).

4 The Tannakian Structure

In this section we assume that F = F̄ is separably closed. Let X_{+}^{∨} be a set of representatives of the L^{+}G-orbits on Gr_{G}. For μ ∈ X_{+}^{∨} we denote by O_{μ} the corresponding L^{+}G-orbit, and by \overline{O}_{μ} its reduced closure with open embeddding j^{μ} : O_{μ} ↪ \overline{O}_{μ}. We equip X_{+}^{∨} with the partial order defined as follows: for every λ, μ ∈ X_{+}^{∨}, we define λ ≤ μ if and only if O_{λ} ⊂ \overline{O}_{μ}.

Proposition 4.1. The category P_{L^{+}G}(Gr_{G}) is semisimple with simple objects the intersection complexes

IC_{μ} = j_{!∗}^{μ}ℚ̄_{ℓ}[dim(O_{μ})], for μ ∈ X_{+}^{∨}.

In particular, if ^{p}j_{∗}^{μ} (resp. ^{p}j_{!}^{μ}) denotes the perverse push forward (resp. perverse extension by zero), then j_{!∗}^{μ} ≃ ^{p}j_{!}^{μ} ≃ ^{p}j_{∗}^{μ}.

Proof. For any μ ∈ X_{+}^{∨}, the étale fundamental group π_{1}^{ét}(O_{μ}) is trivial. Indeed, since \overline{O}_{μ}\O_{μ} is of codimension at least 2 in \overline{O}_{μ}, Grothendieck’s purity theorem implies that π_{1}^{ét}(O_{μ}) = π_{1}^{ét}(\overline{O}_{μ}). The latter group is trivial by [SGA1, XI.1 Corollaire 1.2], because \overline{O}_{λ} is normal (cf. [6]), projective and rational. This shows the claim. Since by [17, Lemme 2.3] the stabilizers of the L^{+}G-action are connected, any L^{+}G-equivariant irreducible local system supported on O_{μ} is isomorphic to the constant sheaf ℚ̄_{ℓ}. Hence, the simple objects in P_{L^{+}G}(Gr_{G}) are the intersection complexes IC_{μ} for μ ∈ X_{+}^{∨}. To show semisimplicity of the Satake category, it is enough to prove

Ext_{P (Gr_{G})}^{1}(IC_{λ}, IC_{μ}) = Hom_{D_{c}^{b}(Gr_{G})}(IC_{λ}, IC_{μ}[1]) =^{!} 0.

We distinguish several cases: Case (i): λ = μ.

Let O_{μ} →^{j} \overline{O}_{μ} ←^{i} \overline{O}_{μ}\O_{μ}, and consider the exact sequence of abelian groups

Hom(IC_{μ}, i_{!}i^{!}IC_{μ}[1]) ⟶ Hom(IC_{μ}, IC_{μ}[1]) ⟶ Hom(IC_{μ}, j_{∗}j^{∗}IC_{μ}[1]) (4.1)

associated to the distinguished triangle i_{!}i^{!}IC_{μ} → IC_{μ} → j_{∗}j^{∗}IC_{μ}. We show that the outer groups in (4.1) are trivial. Indeed, the last group is trivial, since j^{∗}IC_{μ} = ℚ̄_{ℓ}[dim(O_{μ})] gives

Hom(IC_{μ}, j_{∗}j^{∗}IC_{μ}[1]) = Hom(j^{∗}IC_{μ}, j^{∗}IC_{μ}[1]) = Ext^{1}(ℚ̄_{ℓ}, ℚ̄_{ℓ}).

And Ext^{1}(ℚ̄_{ℓ}, ℚ̄_{ℓ}) = H_{ét}^{1}(O_{μ}, ℚ̄_{ℓ}) = 0, because O_{μ} is simply connected. To show that the first group

Hom(IC_{μ}, i_{!}i^{!}IC_{μ}[1]) = Hom(i^{∗}IC_{μ}, i^{!}IC_{μ}[1])

is trivial, note that i^{∗}IC_{μ} lives in perverse degrees ≤ −1 because the 0th perverse cohomology vanishes, since IC_{μ} is a middle perverse extension along j. Hence, the Verdier dual D(i^{∗}IC_{μ})[1] = i^{!}IC_{μ}[1] lives in perverse degrees ≥ 0. This proves case (i). Case (ii): λ ≠ μ and either λ ≤ μ or μ ≤ λ. If λ ≤ μ, let i : \overline{O}_{λ} ↪ \overline{O}_{μ} be the closed embedding. Then

Hom(i_{∗}IC_{λ}, IC_{μ}[1]) = Hom(IC_{λ}, i^{!}IC_{μ}[1]),

and this vanishes, since i^{!}IC_{μ}[1] lives in perverse degrees ≥ 1 or equivalently, the Verdier dual D(i^{!}IC_{μ}) = i^{∗}IC_{μ} lives in perverse degrees ≤ −2. Indeed, by a theorem of Lusztig [12, Theorem 11c], i^{∗}IC_{μ} is concentrated in even perverse degrees, and the 0th perverse cohomology vanishes, since IC_{μ} is a middle perverse extension. If μ ≤ λ, let i : \overline{O}_{μ} ↪ \overline{O}_{λ} the closed embedding. Then

Hom(IC_{λ}, i_{∗}IC_{μ}[1]) = Hom(i^{∗}IC_{λ}, IC_{μ}[1])

vanishes, since i^{∗}IC_{λ} lives in perverse degrees ≤ −2 as before. This proves case (ii). Case (iii): λ ≰ μ and μ ≰ λ. We may assume that λ and μ are contained in the same connected component of Gr_{G}. Choose some ν ∈ X_{+}^{∨} with λ, μ ≤ ν. Consider the cartesian diagram

\begin{matrix}\overline{O}_{λ} ×_{\overline{O}_{ν}} \overline{O}_{μ} & \overline{O}_{μ} \\ \overline{O}_{λ} & \overline{O}_{ν}. \end{matrix} ι_{1} i_{1} ι_{2} i_{2}

Then adjunction gives

Hom(i_{1,∗}IC_{λ}, i_{2,∗}IC_{μ}[1]) = Hom(i_{2}^{∗}i_{1,∗}IC_{λ}, IC_{μ}[1]),  (4.2)

and i_{2}^{∗}i_{1,∗}IC_{λ} ≃ ι_{1,∗}ι_{2}^{∗}IC_{λ} by proper base change. Hence (4.2) equals Hom(ι_{2}^{∗}IC_{λ}, ι_{1}^{!}IC_{μ}[1]) which vanishes. This proves case (iii), hence the proposition.

The affine group scheme L^{+}𝔾_{m} acts on Gr_{G} as follows. For x ∈ L^{+}𝔾_{m}(R), denote by v_{x} the automorphism of Spec(R[[t]]) induced by multiplication with x. If F is a G-torsor over Spec(R[[t]]), we denote by v_{x}^{∗}F the pullback of F along v_{x}. Let (F, β) ∈ Gr_{G}(R). Then the action of L^{+}𝔾_{m} on Gr_{G} is given by

(F, β) ⟼ (v_{x^{−1}}^{∗}F, v_{x^{−1}}^{∗}β),

and is called the Virasoro action. Note that every L^{+}G-orbit in Gr_{G} is stable under L^{+}𝔾_{m}. The semidirect product L^{+}G⋊L^{+}𝔾_{m} acts on Gr_{G}, and the action on each orbit factors through a smooth connected affine group scheme. Hence, we may consider the category P_{L^{+}G⋊L^{+}𝔾_{m}}(Gr_{G}) of L^{+}G ⋊ L^{+}𝔾_{m}-equivariant perverse sheaves on Gr_{G}.

Corollary 4.2. The forgetful functor

P_{L^{+}G⋊L^{+}𝔾_{m}}(Gr_{G}) ⟶ P_{L^{+}G}(Gr_{G})

is an equivalence of categories. In particular, the category P_{L^{+}G}(Gr_{G}) does not depend on the choice of the parameter t.

Proof. By Proposition 4.1 above, every L^{+}G-equivariant perverse sheaf is a direct sum of intersection complexes, and these are L^{+}𝔾_{m}-equivariant.

Remark 4.3. If X = 𝔸_{F}^{1} is the base curve, then the global affine Grassmannian Gr_{X} splits as Gr_{X} ≃ Gr_{G} × X. Corollary 4.2 shows that we can work over an arbitrary curve X as follows. Let X be the functor on the category of F -algebras R parametrizing tuples (x, s) with

\begin{cases}x ∈ X(R) is a point; \\ s is a continuous isomorphism of R-modules Ô_{X_{R},x} →^{≃} R[[t]], \end{cases}

where Ô_{X_{R},x} is the completion of the R-module O_{X_{R},x} along the maximal ideal 𝔪_{x} at x. The affine group scheme L^{+}𝔾_{m} operates from left on X by (g, (x, s)) ↦ (x, gs). The projection p : X → X,(x, s) ↦ x gives X the structure of a L^{+}𝔾_{m}-torsor. Then Gr_{X} ≃ Gr_{G}×^{L^{+}𝔾_{m}} X, and we get a diagram of L^{+}𝔾_{m}- torsors

\begin{aligned}& Gr_{G} × X \\ Gr_{G} × X & & Gr_{X}. \end{aligned} ; \begin{matrix}p \\ q \end{matrix}

For any A ∈ P_{L^{+}G}(Gr_{G}), the perverse sheaf A ⊠ ℚ̄_{ℓ}[1] on Gr_{G} × X is L^{+}𝔾_{m}- equivariant by Corollary 4.2. Hence, p^{∗}(A ⊠ ℚ̄_{ℓ}[1]) descends along q to a perverse sheaf A˜⊠ℚ̄_{ℓ}[1] on Gr_{X}.

We are going to define a fiber functor on P_{L^{+}G}(Gr_{G}). Denote by

ω(-) = ⨁_{i∈ℤ} R^{i}Γ(Gr_{G}, -) : P_{L^{+}G}(Gr_{G}) → Vecℚ̄_{ℓ}  (4.3)

the cohomology functor with values in the category of finite dimensional ℚ̄_{ℓ}- vector spaces.

Lemma 4.4. The functor ω : P_{L^{+}G}(Gr_{G}) → Vecℚ̄_{ℓ} is additive, exact and faithful.

Proof. Additivity is immediate. Exactness follows from Proposition 4.1, since every exact sequence splits, and ω is additive. To show faithfulness, it is enough, again by Proposition 4.1, to show that the intersection cohomology of the Schubert varieties is non-zero. Indeed, we claim that the intersection cohomology of any projective variety T is non-zero. Embedding T into projective space and projecting down on hyperplanes, we obtain a generically finite morphism π : T → ℙ^{n}. Using the decomposition theorem, we see that the intersection complex of ℙ^{n} appears as a direct summand in π_{∗}IC_{T}. Hence, the intersection cohomology of T is non-zero. This proves the lemma.

Corollary 4.5. The tuple (P_{L^{+}G}(Gr_{G}), ⋆) is a neutralized Tannakian category with fiber functor ω : P_{L^{+}G}(Gr_{G}) → Vecℚ̄_{ℓ}.

Proof. We check the criterion in [5, Prop. 1.20]: The category (P_{L^{+}G}(Gr_{G}), ⋆) is abelian ℚ̄_{ℓ}-linear (cf. Appendix A below) and by Theorem 3.1 (ii) above symmetric monoidal. To prove that ω is a fiber functor, we must show that ω is an additive exact faithful tensor functor. Lemma 4.4 shows that ω is additive exact and faithful, and Theorem 3.1 (ii) shows that ω is symmetric monoidal. It remains to show that (P_{L^{+}G}(Gr_{G}), ⋆) has a unit object and that any one dimensional object has an inverse. The unit object is the constant sheaf IC_{0} = ℚ̄_{ℓ} concentrated in the base point e_{0}. We have End(IC_{0}) = ℚ̄_{ℓ}, and dim(ω(IC_{0})) = 1. Now, let A ∈ P_{L^{+}G}(Gr_{G}) with dim(ω(A)) = 1. Then A is supported on a L^{+}G-invariant closed point z_{0} ∈ Gr_{G}. There exists z in the center of LG such that z · z_{0} = e_{0} is the basepoint. If z′_{0} = z · e_{0}, then the intersection cohomology complex A′ supported on z′_{0} satisfies A ⋆ A′ = IC_{0}. This shows the corollary.

5 The Geometric Satake Equivalence

In this section we assume that F = F̄ is separably closed. Denote by H = Aut^{⋆}(ω) the affine ℚ̄_{ℓ}-group scheme of tensor automorphisms defined by Corollary 4.5.

Theorem 5.1. The group scheme H is a connected reductive group over ℚ̄_{ℓ} which is dual to G in the sense of Langlands, i.e. if we denote by Ĝ the Langlands dual group with respect to some pinning of G, then there exists an isomorphism H ≃ Ĝ determined uniquely up to inner automorphisms.

We fix some notation. Let T be a maximal split torus of G and B a Borel subgroup containing T with unipotent radical U. We denote by ⟨-, -⟩ the natural pairing between X = Hom(T, 𝔾_{m}) and X^{∨} = Hom(𝔾_{m}, T). Let R ⊂ X be the root system associated to (G, T), and R_{+} be the set of positive roots corresponding to B. Let R^{∨} ⊂ X^{∨} the dual root system with the bijection R → R^{∨}, α ↦ α^{∨}. Denote by R_{+}^{∨} the set of positive coroots. Let W the Weyl group of (G, T). Consider the half sum of all positive roots

ρ = \frac{1}{2} ∑_{α∈R_{+}} α.

Let Q^{∨} (resp. Q_{+}^{∨}) the subgroup (resp. submonoid) of X^{∨} generated by R^{∨} (resp. R_{+}^{∨}). We denote by

X_{+}^{∨} = {μ ∈ X^{∨} | ⟨α, μ⟩ ≥ 0, ∀α ∈ R_{+}}

the cone of dominant cocharacters with the partial order on X^{∨} defined as follows: λ ≤ μ if and only if μ − λ ∈ Q_{+}^{∨}. Note that (X_{+}^{∨}, ≤) identifies with the partially ordered set of orbit representatives in Section 4 as follows: for every μ ∈ X_{+}^{∨}, let t^{μ} the corresponding element in LT (F), and denote by e_{0} ∈ Gr_{G} the base point. Then μ ↦ t^{μ} · e_{0} gives the bijection of partial ordered sets, i.e. the orbit closures satisfy

\overline{O}_{μ} = ∐_{λ≤μ} O_{λ}, (Cartan stratification)

where O_{λ} denotes the L^{+}G-orbit of t^{λ} · e_{0} (cf. [17, §2]). For every ν ∈ X^{∨}, consider the LU-orbit S_{ν} = LU · t^{ν}e_{0} inside Gr_{G} (cf. [17, §3]). Then S_{ν} is a locally closed ind-subscheme of Gr_{G}, and for every μ ∈ X_{+}^{∨}, there is a locally closed stratification

\overline{O}_{μ} = ∐_{ν∈X^{∨}} S_{ν} ∩ \overline{O}_{μ}. (Iwasawa stratification)

For μ ∈ X_{+}^{∨}, let

Ω(μ) =^{def} {ν ∈ X^{∨} | wν ≤ μ, ∀w ∈ W }.

Proposition 5.2. For every ν ∈ X^{∨} and μ ∈ X_{+}^{∨} the stratum S_{ν} ∩ \overline{O}_{μ} is non-empty if and only if ν ∈ Ω(μ), and in this case it is pure of dimension ⟨ρ, μ + ν⟩.

Proof. The schemes G, B, T and all the associated data are already defined over a finitely generated ℤ-algebra. By generic flatness, we reduce to the case where F = 𝔽_{q} is a finite field. The proposition is proven in [8, Proof of Lemma 2.17.4], which relies on [17, Theorem 3.1].

For every sequence μ_{•} = (μ_{1}, …, μ_{k}) of dominant cocharacters, consider the projective variety over F

\overline{O}_{μ_{•}} =^{def} p^{−1}(\overline{O}_{μ_{1}}) ×^{L^{+}G} … ×^{L^{+}G} p^{−1}(\overline{O}_{μ_{k−1}}) ×^{L^{+}G} \overline{O}_{μ_{k}},

inside LG×^{L^{+}G} …×^{L^{+}G} Gr_{G}, where p : LG → Gr_{G} denotes the quotient map. The quotient exists, by the ind-properness of Gr_{G} and Lemma A.4 below. Now let |μ_{•}| = μ_{1} + … + μ_{k}. Then the restriction m_{μ_{•}} = m|_{\overline{O}_{μ•}} of the k-fold convolution morphism factors as

m_{μ_{•}} : \overline{O}_{μ_{•}} ⟶ \overline{O}_{|μ_{•}|},

and is an isomorphism over O_{|μ_{•}|} ⊂ \overline{O}_{|μ_{•}|}.

Corollary 5.3. For every λ ∈ X_{+}^{∨} with λ ≤ |μ_{•}| and x ∈ O_{λ}(F), one has

dim(m_{μ_{•}}^{−1}(x)) ≤ ⟨ρ, |μ_{•}| − λ⟩,

i.e. the convolution morphism is semismall.

Proof. The proof of [17, Lemme 9.3] carries over word by word, and we obtain that dim(m_{μ_{•}}^{−1}(O_{λ})) ≤ ⟨ρ, |μ_{•}| + λ⟩. Since m_{μ_{•}} is L^{+}G-equivariant and dim(O_{λ}) = ⟨2ρ, λ⟩, the corollary follows.

The convolution IC_{μ_{1}} ⋆ … ⋆ IC_{μ_{n}} is a L^{+}G-equivariant perverse sheaf, and by Proposition 4.1, we can write

IC_{μ_{1}} ⋆ … ⋆ IC_{μ_{n}} ≃ ⨁_{λ≤|μ_{•}|} V_{μ_{•}}^{λ} ⊗ IC_{λ},  (5.1)

where V_{μ_{•}}^{λ} are finite dimensional ℚ̄_{ℓ}-vector spaces.

Lemma 5.4. For every λ ∈ X_{+}^{∨} with λ ≤ |μ_{•}| and x ∈ O_{λ}(F), the vector space V_{μ_{•}}^{λ} has a canonical basis indexed by the irreducible components of m_{μ_{•}}^{−1}(x) of exact dimension ⟨ρ, |μ_{•}| − λ⟩.

Proof. We follow the argument in Haines [9]. We claim that IC_{μ_{•}} = IC_{μ_{1}} ˜⊠ … ˜⊠ IC_{μ_{k}} is the intersection complex on \overline{O}_{μ_{•}}. Indeed, this can be checked locally in the smooth topology, and then easily follows from the definitions. Hence, the left hand side of (5.1) is equal to m_{μ_{•},∗}(IC_{μ_{•}}). If d = − dim(O_{λ}), then taking the d-th stalk cohomology at x in (5.1) gives by proper base change

R^{d}Γ(m_{μ_{•}}^{−1}(x), IC_{μ_{•}}) ≃ V_{μ_{•}}^{λ}.

Since m_{μ_{•}} : \overline{O}_{μ_{•}} → \overline{O}_{|μ_{•}|} is semismall, the cohomology R^{d}Γ(m_{μ_{•}}^{−1}(x), IC_{μ_{•}}) admits by [9, Lemma 3.2] a canonical basis indexed by the top dimensional irreducible components. This proves the lemma.

In the following, we consider \overline{O}_{μ_{•}} as a closed projective subvariety of

\overline{O}_{μ_{1}} × \overline{O}_{μ_{1}+μ_{2}} × … × \overline{O}_{μ_{1}+...+μ_{k}},

via (g_{1}, …, g_{k}) ↦ (g_{1}, g_{1}g_{2}, …, g_{1} … g_{k}). The lemma below is the geometric analogue of the PRV-conjecture.

Lemma 5.5. For every λ ∈ X_{+}^{∨} of the form λ = ν_{1} + … + ν_{k} with ν_{i} ∈ W μ_{i} for i = 1, …, k, the perverse sheaf IC_{λ} appears as a direct summand in IC_{μ_{1}} ⋆ … ⋆ IC_{μ_{k}}.

Proof. Let ν = w(ν_{2} +…+ν_{k}) be the unique dominant element in the W -orbit of ν_{2} + … + ν_{k}. Then λ = ν_{1} + w^{−1}ν. Hence, by induction, we may assume k = 2. By Lemma 5.4, it is enough to show that there exists x ∈ O_{λ}(F) such that m_{μ_{•}}^{−1}(x) is of exact dimension ⟨ρ, |μ_{•}| − λ⟩. Let w ∈ W such that wν_{1} is dominant, and consider wλ = wν_{1} + wν_{2}. We denote by S_{wν_{•}} ∩ \overline{O}_{μ_{•}} the intersection inside \overline{O}_{μ_{1}} × \overline{O}_{μ_{1}+μ_{2}}

S_{wν_{•}} ∩ \overline{O}_{μ_{•}} =^{def} (S_{wν_{1}} × S_{wν_{1}+wν_{2}}) ∩ \overline{O}_{μ_{•}}.

The convolution is then given by projection on the second factor. By [17, Lemme 9.1], we have a canonical isomorphism

S_{wν_{•}} ∩ \overline{O}_{μ_{•}} ≃ (S_{wν_{1}} ∩ \overline{O}_{μ_{1}}) × (S_{wν_{2}} ∩ \overline{O}_{μ_{2}}).

Let y = (y_{1}, y_{2}) in (S_{wν_{•}} ∩ \overline{O}_{μ_{•}})(F). Since for i = 1, 2 the elements wν_{i} are conjugate under W to μ_{i}, there exist by [17, Lemme 5.2] elements u_{1}, u_{2} ∈ L^{+}U(F) such that

y_{1} = u_{1}t^{wν_{1}} · e_{0}
y_{2} = u_{1}t^{wν_{1}}u_{2}t^{wν_{2}} · e_{0}.

The dominance of wν_{1} implies t^{wν_{1}}u_{2}t^{−wν_{1}} ∈ L^{+}U(F), and hence Y = S_{wν_{•}} ∩ \overline{O}_{μ_{•}} maps under the convolution morphism onto an open dense subset Y′ in S_{wλ}∩O_{λ}. Denote by h = m_{μ_{•}}|_{Y} the restriction to Y. Both Y, Y′ are irreducible schemes (their reduced loci are isomorphic to affine space), thus by generic flatness, there exists x ∈ Y′(F) such that

dim(h^{−1}(x)) = dim(Y) − dim(Y′) = ⟨ρ, |μ_{•}| + wλ⟩ − ⟨ρ, λ + wλ⟩ = ⟨ρ, |μ_{•}| − λ⟩.

In particular, dim(m_{μ_{•}}^{−1}(x)) ≥ ⟨ρ, |μ_{•}| − λ⟩, and hence equality by Corollary 5.3.

For the proof of Theorem 5.1, we introduce a weaker partial order ⪯ on X_{+}^{∨} defined as follows: λ ⪯ μ if and only if μ − λ ∈ ℝ_{+}Q_{+}^{∨}. Then λ ≤ μ if and only if λ ⪯ μ and their images in X^{∨}/Q^{∨} coincide (cf. Lemma B.2 below).

Proof of Theorem 5.1. We proceed in several steps:

(1) The affine group scheme H is of finite type over ℚ̄_{ℓ}. By [5, Proposition 2.20 (b)] this is equivalent to the existence of a tensor generator in P_{L^{+}G}(Gr_{G}). Now there exist μ_{1}, …, μ_{k} ∈ X_{+}^{∨} which generate X_{+}^{∨} as semigroups. Then IC_{μ_{1}} ⊕ … ⊕ IC_{μ_{k}} is a tensor generator.

(2) The affine group scheme H is connected reductive.

For every μ ∈ X_{+}^{∨} and k ∈ ℕ, the sheaf IC_{kμ} is a direct summand of IC_{μ}^{⋆k}, hence the scheme H is connected by [5, Corollary 2.22]. By [5, Proposition 2.23], the connected algebraic group H is reductive if and only if P_{L^{+}G}(Gr_{G}) is semisimple, and this is true by Proposition 4.1.

(3) The root datum of H is dual to the root datum of G.

Let (X′, R′, Δ′, X^{′∨}, R^{′∨}, Δ^{′∨}) the based root datum of H constructed in Theorem B.1 below. By Lemma B.5 below it is enough to show that we have an isomorphism of partially ordered semigroups

(X_{+}^{∨}, ≤) ⟶^{≃} (X′_{+}, ≤′).  (5.2)

By Proposition 4.1, the map X_{+}^{∨} → X′_{+}, μ ↦ [IC_{μ}], where [IC_{μ}] is the class of IC_{μ} in K_{0}^{+}P_{L^{+}G}(Gr_{G}) is a bijection of sets. For every λ, μ ∈ X_{+}^{∨}, we claim that λ ⪯ μ if and only if [IC_{λ}] ⪯′ [IC_{μ}]. Assume λ ⪯ μ, and choose a finite subset F ⊂ X_{+}^{∨} satisfying Proposition B.3 (iii). Let A = ⊕_{ν∈F} IC_{ν}, and suppose IC_{χ} is a direct summand of IC_{λ}^{⋆k} for some k ∈ ℕ. In particular, χ ≤ kλ and so χ ∈ W F + ∑_{i=1}^{k} W μ. By Lemma 5.5, the sheaf IC_{χ} is a direct summand of IC_{μ}^{⋆k} ⋆ A, which means [IC_{λ}] ⪯′ [IC_{μ}]. Conversely, assume [IC_{λ}] ⪯′ [IC_{μ}]. Using Proposition B.3 (iv) below, this translates, by looking at the support, into the following condition: there exists ν ∈ X_{+}^{∨} such that \overline{O}_{kλ} ⊂ \overline{O}_{kμ+ν} holds for infinitely many k ∈ ℕ. Equivalently, kλ ≤ kμ + ν for infinitely many k ∈ ℕ which implies λ ⪯ μ. For every λ, μ ∈ X_{+}^{∨}, we claim that [IC_{λ}] + [IC_{μ}] = [IC_{λ+μ}] in X′_{+}: by the proof of Theorem B.1 below, [IC_{λ}] + [IC_{μ}] is the class of the maximal element appearing in IC_{λ} ⋆ IC_{μ}. Since the partial orders ⪯, ⪯′ agree, this is [IC_{λ+μ}]. It remains to show that the partial orders ≤, ≤′ agree. The identification X_{+}^{∨} = X′_{+} prolongs to X^{∨} = X′. We claim that Q_{+}^{∨} = Q′_{+} under this identification and hence Q^{∨} = Q′, which is enough by Lemma B.2 below. Let α^{∨} ∈ Q_{+}^{∨} a simple coroot, and choose some μ ∈ X_{+}^{∨} with ⟨α, μ⟩ = 2. Then μ + s_{α}(μ) = 2μ − α^{∨} is dominant, and hence IC_{2μ−α^{∨}} appears by Lemma 5.5 as a direct summand in IC_{μ}^{⋆2}. By Lemma B.4 this means α^{∨} ∈ Q′_{+}, and thus Q_{+}^{∨} ⊂ Q′_{+}. Conversely, assume α′ ∈ Q′_{+} has the property that there exists μ ∈ X′_{+} with 2μ − α′ ∈ X′_{+} and IC_{2μ−α′} appears as a direct summand in IC_{μ}^{⋆2}. Note that every element in Q′_{+} is a sum of these elements. Then 2μ − α′ ≤ 2μ, and hence α′ ∈ Q_{+}^{∨}. This shows Q′_{+} ⊂ Q_{+}^{∨} and finishes the proof of (5.2).

6 Galois Descent

Let F be any field, and G a connected reductive group defined over F. Fix a separable closure F̄, and let Γ_{F} = Gal(F̄/F) be the absolute Galois group. Let Repℚ̄_{ℓ}(Γ_{F}) be the category of finite dimensional continuous ℓ-adic Galois representations. For any object defined over F, we denote by a subscript (-)F̄ its base change to F̄. Consider the functor

Ω : P_{L^{+}G}(Gr_{G}) ⟶ Repℚ̄_{ℓ}(Γ_{F})
A ⟼ ⨁_{i∈ℤ} R^{i}Γ(Gr_{G,F̄}, AF̄).

There are canonical isomorphisms of fpqc-sheaves (LG)F̄ ≃ LGF̄, (L^{+}G)F̄ ≃ L^{+}GF̄ and Gr_{G,F̄} ≃ Gr_{GF̄}. Hence, Ω ≃ ω ∘ (-)F̄, cf. (4.3). The absolute Galois group Γ_{F} operates on the Tannakian category P_{L^{+}GF̄} (Gr_{GF̄}) by tensor equivalences compatible with the fiber functor ω. Hence, we may form the semidirect product ^{L}G = Aut^{⋆}(ω)(ℚ̄_{ℓ}) ⋊ Γ_{F} considered as a topological group as follows. The group Aut^{⋆}(ω)(ℚ̄_{ℓ}) is equipped with the ℓ-adic topology, the Galois Γ_{F} group with the profinite topology and ^{L}G with the product topology. Note that Γ_{F} acts continuously on Aut^{⋆}(ω)(ℚ̄_{ℓ}) by Proposition 6.6 below. Let Repc ; ℚ̄_{ℓ}(^{L}G) be the full subcategory of the category finite dimensional continuous ℓ-adic representations of LG such that the restriction to Aut^{⋆}(ω)(ℚ̄_{ℓ}) is algebraic.

Theorem 6.1. The functor Ω is an equivalence of abelian tensor categories

Ω : P_{L^{+}G}(Gr_{G}) ⟶ Repc ; ℚ̄_{ℓ}(^{L}G)
A ⟼ Ω(A).

The proof of Theorem 6.1 proceeds in several steps.

Lemma 6.2. Let H be an affine group scheme over a field k. Let Rep_{k}(H) be the category of algebraic representations of H, and let Rep_{k}(H(k)) be the category of finite dimensional representations of the abstract group H(k). Assume that H is reduced and that H(k) ⊂ H is dense. Then the functor

Ψ : Rep_{k}(H) ⟶ Rep_{k}(H(k))
ρ ⟼ ρ(k)

is a fully faithful embedding.

□

We recall some facts on the Tannakian formalism from [20]. Let (C, ⊗) be a neutralized Tannakian category over a field k with fiber functor v. We define a monoidal category Aut^{⊗}(C, v) as follows. Objects are pairs (σ, α), where σ : C → C is a tensor automorphism and α : v ∘ σ → v is a natural isomorphism of tensor functors. Morphisms between (σ, α) and (σ′, α′) are natural tensor isomorphisms between σ and σ′ that are compatible with α, α′ in an obvious way. The monoidal structure is given by compositions. Since v is faithful, Aut^{⊗}(C, v) is equivalent to a set, and in fact is a group. Let H = Aut_{C}^{⊗}(v), the Tannakian group defined by (C, v). There is a canonical action of Aut^{⊗}(C, v) on H by automorphisms as follows. Let (σ, α) be in Aut^{⊗}(C, v) . Let R be a k-algebra, and let h : v_{R} → v_{R} be a R-point of H. Then (σ, α) · h is the following composition

v_{R} ⟶^{α^{−1}} v_{R} ∘ σ ⟶^{h∘id} v_{R} ∘ σ ⟶^{α} v_{R}.

Let Γ be an abstract group. Then an action of Γ on (C, v) is by definition a group homomorphism act : Γ → Aut^{⊗}(C, v). Assume that Γ acts on (C, v). Then we define C^{Γ}, the category of Γ-equivariant objects in C as follows. Objects are (X, {c_{γ}}_{γ∈Γ}), where X is an object in C and c_{γ} : act_{γ}(X) ≃ X is an isomorphism, satisfying the natural cocycle condition, i.e. c_{γ′γ} = c_{γ′} ∘act_{γ′}(c_{γ}). The morphisms between (X, {c_{γ}}_{γ∈Γ}) and (X′, {c′_{γ}}_{γ∈Γ}) are morphisms between X and X′, compatible with c_{γ}, c′_{γ} in an obvious way.

Lemma 6.3. Let Γ be a group acting on (C, v).

(i) The category C^{Γ} is an abelian tensor category. (ii) Assume that H is reduced and that k is algebraically closed. The functor v is an equivalence of abelian tensor categories

C^{Γ} ≃ Rep_{k}^{o}(H(k) ⋊ Γ)

where Rep_{k}^{o}(H(k) ⋊ Γ) is the full subcategory of finite dimensional representations of the abstract group H(k) ⋊ Γ such that the restriction to H(k) is algebraic.

Remark 6.4. In fact, the category C^{Γ} is neutralized Tannakian with fiber functor v. If Γ is finite, then Aut_{C^{Γ}}^{⊗}(v) ≃ H ⋊ Γ. However, if Γ is not finite, then Aut_{C^{Γ}}^{⊗}(v) is in general not H ⋊ Γ, where the latter is regarded as an affine group scheme.

Proof of Lemma 6.3. The monoidal structure on C^{Γ} is defined as

(X, {c_{γ}}_{γ∈Γ}) ⊗ (X′, {c′_{γ}}_{γ∈Γ}) = (X′′, {c′′_{γ}}_{γ∈Γ}),

where X′′ = X ⊗ X′ and c′′_{γ} : act_{γ}(X′′) → X′′ is the composition

act_{γ}(X ⊗ X′) ≃ act_{γ}(X) ⊗ act_{γ}(X′) ⟶^{c_{γ}⊗c′_{γ}} X ⊗ X′.

This gives C^{Γ} the structure of an abelian tensor category. Now assume that H is reduced and that k is algebraically closed. It is enough to show that as tensor categories

Ψ : Rep_{k}(H)^{Γ} ⟶^{≃} Rep_{k}^{o}(H(k) ⋊ Γ)

compatible with the forgetful functors. Let ((V, ρ), {c_{γ}}_{γ∈Γ}) ∈ Rep_{k}(H)^{Γ}. Then we define (V, ρ_{Γ}) ∈ Rep_{k}^{o}(H(k) ⋊ Γ) by

(h, γ) ⟼ ρ(h) ∘ α_{h}(V) ∘ v ∘ c_{γ}^{−1} ∈ GL(V), where α_{h} : v ∘ σ_{h} ≃ v is induced by the action of Γ as above. Using the cocycle relation, one checks that this is indeed a representation. By Lemma 6.2, the natural map

Hom_{H}(ρ, ρ′) ⟶ Hom_{H(k)}(ρ(k), ρ′(k))

is bijective. Taking Γ-invariants shows that the functor Ψ is fully faithful. Essential surjectivity is obvious.

Now we specialize to the case (C, ⊗) = (P_{L^{+}GF̄} (Gr_{G,F̄}), ⋆) with fiber functor v = ω. Then the absolute Galois group Γ = ΓF̄ acts on this Tannakian category (cf. Appendix A.1).

Proof of Theorem 6.1. The functor Ω is fully faithful. Let P_{L^{+}GF̄} (Gr_{G,F̄})^{Γ,c} be the full subcategory of P_{L^{+}GF̄} (Gr_{G,F̄})^{Γ} consisting of perverse sheaves together with a continuous descent datum (cf. Appendix A.1). By Lemma A.6, the functor A ↦ AF̄ is an equivalence of abelian categories P_{L^{+}G}(Gr_{G}) ≃ P_{L^{+}GF̄} (Gr_{G,F̄})^{Γ,c}. Hence, we get a commutative diagram

P_{L^{+}GF̄} (Gr_{G,F̄})^{Γ}  Repo ; ℚ̄_{ℓ}(^{L}G)

P_{L^{+}G}(Gr_{G})  Repc ; ℚ̄_{ℓ}(^{L}G), ω Ω A ↦ AF̄

where ω is an equivalence of categories by Lemma 6.3 (ii), and where the vertical arrows are fully faithful. Hence, Ω is fully faithful. The functor Ω is essentially surjective. Let ρ be in Repc ; ℚ̄_{ℓ}(^{L}G). Without loss of generality, we assume that ρ is indecomposable. Let H = Aut^{∗}(ω). By Proposition 4.1, the restriction ρ|_{H} is semisimple. Denote by A the set of isotypic components of ρ|_{H}. Then Γ_{F} operates transitively on A, and for every a ∈ A its stabilizer in Γ_{F} is the absolute Galois group Γ_{E} for some finite separable extension E/F. By Galois descent along finite extensions, we may assume that E = F, and hence that ρ|_{H} has only one isotypic component. Let ρ_{0} be the simple representation occuring in ρ|_{H}. Then Hom_{H}(ρ_{0}, ρ) is a continuous Γ-representation, and the natural morphism

ρ_{0} ⊗ Hom_{H}(ρ_{0}, ρ) ⟶ ρ

given by v ⊗ f ↦ f(v) is an isomorphism of ^{L}G-representations. Let IC_{X} be the simple perverse sheaf on Gr_{G,F̄} with ω(IC_{X}) ≃ ρ_{0}. Since ρ has only one isotypic component, the support X = supp(IC_{X}) is Γ-invariant, and hence defined over F. Denote by V the local system on Spec(F) given by the Γrepresentation Hom_{H}(ρ_{0}, ρ). Then IC_{X} ⊗ V is an object in P_{L^{+}G}(Gr_{G}) such that Ω(IC_{X} ⊗ V) ≃ ρ_{0} ⊗ Hom_{H}(ρ_{0}, ρ). This proves the theorem.

The proof of Theorem 6.1 also shows the following fact.

Corollary 6.5. Let A ∈ P_{L^{+}G}(Gr_{G}) indecomposable. Let {X_{i}}_{i∈I} be the set of irreducible components of supp(AF̄). Denote by E the minimal finite separable extension of F such that X_{i} is defined over E for all i ∈ I. Then as perverse sheaves on Gr_{E}

A_{E} ≃ ⨁_{i∈I} IC_{X_{i}} ⊗ V_{i},

where V_{i} are indecomposable local systems on Spec(E).

□

We briefly explain the connection to the full L-group. For more details see [20]. Let Ĝ be the reductive group over ℚ̄_{ℓ} dual to GF̄ in the sense of Langlands, i.e. the root datum of Ĝ is dual to the root datum of GF̄. There are two natural actions of Γ_{F} on Ĝ as follows. Up to the choice of a pinning (Ĝ, B̂, T̂, X̂) of Ĝ, we have an action act^{alg} via

act^{alg} : Γ_{F} → Out(GF̄) ≃ Out(Ĝ) ≃ Aut(Ĝ, B̂, T̂, X̂) ⊂ Aut(Ĝ),  (6.1)

where Out(-) denotes the outer automorphisms. On the other hand, we have an action act^{geo} : ΓF̄ → Aut(Ĝ) via the Tannakian equivalence from Theorem 5.1. The relation between act^{geo} and act^{alg} is as follows. Let cycl : Γ_{F} → ℤ_{ℓ}^{×} be the cyclotomic character of Γ_{F} defined by the action of Γ_{F} on the ℓ^{∞}-roots of unity of F̄. Let Ĝ_{ad} be the adjoint group of Ĝ. Let ρ be the half sum of positive coroots of Ĝ, which gives rise to a one-parameter group ρ : 𝔾_{m} → Ĝ_{ad}. We define a map

χ : Γ_{F} ⟶^{cycl} ℤ_{ℓ}^{×} ⟶^{ρ} Ĝ_{ad}(ℚ̄_{ℓ}),

which gives a map Ad_{χ} : Γ_{F} → Aut(Ĝ) to the inner automorphism of Ĝ.

Proposition 6.6 ([20] Proposition A.4). For all γ ∈ Γ_{F},

act^{geo}(γ) = act^{alg}(γ) ∘ Ad_{χ}(γ).
□

Remark 6.7. Proposition 6.6 shows that act^{geo} only depends on the quasi-split form of G, since the same is true for act^{alg}. In particular, the Satake category P_{L^{+}G}(Gr_{G}) only depends on the quasi-split form of G whereas the ind-scheme Gr_{G} does depend on G.

Let^{L}G^{alg} = Ĝ(ℚ̄_{ℓ})⋊_{act}_{alg} Γ_{F} be the full L-group. Set^{L}G^{geo} = Ĝ(ℚ̄_{ℓ})⋊_{act}^{geo}Γ_{F}.

Corollary 6.8 ([20] Corollary A.5). The map (g, γ) ↦ (Ad_{χ(γ^{−1})}(g), γ) gives an isomorphism ^{L}G^{alg} →^{≃} ^{L}G^{geo}.

□

Combining Corollary 6.8 with Theorem 6.1, we obtain the following corollary.

Corollary 6.9. There is an equivalence of abelian tensor categories

P_{L^{+}G}(Gr_{G}) ≃ Repc ; ℚ̄_{ℓ}(^{L}G^{alg}),

where Repc ; ℚ̄_{ℓ}(^{L}G^{alg}) denotes the full subcategory of the category of finite dimensional continuous ℓ-adic representations of ^{L}G^{alg} such that the restriction to Ĝ(ℚ̄_{ℓ}) is algebraic.

□

A Perverse Sheaves

For the construction of the category of ℓ-adic perverse sheaves, we refer to the work of Y. Laszlo and M. Olsson [13]. In this appendix we explain our conventions on perverse sheaves on ind-schemes. Let F be an arbitrary field. Fix a prime ℓ ≠ char(F), and denote by ℚ_{ℓ} the field of ℓ-adic numbers with algebraic closure ℚ̄_{ℓ}. For any separated scheme T of finite type over F, we consider the bounded derived category D_{c}^{b}(T, ℚ̄_{ℓ}) of constructible ℓ-adic sheaves on T. Let P (T) be the abelian ℚ̄_{ℓ}-linear full subcategory of ℓ-adic perverse sheaves, i.e. the heart of the perverse t-structure on the triangulated category D_{c}^{b}(T, ℚ̄_{ℓ}). Now let (T)_{i∈I} be an inductive system of separated schemes of finite type over F with closed immersions as transition morphisms. A fpqc-sheaf T on the category of F -algebras is called a strict ind-scheme of ind-finite type over F if there is an isomorphism of fpqc-sheaves T ≃ \underrightarrow{lim}_{i} T_{i}, for some system (T)_{i∈I} as above. The inductive system (T)_{i∈I} is called an ind-presentation of T . For i ≤ j, push forward gives transition morphisms D_{c}^{b}(T_{i}, ℚ̄_{ℓ}) → D_{c}^{b}(T_{j}, ℚ̄_{ℓ}) which restrict to P (T_{i}) → P (T_{j}), because push forward along closed immersions is t-exact.

Definition A.1. Let T be a strict ind-scheme of ind-finite type over F, and (T_{i})_{i∈I} be an ind-presentation.

(i) The bounded derived category of constructible ℓ-adic complexes D_{c}^{b}(T , ℚ̄_{ℓ}) on T is the inductive limit

D_{c}^{b}(T , ℚ̄_{ℓ}) =^{def} \underrightarrow{lim}_{i} D_{c}^{b}(T_{i}, ℚ̄_{ℓ}).

(ii) The category of ℓ-adic perverse sheaves P (T ) on T is the inductive limit

P (T ) =^{def} \underrightarrow{lim}_{i} P (T_{i}).

The definition is independent of the chosen ind-presentation of T . The category D_{c}^{b}(T , ℚ̄_{ℓ}) inherits a triangulation and a perverse t-structure from the D_{c}^{b}(T_{i}, ℚ̄_{ℓ})’s. The heart with respect to the perverse t-structure is the abelian ℚ̄_{ℓ}-linear full subcategory P (T ).

If f : T → S is a morphism of strict ind-schemes of ind-finite type over F, we have the Grothendieck operations f_{∗}, f_{!}, f^{∗}, f^{!}, and the usual constructions carry over after the choice of ind-presentations. In Section 3.3 we work with equivariant objects in the category of perverse sheaves. The context is as follows. Let f : T → S be a morphism of separated schemes of finite type, and let H be a smooth affine group scheme over S with geometrically connected fibers acting on f : T → S. Then a perverse sheaf A on T is called H-equivariant if there is an isomorphism in the derived category

θ : a^{∗}A ≃ p^{∗}A,  (A.1)

where a : H ×_{S} T → T (resp. p : H ×_{S} T → T) is the action (resp. projection on the second factor). A few remarks are in order: if the isomorphism (A.1) exists, then it can be rigidified such that e_{T}^{∗} θ is the identity, where e_{T} : T → H ×_{S} T is the identity section. A rigidified isomorphism θ automatically satisfies the cocycle relation due to the fact that H has geometrically connected fibers. The subcategory P_{H}(T) of P (T) of H-equivariant objects together with Hequivariant morphisms is called the category of H-equivariant perverse sheaves on T.

Lemma A.2 ([13] Remark 5.5). Consider the stack quotient H\T, an Artin stack of finite type over S. Let p : T → H\T be the quotient map of relative dimension d = dim(T/S). Then the pull back functor

p^{∗}[d] : P (H\T) ⟶ P_{H}(T),

is an equivalence of categories. In particular, P_{H}(T) is abelian and ℚ̄_{ℓ}-linear.

□

Now let T be a strict ind-scheme of ind-finite type, and f : T → S a morphism to a separated scheme of finite type. Fix an ind-presentation (T_{i})_{i∈I} of T . Let (H_{i})_{i∈I} be an inverse system of smooth affine group scheme with geometrically connected fibers. Let H = \underleftarrow{lim}_{i} H_{i} be the inverse limit, an affine group scheme over S, because the transition morphism are affine. Assume that H acts on f : T → S such that the action restricts to the inductive system (f|_{T_{i}})_{i∈I}. Assume that the H-action factors through H_{i} on f|_{T_{i}} for every i ∈ I.

Definition A.3. Let f : T → S, (T_{i})_{i∈I} and H as above. The category P_{H}(T ) of H-equivariant perverse sheaves on T is the inductive limit

P_{H}(T ) =^{def} \underrightarrow{lim}_{i} P_{H_{i}}(T_{i}).

It follows from Lemma A.2 that the category P_{H}(T ) is an abelian ℚ̄_{ℓ}-linear category. The following lemma is used throughout the text.

Lemma A.4. Let T → S be a H-torsor, and let Y be a S-scheme with H-action. Assume that the action of H on Y factors over H_{i} for i >> 0. Then there is a canonical isomorphism of fpqc-sheaves

T ×^{H} Y ⟶^{≃} T^{(i)} ×^{H_{i}} Y,

where T^{(i)} = T ×^{H} H_{i}.

□

Remark A.5. In particular, if T^{(i)} ×^{H_{i}} Y is representable of finite type, then is T ×^{H} Y is representable of finite type.

A.1 Galois Descent of Perverse Sheaves

Fix a separable closure F̄ of F. Let Γ = Gal(F̄/F) be the absolute Galois group. For any complex of sheaves A on T, we denote by AF̄ its base change to TF̄ = T ⊗ F̄. We define the category of perverse sheaves with continuous Γdescent datum P (TF̄)^{Γ,c} as follows. The objects are pairs (A, {c_{γ}}_{γ∈Γ}), where A ∈ P (TF̄) and {c_{γ}}_{γ∈Γ} is a family of isomorphisms

c_{γ} : γ_{∗}A ⟶^{≃} A,

satisfying the cocycle condition c_{γ′γ} = c_{γ′} ∘ γ′_{∗}(c_{γ}) such that the datum is continuous in the following sense. For every i ∈ ℤ and every locally closed subscheme S ⊂ T such that the standard cohomology sheaf H^{i}(A)|_{S} is a local system, and for every U → S étale, with U separated quasi-compact, the induced ℓ-adic Galois representation on the UF̄-sections

Γ ⟶ GL(H^{i}(A)(UF̄)),

is continuous. The morphisms in P (TF̄)^{Γ,c} are morphisms in P (TF̄) compatible with the c_{γ}’s. For every A ∈ P (T), its pullback AF̄ admits a canonical continuous descent datum. Hence, we get a functor

Φ : P (T) ⟶ P (TF̄)^{Γ,c}
A ⟼ AF̄.

Lemma A.6 (SGA 7, XIII, 1.1). The functor Φ is an equivalence of categories.

□

B Reconstruction of Root Data

Let G a split connected reductive group over an arbitrary field k. Denote by Rep_{G} the Tannakian category of algebraic representations of G. If k is algebraically closed of characteristic 0, then D. Kazhdan, M. Larsen and Y. Varshavsky [10, Corollary 2.5] show how to reconstruct the root datum of G from the Grothendieck semiring K_{0}^{+}[G] = K_{0}^{+}Rep_{G}. In fact, their construction works over arbitrary fields. This relies on the conjecture of Parthasarathy, Ranga-Rao and Varadarajan (PRV-conjecture) proven by S. Kumar [11] (char(k) = 0) and O. Mathieu [15] (char(k) > 0).

Theorem B.1. The root datum of G can be reconstructed from the Grothendieck semiring K_{0}^{+}[G].

This means, if H is another split connected reductive group over k, and if φ : K_{0}^{+}[H] → K_{0}^{+}[G] is an isomorphism of Grothendieck semirings, then there exists an isomorphism of group schemes ϕ : H → G determined uniquely up to inner automorphism such that ϕ = K_{0}^{+}[φ]. Let T be a maximal split torus of G and B a Borel subgroup containing T. We denote by ⟨-, -⟩ the natural pairing between X = Hom(T, 𝔾_{m}) and X^{∨} = Hom(𝔾_{m}, T). Let R ⊂ X be the root system associated to (G, T), and R_{+} be the set of positive roots corresponding to B. Let R^{∨} ⊂ X^{∨} the dual root system with the bijection R → R^{∨}, α ↦ α^{∨}. Denote by R_{+}^{∨} the set of positive coroots. Let W the Weyl group of (G, T). Consider the half sum of all positive roots

ρ = \frac{1}{2} ∑_{α∈R_{+}} α.

Let Q (resp. Q_{+}) the subgroup (resp. submonoid) of X generated by R (resp. R_{+}). We denote by

X_{+} = {μ ∈ X | ⟨μ, α⟩ ≥ 0, ∀α ∈ R_{+}^{∨}}

the cone of dominant characters. We consider partial orders ≤ and ⪯ on X defined as follows. For λ, μ ∈ X, we define λ ≤ μ if and only if μ − λ ∈ Q_{+}, and we define λ ⪯ μ if and only if μ − λ = ∑_{α∈Δ} x_{α}α with x_{α} ∈ ℝ_{≥0}. The latter order is weaker than the former order in the sense that λ ≤ μ implies λ ⪯ μ, but in general not conversely.

Lemma B.2 ([18]). For every λ, μ ∈ X_{+}, then λ ≤ μ if and only if λ ⪯ μ and the images of λ, μ in X/Q agree.

Let

Dom_{⪯μ} = {ν ∈ X_{+} | ν ⪯ μ}.

For a finite subset F of the euclidean vector space E = X ⊗ ℝ, we denote by Conv(F) its convex hull.

Proposition B.3. For λ, μ ∈ X_{+}, the following conditions are equivalent:

(i) λ ⪯ μ (ii) Conv(W λ) ⊂ Conv(W μ) (iii) There exists a finite subset F ⊂ X_{+} such that for all k ∈ ℕ:

Dom_{⪯kλ} ⊂ W F + ∑_{i=1}^{k} W μ

(iv) There exists a representation U such that for every k ∈ ℕ, every irreducible subquotient of V_{λ}^{⊗k} is a subquotient of V_{μ}^{⊗k} ⊗ U.

Proof. The equivalence of (i) and (ii) is well-known. The implication (ii)⇒(iii) follows from [10, Lemma 2.4]. Assume (iii), we show that (iv) holds: let U = ⊕_{ν∈F} V_{ν}, and suppose V_{χ} is a irreducible subquotient of V_{λ}^{⊗k}, in particular χ ≤ kλ. By (iii), χ has the form wν + ∑_{i=1}^{k} w_{i}μ with w, w_{1}, …, w_{k} ∈ W and ν ∈ F. Using the PRV-conjecture [4, Theorem 4.3.2], we conclude that V_{χ} is a subquotient of V_{μ}^{⊗k} ⊗ V_{ν}, hence also of V_{μ}^{⊗k} ⊗ U. This shows (iv). The implication (iv)⇒(i) is shown in [10, Proposition 2.2].

For μ ∈ X_{+}, let v_{μ} be the corresponding element in K_{0}^{+}[G]. Let Q_{+} ⊂ X be the semigroup generated by the set

{α ∈ X | ∃ μ ∈ X_{+} : 2μ − α ∈ X_{+} and v_{μ}^{2} − v_{2μ−α} ∈ K_{0}^{+}[G]}.

Lemma B.4. There is an equality of semigroups Q_{+} = Q_{+}.

Proof. It is obvious that Q_{+} ⊂ Q_{+}, and we show that Q_{+} contains the simple roots. Let α be a simple root, and choose some μ ∈ X such that ⟨μ, α^{∨}⟩ = 2. Then 2μ−α paired with any simple root is positive, and hence μ+s_{α}(μ) = 2μ−α is dominant. By the P RV -conjecture [4, Theorem 4.3.2], the representation V_{2μ−α} appears as an irreducible subquotient in V_{μ}^{⊗2}, i.e. v_{μ}^{2} − v_{2μ−α} ∈ K_{0}^{+}[G].

The proof of Theorem B.1 goes along the lines of [10, Corollary 2.5].

Proof of Theorem B.1. By Lemma B.5 below it is enough to construct the partially ordered semigroup (X_{+}, ≤) of dominant weights. The underlying set of dominant weights X_{+} is the set of irreducible objects in K_{0}^{+}[G]. Then the partial order ⪯ on X_{+} is characterized by Proposition B.3 as follows: for λ, μ ∈ X_{+}, one has λ ⪯ μ if and only if there exists a u ∈ K_{0}^{+}[G] such that for all k ∈ ℕ and ν ∈ X_{+},

v_{λ}^{k} − v_{ν} ∈ K_{0}^{+}[G] ⟹ v_{μ}^{k} · u − v_{ν} ∈ K_{0}^{+}[G].

The semigroup structure on X_{+} is given by: for λ, μ ∈ X_{+}, one has ν = λ+μ if and only if ν is the unique dominant weight which is maximal (w.r.t. ⪯) with the property that v_{λ} · v_{μ} − v_{ν} ∈ K_{0}^{+}[G]. Now X is the group completion of X_{+}, and by Lemma B.4 we can reconstruct Q_{+} ⊂ X. Then Q is the group completion of Q_{+}, and by Lemma B.2 we can reconstruct ≤. This shows that the root datum of G can be reconstructed from K_{0}^{+}[G]. Now if H is another split connected reductive group over k, and φ : K_{0}^{+}[H] → K_{0}^{+}[G] an isomorphism of Grothedieck semirings, then the argument above shows that there is an isomorphism of partially ordered semigroups

(X_{+}^{H}, ≤^{H}) ⟶ (X_{+}^{G}, ≤^{G})  (B.1)

inducing φ on Grothendieck semirings. By Lemma B.5 below, the morphism B.1 prolongs to an isomorphism of the associated based root data. Hence, there exists an isomorphism of group schemes ϕ : H → G inducing the isomorphism of based root data. In particular, φ = K_{0}^{+}[ϕ], and such an isomorphism ϕ is uniquely determined up to inner automorphism. This finishes the proof of Theorem B.1.

Lemma B.5. Let B = (X, R, Δ, X^{∨}, R^{∨}, Δ^{∨}) any based root datum. Denote by (X_{+}, ≤) the partially ordered semigroup of dominant weights. Then the root datum B can be reconstructed from (X_{+}, ≤), i.e. if B′ = (X′, R′, Δ′, X^{′∨}, R^{′∨}, Δ^{′∨}) is another based root datum with associated dominant weights (X′_{+}, ≤′), then any ismorphism (X, ≤) → (X′, ≤′) of partially ordered semigroups prolongs to an isomorphism B → B′ of based root data.

Proof. The weight lattice X is the group completion of X_{+}, a finite free ℤmodule. The dominance order ≤ extends uniquely to X, also denoted ≤. Then X^{∨} = Hom_{ℤ}(X, ℤ) is the coweight lattice, and the natural pairing X ×X^{∨} → ℤ identifies with ⟨-, -⟩. The reconstruction of the roots and coroots proceeds in several steps:

(1) The set of simple roots Δ ⊂ X: A weight α ∈ X\{0} is in Δ if and only if 0 ≤ α, and α is minimal with this property.

(2) The set of simple coroots Δ^{∨} ⊂ X^{∨}: An element of X^{∨} is uniquely determined by its value on X_{+}. Fix α ∈ Δ with corresponding simple coroot α^{∨}. Then for any μ ∈ X_{+}, the value ⟨μ, α^{∨}⟩ is the unique number m ∈ ℕ such that 2μ − mα is dominant, but 2μ − (m + 1)α is not. Indeed, we have

⟨2μ − mα, α^{∨}⟩ ≥ 0 ⇔ ⟨μ, α^{∨}⟩ ≥ m,

and, for every other simple coroot β^{∨} ≠ α^{∨} and every n ∈ ℕ,

⟨2μ − nα, β^{∨}⟩ = 2⟨μ, β^{∨}⟩ − n⟨α, β^{∨}⟩ ≥ 2⟨μ, β^{∨}⟩ ≥ 0,

since ⟨α, β^{∨}⟩ ≤ 0. Hence, ⟨2μ − (m + 1)α, α^{∨}⟩ < 0 and so m = ⟨μ, α^{∨}⟩.

(3) The sets of roots R and coroots R^{∨}: The Weyl group W ⊂ Aut_{ℤ}(X) is the finite subgroup generated by the reflections s_{α,α^{∨}} associated to the pair (α, α^{∨}) ∈ Δ × Δ^{∨}. Then R = W · Δ, i.e., the roots are given by the translates of the simple roots under W. Since Aut_{ℤ}(X^{∨}) = Aut_{ℤ}(X)^{op}, the Weyl group W acts on X^{∨} and R^{∨} = W · Δ^{∨}. This proves the lemma.

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Timo Richarz

Mathematisches Institut der Universität Bonn Endenicher Allee 60 53115 Bonn, Germany