Hecke Operators on Quasimaps

into Horospherical Varieties

Dennis Gaitsgory and David Nadler

Received: November 10, 2006 Revised: January 30, 2009

Communicated by Edward Frenkel

Abstract. Let G be a connected reductive complex algebraic group. This paper and its companion [GN] are devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical Gvariety X. The space Z may be thought of as an algebraic model for the loop space of X. The theory we develop associates to X a connected reductive complex algebraic subgroup Ȟ of the dual group Ǧ. The construction of Ȟ is via Tannakian formalism: we identify a certain tensor category Q(Z) of perverse sheaves on Z with the category of finite-dimensional representations of Ȟ.

In this paper, we focus on horospherical varieties, a class of varieties closely related to flag varieties. For an affine horospherical G-variety X_{horo}, the category Q(Z_{horo}) is equivalent to a category of vector spaces graded by a lattice. Thus the associated subgroup Ȟ_{horo} is a torus. The case of horospherical varieties may be thought of as a simple example, but it also plays a central role in the general theory. To an arbitrary affine spherical G-variety X, one may associate a horospherical variety X_{horo}. Its associated subgroup Ȟ_{horo} turns out to be a maximal torus in the subgroup Ȟ associated to X.

2000 Mathematics Subject Classification: 22E67, 14M17 Keywords and Phrases: Loop spaces, spherical varieties, Langlands duality

1. Introduction

Let G be a connected reductive complex algebraic group. In this paper and its companion [GN], we study the space Z of meromorphic quasimaps from a curve into an affine spherical G-variety X. A G-variety X is said to be spherical if a Borel subgroup of G acts on X with a dense orbit. Examples include flag varieties, symmetric spaces, and toric varieties. A meromorphic quasimap consists of a point of the curve, a G-bundle on the curve, and a meromorphic section of the associated X-bundle with a pole only at the distinguished point. The space Z may be thought of as an algebraic model for the loop space of X. The theory we develop identifies a certain tensor category Q(Z) of perverse sheaves on Z with the category of finite-dimensional representations of a connected reductive complex algebraic subgroup Ȟ of the dual group Ǧ. Our method is to use Tannakian formalism: we endow Q(Z) with a tensor product, a fiber functor to vector spaces, and the necessary compatibility constraints so that it must be equivalent to the category of representations of such a group. Under this equivalence, the fiber functor corresponds to the forgetful functor which assigns to a representation of Ȟ its underlying vector space. In the paper [GN], we define the category Q(Z), and endow it with a tensor product and fiber functor. This paper provides a key technical result needed for the construction of the fiber functor. Horospherical G-varieties form a special class of G-varieties closely related to flag varieties. A subgroup S ⊂ G is said to be horospherical if it contains the unipotent radical of a Borel subgroup of G. A G-variety X is said to be horospherical if for each point x ∈ X, its stabilizer S_{x} ⊂ G is horospherical. When X is an affine horospherical G-variety, the subgroup Ȟ we associate to it turns out to be a torus. To see this, we explicitly calculate the functor which corresponds to the restriction of representations from Ǧ. Representations of Ǧ naturally act on the category Q(Z) via the geometric Satake correspondence. The restriction of representations is given by applying this action to the object of Q(Z) corresponding to the trivial representation of Ȟ. The main result of this paper describes this action in the horospherical case. The statement does not mention Q(Z), but rather what is needed in [GN] where we define and study Q(Z). In the remainder of the introduction, we first describe a piece of the theory of geometric Eisenstein series which the main result of this paper generalizes. This may give the reader some context from which to approach the space Z and our main result. We then define Z and state our main result. Finally, we collect notation and preliminary results needed in what follows. Throughout the introduction, we use the term space for objects which are strictly speaking stacks and ind-stacks.

1.1. Background. One way to approach the results of this paper is to interpret them as a generalization of a theorem of Braverman-Gaitsgory [BG, Theorem 3.1.4] from the theory of geometric Eisenstein series. Let C be a smooth complete complex algebraic curve. The primary aim of the geometric Langlands program is to construct sheaves on the moduli space Bun_{G} of G-bundles on C which are eigensheaves for Hecke operators. These are the operators which result from modifying G-bundles at prescribed points of the curve C. Roughly speaking, the theory of geometric Eisenstein series constructs sheaves on Bun_{G} starting with local systems on the moduli space Bun_{T}, where T is the universal Cartan of G. When the original local system is sufficiently generic, the resulting sheaf is an eigensheaf for the Hecke operators. At first glance, the link between Bun_{T} and Bun_{G} should be the moduli stack Bun_{B} of B-bundles on C, where B ⊂ G is a Borel subgroup with unipotent radical U ⊂ B and reductive quotient T = B/U. Unfortunately, naively working with the natural diagram

Bun_{B} → Bun_{G}
↓
Bun_{T}

leads to difficulties: the fibers of the horizontal map are not compact. The eventual successful construction depends on V. Drinfeld’s relative compactification of Bun_{B} along the fibers of the map to Bun_{G}. The starting point for the compactification is the observation that Bun_{B} also classifies data

(𝒫_{G} ∈ Bun_{G}, 𝒫_{T} ∈ Bun_{T} , σ : 𝒫_{T} → 𝒫_{G}×^{G}G/U)

where σ is a T -equivariant bundle map to the 𝒫_{G}-twist of G/U. From this perspective, it is natural to be less restrictive and allow maps into the 𝒫_{G}-twist of the fundamental affine space

\overline{G/U} = Spec(ℂ[G]^{U}).

Here ℂ[G] denotes the ring of regular functions on G, and ℂ[G]^{U} ⊂ ℂ[G] the (right) U-invariants. Following V. Drinfeld, we define the compactification \overline{Bun}_{B} to be that classifying quasimaps

(𝒫_{G} ∈ Bun_{G}, 𝒫_{T} ∈ Bun_{T} , σ : 𝒫_{T} → 𝒫_{G}×^{G}\overline{G/U})

where σ is a T -equivariant bundle map which factors

σ|_{C′} : 𝒫_{T} |_{C′} → 𝒫_{G}×^{G}G/U|_{C′} → 𝒫_{G}×^{G}\overline{G/U}|_{C′},

for some open curve C′ ⊂ C. Of course, the quasimaps that satisfy

σ : 𝒫_{T} → 𝒫_{G}×^{G}G/U

form a subspace canonically isomorphic to Bun_{B}. Since the Hecke operators on Bun_{G} do not lift to \overline{Bun}_{B}, it is useful to introduce a version of \overline{Bun}_{B} on which they do. Following [BG, Section 4], we define the space _{∞}\overline{Bun}_{B} to be that classifying meromorphic quasimaps

(c ∈ C, 𝒫_{G} ∈ Bun_{G}, 𝒫_{T} ∈ Bun_{T} , σ : 𝒫_{T} |_{C\c} → 𝒫_{G}×^{G}\overline{G/U}|_{C\c})

where σ is a T -equivariant bundle map which factors

σ|_{C′} : 𝒫_{T} |_{C′} → 𝒫_{G}×^{G}G/U|_{C′} → 𝒫_{G}×^{G}\overline{G/U}|_{C′},

for some open curve C′ ⊂ C \ c. We call c ∈ C the pole point of the quasimap. Given a meromorphic quasimap with G-bundle 𝒫_{G} and pole point c ∈ C, we may modify 𝒫_{G} at c and obtain a new meromorphic quasimap. In this way, the Hecke operators on Bun_{G} lift to _{∞}\overline{Bun}_{B}. Now the result we seek to generalize [BG, Theorem 3.1.4] describes how the Hecke operators act on a distinguished object of the category P(_{∞}\overline{Bun}_{B}) of perverse sheaves with ℂ-coefficients on _{∞}\overline{Bun}_{B}. Let Λ_{G} = Hom(ℂ^{×}, T) be the coweight lattice, and let Λ_{G}^{+} ⊂ Λ be the semigroup of dominant coweights of G. For λ ∈ Λ_{G}^{+}, we have the Hecke operator

H_{G}^{λ} : P(_{∞}\overline{Bun}_{B}) → P(_{∞}\overline{Bun}_{B})

given by convolving with the simple spherical modification of coweight λ. (See [BG, Section 4] or Section 5 below for more details.) For μ ∈ Λ_{G}, we have the locally closed subspace _{∞}\overline{Bun}_{B}^{μ} ⊂ _{∞}\overline{Bun}_{B} that classifies data for which the map

𝒫_{T} (μ · c)|_{C\c} →^{σ} 𝒫_{G}×^{G}\overline{G/U}|_{C\c}

extends to a holomorphic map

𝒫_{T} (μ · c) →^{σ} 𝒫_{G}×^{G}\overline{G/U}

which factors

𝒫_{T} (μ · c) →^{σ} 𝒫_{G}×^{G}G/U → 𝒫_{G}×^{G}\overline{G/U}.

We write _{∞}\overline{Bun}_{B}^{≤μ} ⊂ _{∞}\overline{Bun}_{B} for the closure of _{∞}\overline{Bun}_{B}^{μ} ⊂ _{∞}\overline{Bun}_{B}, and

IC_{_{∞}\overline{Bun}_{B}}^{≤μ} ∈ P(_{∞}\overline{Bun}_{B})

for the intersection cohomology sheaf of _{∞}\overline{Bun}_{B}^{≤μ} ⊂ _{∞}\overline{Bun}_{B}.

Theorem 1.1.1. [BG, Theorem 3.1.4] For λ ∈ Λ_{G}^{+}, there is a canonical isomorphism

H_{G}^{λ}(IC_{_{∞}\overline{Bun}_{B}}^{≤0}) ≃ ∑_{μ∈Λ_{T}} IC_{_{∞}\overline{Bun}_{B}}^{≤μ} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)

Here we write V λ ; Ǧ for the irreducible representation of the dual group Ǧ with highest weight λ ∈ Λ_{G}^{+}, and V μ ; Ť for the irreducible representation of the dual torus Ť of weight μ ∈ Λ_{G}. In the same paper of Braverman-Gaitsgory [BG, Section 4], there is a generalization [BG, Theorem 4.1.5] of this theorem from the Borel subgroup B ⊂ G to other parabolic subgroups P ⊂ G. We recall and use this generalization in Section 5 below. It is the starting point for the results of this paper.

1.2. Main result. The main result of this paper is a version of [BG, Theorem 3.1.4] for X an arbitrary affine horospherical G-variety with a dense G-orbit X̊ ⊂ X. For any point in the dense G-orbit X̊ ⊂ X, we refer to its stabilizer S ⊂ G as the generic stabilizer of X. All such subgroups are conjugate to each other. By choosing such a point, we obtain an identification X̊ ≃ G/S. To state our main theorem, we first introduce some more notation. Satz 2.1 of [Kn] states that the normalizer of a horospherical subgroup S ⊂ G is a parabolic subgroup P ⊂ G with the same derived group [P, P] = [S, S]. We write A for the quotient torus P/S, and Λ_{A} = Hom(ℂ^{×}, A) for its coweight lattice. Similarly, for the identity component S^{0} ⊂ S, we write A_{0} for the quotient torus P/S_{0}, and Λ_{A_{0}} = Hom(ℂ^{×}, A_{0}) for its coweight lattice. The natural maps T → A_{0} → A induce maps of coweight lattices

Λ_{T} →^{q} Λ_{A_{0}} →^{i} Λ_{A},

where q is a surjection, and i is an injection. For a conjugate of S, the associated tori are canonically isomorphic to those associated to S. Thus when S is the generic stabilizer of a horospherical G-variety X, the above tori, lattices and maps are canonically associated to X. For an affine horospherical G-variety X with dense G-orbit X̊ ⊂ X, we define the space Z to be that classifying mermorphic quasimaps into X. Such a quasimap consists of data

(c ∈ C, 𝒫_{G} ∈ Bun_{G}, σ : C \ c → 𝒫_{G}×^{G}X|_{C\c})

where σ is a section which factors

σ|_{C′} : C′ → 𝒫_{G}×^{G}X̊|_{C′} → 𝒫_{G}×^{G}X|_{C′},

for some open curve C′ ⊂ C \ c. Given a meromorphic quasimap into X with G-bundle 𝒫_{G} and pole point c ∈ C, we may modify 𝒫_{G} at c and obtain a new meromorphic quasimap. But in this context the resulting Hecke operators on Z do not in general preserve the category of perverse sheaves. Instead, we must consider the bounded derived category Sh(Z) of sheaves of ℂ-modules on Z. For λ ∈ Λ_{G}^{+}, we have the Hecke operator

H_{G}^{λ} : Sh(Z) → Sh(Z)

given by convolving with the simple spherical modification of coweight λ. (See Section 5 below for more details.) For κ ∈ Λ_{A_{0}}, we have a locally closed subspace Z^{κ} ⊂ Z consisting of meromorphic quasimaps that factor

σ : C \ c → 𝒫_{G}×^{G}X̊|_{C\c} → 𝒫_{G}×^{G}X|_{C\c}

and have a singularity of type κ at c ∈ C. (See Section 3.5 below for more details.) We write Z^{≤κ} ⊂ Z for the closure of Z^{κ} ⊂ Z, and

IC_{Z}^{≤κ} ∈ Sh(Z)

for its intersection cohomology sheaf. Our main result is the following.

Theorem 1.2.1. For λ ∈ Λ_{G}^{+}, there is an isomorphism

H_{G}^{λ}(IC_{Z}^{≤0}) ≃ ∑_{κ∈Λ_{A_{0}}} ∑_{μ∈Λ_{T} ,q(μ)=κ} IC_{Z}^{≤κ} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)[⟨2ρ̌_{M}, μ⟩].

Here the torus A_{0} and its coweight lattice Λ_{A_{0}} are those associated to the generic stabilizer S ⊂ G. We write M for the Levi quotient of the normalizer P ⊂ G of the generic stabilizer S ⊂ G, and 2ρ̌_{M} for the sum of the positive roots of M. In the context of the companion paper [GN], the theorem translates into the following fundamental statement. The tensor category Q(Z) associated to X is the category of semisimple perverse sheaves with simple summands IC_{Z}^{≤κ}, for κ ∈ Λ_{A_{0}}, and the dual subgroup Ȟ associated to X is the subtorus Spec ℂ[Λ_{A_{0}}] ⊂ Ť.

1.3. Notation. Throughout this paper, let G be a connected reductive complex algebraic group, let B ⊂ G be a Borel subgroup with unipotent radical U(B), and let T = B/U(B) be the abstract Cartan. Let Λ̌_{G} denote the weight lattice Hom(T, ℂ^{×}), and Λ̌_{G}^{+} ⊂ Λ̌_{G} the semigroup of dominant weights. For λ ∈ Λ̌_{G}^{+}, we write V_{G}^{λ} for the irreducible representation of G of highest weight λ. Let Λ_{G} denote the coweight lattice Hom(ℂ^{×}, T), and Λ_{G}^{+} ⊂ Λ_{G} the semigroup of dominant coweights. For λ ∈ Λ_{G}^{+}, let V λ ; Ǧ denote the irreducible representation of the dual group Ǧ of highest weight λ. Let Λ_{G}^{pos} ⊂ Λ_{G} denote the semigroup of coweights in Λ_{G} which are non-negative on Λ̌_{G}^{+}, and let R_{G}^{pos} ⊂ Λ_{G}^{pos} denote the semigroup of positive coroots. Let P ⊂ G be a parabolic subgroup with unipotent radical U(P), and let M be the Levi factor P/U(P). We have the natural map

ř: Λ̌_{M/[M,M]} → Λ̌_{G}

of weights, and the dual map

r : Λ_{G} → Λ_{M/[M,M]}

of coweights. Let Λ̌_{G,P}^{+} ⊂ Λ̌_{M/[M,M]} denote the inverse image ř^{−1}(Λ̌_{G}^{+}). Let Λ_{G,P}^{pos} ⊂ Λ_{M/[M,M]} denote the semigroup of coweights in Λ_{M/[M,M]} which are nonnegative on Λ̌_{G,P}^{+} . Let R_{G,P}^{pos} ⊂ Λ_{G,P}^{pos} denote the image r(R_{G}^{pos}). Let 𝒲_{M} denote the Weyl group of M, and let 𝒲_{M} Λ̌_{G}^{+} ⊂ Λ̌_{G} denote the union of the translates of Λ̌_{G}^{+} by 𝒲_{M}. Let Λ̃_{G,P}^{pos} ⊂ Λ_{M}^{+} denote the semigroup of dominant coweights of M which are nonnegative on 𝒲_{M} Λ̌_{G}^{+}. Finally, let ⟨·, ·⟩ : Λ̌_{G} × Λ_{G} → ℤ denote the natural pairing, and let ρ̌_{M} ∈ Λ̌_{G} denote half the sum of the positive roots of M.

1.4. Bundles and Hecke correspondences. Let C be a smooth complete complex algebraic curve. For a connected complex algebraic group H, let Bun_{H} be the moduli stack of H-bundles on C. Objects of Bun_{H} will be denoted by 𝒫_{H}. Let ℋ_{H} be the Hecke ind-stack that classifies data

(c ∈ C, 𝒫_{H}^{1}, 𝒫_{H}^{2} ∈ Bun_{H}, α : 𝒫_{H}^{1}|_{C\c} →^{∼} 𝒫_{H}^{2}|_{C\c})

where α is an isomorphism of H-bundles. We have the maps

Bun_{H} ←^{h\overleftarrow{H}} ℋ_{H} →^{h\overrightarrow{H}} Bun_{H}

defined by

h\overleftarrow{H}(c, 𝒫_{H}^{1}, 𝒫_{H}^{2}, α) = 𝒫_{H}^{1} h\overrightarrow{H}(c, 𝒫_{H}^{1}, 𝒫_{H}^{2}, α) = 𝒫_{H}^{2},

and the map

π : ℋ_{H} → C

defined by

π(c, 𝒫_{H}^{1}, 𝒫_{H}^{2}, α) = c.

It is useful to have another description of the Hecke ind-stack ℋ_{H} for which we introduce some more notation. Let 𝒪 be the ring of formal power series ℂ[[t]], let 𝒦 be the field of formal Laurent series ℂ((t)), and let D be the formal disk Spec(𝒪). For a point c ∈ C, let 𝒪_{c} be the completed local ring of C at c, and let D_{c} be the formal disk Spec(𝒪_{c}). Let Aut(𝒪) be the group-scheme of automorphisms of the ring 𝒪. Let H(𝒪) be the group of 𝒪-valued points of H, and let H(𝒦) be the group of 𝒦-valued points of H. Let Gr_{H} be the affine Grassmannian of H. It is an ind-scheme whose set of ℂ-points is the quotient H(𝒦)/H(𝒪). Now consider the (H(𝒪) ⋊ Aut(𝒪))-torsor

{ \atop Bun_{H} ×C} → Bun_{H} ×C

that classifies data

(c ∈ C, 𝒫_{H} ∈ Bun_{H}, β : D × H →^{∼} 𝒫_{H}|_{D_{c}}, γ : D →^{∼} D_{c})

where β is an isomorphism of H-bundles, and γ is an identification of formal disks. We have an identification

ℋ_{H} ≃ { \atop Bun_{H} ×C} ×^{(H(𝒪)⋊Aut(𝒪))} Gr_{H}

such that the projection h\overrightarrow{H} corresponds to the obvious projection from the twisted product to Bun_{H}. For H reductive, the (H(𝒪) ⋊ Aut(𝒪))-orbits Gr_{H}^{λ} ⊂ Gr_{H} are indexed by λ ∈ Λ_{H}^{+}. For λ ∈ Λ_{H}^{+}, we write ℋ_{H}^{λ} ⊂ ℋ_{H} for the substack

ℋ_{H}^{λ} ≃ { \atop Bun_{H} ×C} ×^{(H(𝒪)⋊Aut(𝒪))} Gr_{H}^{λ} .

For a parabolic subgroup P ⊂ H, the connected components S_{P,θ} ⊂ Gr_{P} are indexed by θ ∈ Λ_{P} /Λ_{[P,P]^{sc}}, where [P, P]^{sc} denotes the simply connected cover of [P, P]. For θ ∈ Λ_{P} /Λ_{[P,P]^{sc}}, we write 𝒮_{P,θ} ⊂ ℋ_{P} for the ind-substack

𝒮_{P,θ} ≃ { \atop Bun_{P} ×C} ×^{(P (𝒪)⋊Aut(𝒪))} S_{P,θ}.

For θ ∈ Λ_{P} /Λ_{[P,P]^{sc}}, and λ ∈ Λ_{H}^{+}, we write 𝒮_{P,θ}^{λ} ⊂ ℋ_{P} for the ind-substack

𝒮_{P,θ}^{λ} ≃ { \atop Bun_{P} ×C} ×^{(P (𝒪)⋊Aut(𝒪))} S_{P,θ}^{λ}

where S_{P,θ}^{λ} denotes the intersection S_{P,θ} ∩ Gr_{H}^{λ}. For any ind-stack 𝒵 over Bun_{H} ×C, we have the (H(𝒪) ⋊ Aut(𝒪))-torsor

𝒵̂ → 𝒵

obtained by pulling back the (H(𝒪) ⋊ Aut(𝒪))-torsor

{ \atop Bun_{H} ×C} → Bun_{H} ×C.

We also have the Cartesian diagram

ℋ_{H} ×_{Bun_{H} ×C} 𝒵 →^{h\overrightarrow{H}}  𝒵
↓  ↓
ℋ_{H}  →^{h\overrightarrow{H}} Bun_{H}

and an identification

ℋ_{H} ×_{Bun_{H} ×C} 𝒵 ≃ 𝒵̂ ×^{(H(𝒪)⋊Aut(𝒪))} Gr_{H}

such that the projection h\overrightarrow{H} corresponds to the obvious projection from the twisted product to 𝒵. For ℱ ∈ Sh(𝒵), and 𝒫 ∈ P_{(H(𝒪)⋊Aut(𝒪))} (Gr_{H}), we may form the twisted product

(ℱ ˜⊠𝒫)^{r} ∈ Sh(ℋ_{H} ×_{Bun_{H} ×C} 𝒵).

with respect to the map h\overrightarrow{H}. In particular, for λ ∈ Λ_{H}^{+}, we may take 𝒫 to be the intersection cohomology sheaf 𝒜_{G}^{λ} of the closure \overline{Gr}_{H}^{λ} ⊂ Gr_{H} of the (H(𝒪) ⋊ Aut(𝒪))-orbit Gr_{H}^{λ} ⊂ Gr_{H}.

2. Affine horospherical G-varieties

A subgroup S ⊂ G is said to be horospherical if it contains the unipotent radical of a Borel subgroup of G. A G-variety X is said to be horospherical if for each point x ∈ X, its stabilizer S_{x} ⊂ G is horospherical. A G-variety X is said to be spherical if a Borel subgroup of G acts on X with a dense orbit. Note that a horospherical G-variety contains a dense G-orbit if and only if it is spherical.

Let X be an affine G-variety. As a representation of G, the ring of regular functions ℂ[X] decomposes into isotypic components

ℂ[X] ≃ ∑_{λ∈Λ̌_{G}^{+}} ℂ[X]_{λ}.

We say that ℂ[X] is graded if

ℂ[X]_{λ}ℂ[X]_{μ} ⊂ ℂ[X]_{λ+μ},

for all λ, μ ∈ Λ̌_{G}^{+}. We say that ℂ[X] is simple if the irreducible representation V^{λ} of highest weight λ occurs in ℂ[X]_{λ} with multiplicity 0 or 1, for all λ ∈ Λ̌_{G}^{+}.

Proposition 2.0.1. Let X be an affine G-variety.

(1) [Pop, Proposition 8, (3)] X is horospherical if and only if ℂ[X] is graded.

(2) [Pop, Theorem 1] X is spherical if and only if ℂ[X] is simple.

We see by the proposition that affine horospherical G-varieties containing a dense G-orbit are classified by finitely-generated subsemigroups of Λ̌_{G}^{+}. To such a variety X, one associates the subsemigroup

Λ̌_{X}^{+} ⊂ Λ̌_{G}^{+}

of dominant weights λ with dim ℂ[X]_{λ} > 0.

2.1. Structure of generic stabilizer.

Theorem 2.1.1. [Kn, Satz 2.2] If X is an irreducible horospherical G-variety, then there is an open G-invariant subset W ⊂ X, and a G-equivariant isomorphism W ≃ G/S × Y, where S ⊂ G is a horospherical subgroup, and Y is a variety on which G acts trivially.

Note that for any two such open subsets W ⊂ X and isomorphisms W ≃ G/S × Y, the subgroups S ⊂ G are conjugate. We refer to such a subgroup S ⊂ G as the generic stabilizer of X.

Lemma 2.1.2. [Kn, Satz 2.1] If S ⊂ G is a horospherical subgroup, then its normalizer is a parabolic subgroup P ⊂ G with the same derived group [P, P] = [S, S] and unipotent radical U(P) = U(S).

Note that the identity component S^{0} ⊂ S is also horospherical with the same derived group [S^{0}, S^{0}] = [S, S] and unipotent radical U(S^{0}) = U(S). Let S ⊂ G be a horospherical subgroup with identity component S^{0} ⊂ S, and normalizer P ⊂ G. We write A for the quotient torus P/S, and Λ_{A} for its coweight lattice Hom(ℂ^{×}, A). Similarly, we write A_{0} for the quotient torus P/S^{0}, and Λ_{A_{0}} for its coweight lattice Hom(ℂ^{×}, A_{0}). The natural maps

T → A_{0} → A

induce maps of coweight lattices

Λ_{T} →^{q} Λ_{A_{0}} →^{i} Λ_{A},

where q is a surjection, and i is an injection. For a conjugate of S, the associated tori, lattices, and maps are canonically isomorphic to those associated to S.

Thus when S is the generic stabilizer of a horospherical G-variety X, the tori, lattices and maps are canonically associated to X. We shall need the following finer description of which subgroups S ⊂ G may appear as the generic stabilizer of an affine horospherical G-variety. To state it, we introduce some more notation used throughout the paper. For a horospherical subgroup S ⊂ G with identity component S^{0} ⊂ S, and normalizer P ⊂ G, let M be the Levi quotient P/U(P), let M_{S} be the Levi quotient S/U(S), and let M_{S}^{0} be the identity component of M_{S}. The natural maps

S^{0} → S → P

induce isomorphisms of derived groups

[M_{S}^{0}, M_{S}^{0}] →^{∼} [M_{S}, M_{S}] →^{∼} [M, M].

We write Λ_{M/[M,M]} for the coweight lattice of the torus M/[M, M], and Λ_{M_{S}^{0}/[M_{S},M_{S}]} for the coweight lattice of the torus M_{S}^{0}/[M_{S}, M_{S}]. The natural maps

M_{S}^{0}/[M_{S}, M_{S}] → M/[M, M] → A_{0}

induce a short exact sequence of coweight lattices

0 → Λ_{M_{S}^{0}/[M_{S},M_{S}]} → Λ_{M/[M,M]} → Λ_{A_{0}} → 0.

Proposition 2.1.3. Let S ⊂ G be a horospherical subgroup. Then S is the generic stabilizer of an affine horospherical G-variety containing a dense Gorbit if and only if

Λ_{M_{S}^{0}/[M_{S},M_{S}]} ∩ Λ_{G,P}^{pos} = ⟨0⟩.

Proof. The proof of the proposition relies on the following lemma. Let V̌ be a finite-dimensional real vector space, and let V̌^{+} be an open set in V̌ which is preserved by the action of ℝ^{>0}. Let V be the dual of V̌, and let V^{pos} be the closed cone of covectors in V that are nonnegative on all vectors in V̌^{+}. For a linear subspace W̌ ⊂ V̌, we write W̌^{⊥} ⊂ V for its orthogonal.

Lemma 2.1.4. The map W̌ ↦ W̌^{⊥} provides a bijection from the set of all linear subspaces W̌ ⊂ V̌ such that W̌ ∩V̌^{+} ≠ ∅ to the set of all linear subspaces W ⊂ V such that W ∩ V^{pos} = ⟨0⟩.

Proof. If W̌ ∩V̌^{+} ≠ ∅, then clearly W̌^{⊥} ∩V^{pos} = ⟨0⟩. Conversely, if W ∩V^{pos} = ⟨0⟩, then since V̌^{+} is open, there is a hyperplane H ⊂ V such that W ⊂ H, and H ∩V^{pos} = ⟨0⟩. Thus H^{⊥} ⊂ W^{⊥}, and H^{⊥} ∩V̌^{+} ≠ ∅, and so W^{⊥} ∩V̌^{+} ≠ ∅. □

Now suppose X is an affine horospherical G-variety with an open G-orbit and generic stabilizer S ⊂ G with normalizer P ⊂ G. Then we have Λ̌_{X}^{+} ⊂ Λ̌_{G,P}^{+} , since otherwise [S, S] would be smaller. We also have that Λ̌_{X}^{+} intersects the interior of Λ̌_{G,P}^{+} , since otherwise [S, S] would be larger. Applying Lemma 2.1.4, we conclude

Λ_{M_{S}^{0}/[M_{S},M_{S}]} ∩ Λ_{G,P}^{pos} = ⟨0⟩.

Conversely, suppose S ⊂ G is a horospherical subgroup with normalizer P ⊂ G. We define X to be the spectrum of the ring ℂ[X] of (right) S-invariants in the ring of regular functions ℂ[G]. Then ℂ[X] is finitely-generated, since S contains the unipotent radical of a Borel subgroup of G. We have Λ̌_{X}^{+} ⊂ Λ̌_{G,P}^{+} , since otherwise [S, S] would be smaller. Suppose

Λ_{M_{S}^{0}/[M_{S},M_{S}]} ∩ Λ_{G,P}^{pos} = ⟨0⟩.

Applying Lemma 2.1.4, we conclude that Λ̌_{X}^{+} intersects the interior of Λ̌_{G,P}^{+} . Therefore S/[S, S] consists of exactly those elements of P/[P, P] annhilated by Λ̌_{X}^{+}, and so S is the generic stabilizer of X. □

2.2. Canonical affine closure. Let S ⊂ G be the generic stabilizer of an affine horospherical G-variety X containing a dense G-orbit. Let ℂ[G] be the ring of regular functions on G, and let ℂ[G]^{S} ⊂ ℂ[G] be the (right) S-invariants. We call the affine variety

\overline{G/S} = Spec(ℂ[G]^{S})

the canonical affine closure of G/U. We have the natural map

\overline{G/S} → X

corresponding to the restriction map

ℂ[X] → ℂ[G/S] ≃ ℂ[G]^{S}.

Since S is horospherical, the ring ℂ[G]^{S} is simple and graded, and so the affine variety \overline{G/S} is spherical and horospherical. Although we do not use the following, it clarifies the relation between X and the canonical affine closure \overline{G/S}.

Proposition 2.2.1. Let X be an affine horospherical G-variety containing a dense G-orbit and generic stabilizer S ⊂ G. The semigroup Λ̌+ ; \overline{G/S} ⊂ Λ̌_{G} is the

intersection of the dominant weights Λ̌_{G}^{+} ⊂ Λ̌_{G} with the group generated by the semigroup Λ̌_{X}^{+} ⊂ Λ̌_{G}.

Proof. Let P ⊂ G be the normalizer of S ⊂ G. The intersection of Λ̌_{G}^{+} and the group generated by Λ̌_{X}^{+} consists of exactly those weights in Λ̌_{G,P}^{+} that annhilate S/[S, S]. □

3. Ind-stacks

As usual, let C be a smooth complete complex algebraic curve.

3.1. Labellings. Fix a pair (Λ, Λ^{pos}) of a lattice Λ and a semigroup Λ^{pos} ⊂ Λ. We shall apply the following to the pair (Λ_{M/[M,M]}, Λ_{G,P}^{pos} ). For θ^{pos} ∈ Λ^{pos}, we write 𝔘(θ^{pos}) for a decomposition

θ^{pos} = ∑_{m} n_{m}θ_{m}^{pos}

where θ_{m}^{pos} ∈ Λ^{pos} \ {0} are pairwise distinct and n_{m} are positive integers.

For θ^{pos} ∈ Λ^{pos}, and a decomposition 𝔘(θ^{pos}), we write C^{𝔘(θ^{pos})} for the partially symmetrized power ∏_{m} C^{(n_{m})} of the curve C. We write C_{0}^{𝔘(θ^{pos})} ⊂ C^{𝔘(θ^{pos})} for the complement of the diagonal divisor. For Θ a pair (θ, 𝔘(θ^{pos})) consisting of θ ∈ Λ, and 𝔘(θ^{pos}) a decomposition of θ^{pos} ∈ Λ^{pos}, we write C^{Θ} for the product C × C^{𝔘(θ^{pos})}. We write C_{0}^{Θ} ⊂ C^{Θ} for the complement of the diagonal divisor. Although C^{Θ} is independent of θ, it is notationally convenient to denote it as we do.

3.2. \overline{Ind-stack} associated to parabolic subgroup. Fix a parabolic subgroup P ⊂ G, and let M be its Levi quotient P/U(P). For our application, P will be the normalizer of the generic stabilizer S ⊂ G of an irreducible affine horospherical G-variety. Let _{∞}\overline{Bun}_{P} be the ind-stack that classifies data

(c ∈ C, 𝒫_{G} ∈ Bun_{G}, 𝒫_{M/[M,M]} ∈ Bun_{M/[M,M]},

σ : 𝒫_{M/[M,M]}|_{C\c} → 𝒫_{G}×^{G}\overline{G/[P, P]}|_{C\c})

where σ is an M/[M, M]-equivariant section which factors

σ|_{C′} : 𝒫_{M/[M,M]}|_{C′} → 𝒫_{G}×^{G}G/[P, P]|_{C′} → 𝒫_{G}×^{G}\overline{G/[P, P]}|_{C′}

for some open curve C′ ⊂ C \ c.

3.2.1. Stratification. Let Θ be a pair (θ, 𝔘(θ^{pos})), with θ ∈ Λ_{M/[M,M]}, and θ^{pos} ∈ Λ_{G,P}^{pos} . We recall that we have a locally closed embedding

j_{Θ} : Bun_{P} ×C_{0}^{Θ} → _{∞}\overline{Bun}_{P}

defined by

j_{Θ}(𝒫_{P} , (c, ∑_{m,n} θ_{m}^{pos} · c_{m,n})) = (c, 𝒫_{P} ×^{P} G, 𝒫_{P} ×^{P} [P, P](−θ · c − ∑_{m,n} θ_{m}^{pos} · c_{m,n}), σ)

where σ is the natural map

𝒫_{P} ×^{P} [P, P](−θ · c − ∑_{m,n} θ_{m}^{pos} · c_{m,n})|_{C\c} → 𝒫_{P} ×^{P} G×^{G}\overline{G/[P, P]}|_{C\c}

induced by the inclusion

𝒫_{P} ×^{P} \overline{P/[P, P]} ⊂ 𝒫_{P} ×^{P} \overline{G/[P, P]} ≃ 𝒫_{P} ×^{P} G×^{G}\overline{G/[P, P]}.

The following is an ind-version of [BG, Propositions 6.1.2 & 6.1.3], or [BFGM, Proposition 1.5], and we leave the proof to the reader.

Proposition 3.2.2. Let Θ be a pair (θ, 𝔘(θ^{pos})), with θ ∈ Λ_{M/[M,M]}, and θ^{pos} ∈ Λ_{G,P}^{pos} . Every closed point of _{∞}\overline{Bun}_{P} belongs to the image of a unique j_{Θ}.

For Θ a pair (θ, 𝔘(θ^{pos})), with θ ∈ Λ_{M/[M,M]}, and θ^{pos} ∈ Λ_{G,P}^{pos} , we write

∞\overline{Bun}_{P}^{Θ} ⊂ _{∞}\overline{Bun}_{P} for the image of j_{Θ}, and _{∞}\overline{Bun}_{P}^{≤Θ} ⊂ _{∞}\overline{Bun}_{P} for the closure of _{∞}\overline{Bun}_{P}^{Θ} ⊂ _{∞}\overline{Bun}_{P}. For Θ a pair (θ, 𝔘(0)), with θ ∈ Λ_{M/[M,M]}, the substack _{∞}\overline{Bun}_{P}^{Θ} ⊂ _{∞}\overline{Bun}_{P} classifies data (c, 𝒫_{G}, 𝒫_{M/[M,M]}, σ) for which the map

𝒫_{M/[M,M]}(θ · c)|_{C\c} →^{σ} 𝒫_{G}×^{G}\overline{G/[P, P]}|_{C\c}

extends to a holomorphic map

𝒫_{M/[M,M]}(θ · c) →^{σ} 𝒫_{G}×^{G}\overline{G/[P, P]}

which factors

𝒫_{M/[M,M]}(θ · c) →^{σ} 𝒫_{G}×^{G}G/[P, P] → 𝒫_{G}×^{G}\overline{G/[P, P]}.

In this case, we write j_{θ} in place of j_{Θ}, _{∞}\overline{Bun}_{P}^{θ} in place of_{∞}\overline{Bun}_{P}^{Θ}, and_{∞}\overline{Bun}_{P}^{≤θ} in place of _{∞}\overline{Bun}_{P}^{≤Θ}. For example, _{∞}\overline{Bun}_{P}^{≤0} ⊂ _{∞}\overline{Bun}_{P} is the closure of the canonical embedding

j_{0} : Bun_{P} ×C → _{∞}\overline{Bun}_{P} .

3.3. ; Ind-stack associated to parabolic subgroup. Fix a parabolic subgroup P ⊂ G, and let M be its Levi quotient P/U(P). As usual, for our application, P will be the normalizer of the generic stabilizer S ⊂ G of an irreducible affine horospherical G-variety. Let _{∞}B̃un_{P} be the ind-stack that classifies data

(c ∈ C, 𝒫_{G} ∈ Bun_{G}, 𝒫_{M} ∈ Bun_{M}, σ : 𝒫_{M}|_{C\c} → 𝒫_{G}×^{G}\overline{G/U(P)}|_{C\c})

where σ is an M-equivariant section which factors

σ|_{C′} : 𝒫_{M}|_{C′} → 𝒫_{G}×^{G}G/U(P)|_{C′} → 𝒫_{G}×^{G}\overline{G/U(P)}|_{C′}

for some open curve C′ ⊂ C \ c.

3.3.1. Stratification. For θ^{pos} ∈ Λ_{G,P}^{pos} , we write 𝔘̃(θ^{pos}) for a collection of (not necessarily distinct) elements θ̃_{m}^{pos} ∈ Λ̃_{G,P}^{pos} \ {0} such that

θ^{pos} = ∑_{m} r(θ̃_{m}^{pos}).

We write r(𝔘̃(θ^{pos})) for the decomposition such a collection defines. Let Θ̃ be a pair (θ̃, 𝔘̃(θ^{pos})) with θ̃ ∈ Λ_{M}^{+}, and θ^{pos} ∈ Λ_{G,P}^{pos} , and let Θ be the associated pair (r(θ̃), r(𝔘̃(θ^{pos}))). We define the Hecke ind-stack

ℋΘ̃ ; M,0 → C_{0}^{Θ}

to be that with fiber over (c, c_{𝔘(θ^{pos})}) ∈ C_{0}^{Θ}, where c_{𝔘(θ^{pos})} = ∑_{m} r(θ̃_{m}^{pos}) · c_{m}, the fiber product

ℋθ̃ ; M|_{c} ×_{Bun_{M}} ∏_{Bun_{M}} ℋθ̃_{m}^{pos} ; M|_{c_{m}}.

The following is an ind-version of [BG, Proposition 6.2.5], or [BFGM, Proposition 1.9], and we leave the proof to the reader.

Proposition 3.3.2. Let Θ̃ be a pair (θ̃, 𝔘̃(θ^{pos})) with θ̃ ∈ Λ_{M}^{+}, and θ^{pos} ∈ Λ_{G,P}^{pos} . On the level of reduced ind-stacks, there is a locally closed embedding

jΘ̃ : Bun_{P} ×_{Bun_{M}}ℋΘ̃ ; M,0 → _{∞}B̃un_{P} .

Every closed point of _{∞}B̃un_{P} belongs to the image of a unique jΘ̃.

For Θ̃ a pair (θ̃, 𝔘̃(θ^{pos})), with θ̃ ∈ Λ_{M}^{+}, and θ^{pos} ∈ Λ_{G,P}^{pos} , we write _{∞}B̃unΘ̃ ; P ⊂

∞B̃un_{P} for the image of jΘ̃, and _{∞}B̃un_{P}^{≤Θ̃} ⊂ _{∞}B̃un_{P} for the closure of

∞B̃unΘ̃ ; P ⊂ _{∞}B̃un_{P}.

For Θ̃ a pair (θ̃, 𝔘̃(0)), with θ̃ ∈ Λ_{M}^{+}, we write jθ̃ in place of jΘ̃, _{∞}B̃unθ̃ ; P in

place of _{∞}B̃unΘ̃ ; P, and _{∞}B̃un_{P}^{≤θ̃} in place of _{∞}B̃un_{P}^{≤Θ̃} For example, _{∞}B̃un_{P}^{≤0} is the closure of the canonical embedding

j˜0 : Bun_{P} ×C → _{∞}B̃un_{P} .

3.4. \overline{Ind-stack} associated to generic stabilizer. Let X be an irreducible affine horospherical G-variety with generic stabilizer S ⊂ G. Recall that the normalizer of S is a parabolic subgroup P ⊂ G with the same derived group [P, P] = [S, S] and unipotent radical U(P) = U(S). Let M be the Levi quotient P/U(P), and let M_{S} be the Levi quotient S/U(S). Let \overline{Z}_{can} be the ind-stack that classifies data

(c ∈ C, 𝒫_{G} ∈ Bun_{G}, 𝒫_{M_{S}/[M_{S},M_{S}]} ∈ Bun_{M_{S}/[M_{S},M_{S}]},

σ : 𝒫_{M_{S}/[M_{S},M_{S}]}|_{C\c} → 𝒫_{G}×^{G}\overline{G/[S, S]}|_{C\c})

where σ is an M_{S}/[M_{S}, M_{S}]-equivariant section which factors

σ|_{C′} : 𝒫_{M_{S}/[M_{S},M_{S}]}|_{C′} → 𝒫_{G}×^{G}G/[S, S]|_{C′} → 𝒫_{G}×^{G}\overline{G/[S, S]}|_{C′}

for some open curve C′ ⊂ C \ c. The following is immediate from the definitions.

Proposition 3.4.1. The diagram

\overline{Z}_{can}  →  _{∞}\overline{Bun}_{P}
↓  ↓
Bun_{M_{S}/[M_{S},M_{S}]} → Bun_{M/[M,M]}

is Cartesian.

3.4.2. Stratification. Let Θ be a pair (θ, 𝔘(θ^{pos})), with θ ∈ Λ_{M/[M,M]}, and θ^{pos} ∈ Λ_{G,P}^{pos} .

We write \overline{Z}_{can}^{Θ} ⊂ \overline{Z}_{can} for the substack which completes the Cartesian diagram

\overline{Z}_{can}^{Θ}  →  _{∞}\overline{Bun}_{P}^{Θ}
↓  ↓
Bun_{M_{S}/[M_{S},M_{S}]} → Bun_{M/[M,M]},

and \overline{Z}_{can}^{≤Θ} ⊂ \overline{Z}_{can} for the closure of \overline{Z}_{can}^{Θ} ⊂ \overline{Z}_{can}. For Θ a pair (θ, 𝔘(0)), with θ ∈ Λ_{M/[M,M]}, we write \overline{Z}_{can}^{θ} in place of \overline{Z}_{can}^{Θ},

and \overline{Z}_{can}^{≤θ} in place of \overline{Z}_{can}^{≤Θ}. For example, \overline{Z}_{can}^{≤0} is the closure of the canonical embedding

Bun_{S} ×C ⊂ \overline{Z}_{can}.

3.5. Naive ind-stack associated to X. Let X be an affine horospherical G-variety with dense G-orbit X̊ ⊂ X and generic stabilizer S ⊂ G. Let Z be the ind-stack that classifies data

(c ∈ C, 𝒫_{G} ∈ Bun_{G}, σ : C \ c → 𝒫_{G}×^{G}X|_{C\c})

where σ is a section which factors

σ|_{C′} : C′ → 𝒫_{G}×^{G}X̊|_{C′} → 𝒫_{G}×^{G}X|_{C′}

for some open curve C′ ⊂ C \ c. For the canonical affine closure \overline{G/S}, we write Z_{can} for the corresponding indstack. We call the ind-stack Z naive, since there is no auxilliary bundle in its definition: it classifies honest sections. Let ^{⋆}Z be the ind-stack that classifies data

(c ∈ C, 𝒫_{G} ∈ Bun_{G}, 𝒫_{M/M_{S}} ∈ Bun_{M/M_{S}}, σ : 𝒫_{M/M_{S}}|_{C\c} → 𝒫_{G}×^{G}X|_{C\c})

where σ is an M/M_{S}-equivariant section which factors

σ|_{C′} : 𝒫_{M/M_{S}}|_{C′} → 𝒫_{G}×^{G}X̊|_{C′} → 𝒫_{G}×^{G}X|_{C′}

for some open curve C′ ⊂ C \c. Here as usual, we write M for the Levi quotient P/U(P) of the normalizer P ⊂ G of the generic stabilizer S ⊂ G, and M_{S} for the Levi quotient S/U(S). For the canonical affine closure \overline{G/S}, we write ^{⋆}Z_{can} for the corresponding ind-stack. The following analogue of Proposition 3.4.1 is immediate from the definitions.

Proposition 3.5.1. The diagram

Z  →  ^{⋆}Z
↓  ↓
Bun_{⟨1⟩} → Bun_{M/M_{S}}

is Cartesian.

3.5.2. Stratification. We shall content ourselves here with defining the substacks of the naive ind-stack Z which appear in our main theorem. (See [GN] for a different perspective involving a completely local definition.) Recall that we write A for the quotient torus P/S, and Λ_{A} for its coweight lattice. Similarly, for the identity component S^{0} ⊂ S, we write A_{0} for the quotient torus P/S^{0}, and Λ_{A_{0}} for its coweight lattice. The natural map A_{0} → A provides an inclusion of coweight lattices Λ_{A_{0}} → Λ_{A}. For κ ∈ Λ_{A}, we shall define a closed substack Z^{≤κ} ⊂ Z. When κ ∈ Λ_{A_{0}}, the closed substack Z^{≤κ} ⊂ Z appears in our main theorem. For κ ∈ Λ_{A}, let ^{⋆}Z^{κ} ⊂ ^{⋆}Z be the locally closed substack that classifies data (c, 𝒫_{G}, 𝒫_{M/M_{S}}, σ) for which the natural map

𝒫_{M/M_{S}}(κ · c)|_{C\c} →^{σ} 𝒫_{G}×^{G}X|_{C\c}

extends to a holomorphic map

𝒫_{M/M_{S}}(κ · c) →^{σ} 𝒫_{G}×^{G}X

which factors

𝒫_{M/M_{S}}(κ · c) →^{σ} 𝒫_{G}×^{G}X̊ → 𝒫_{G}×^{G}X.

We write ^{⋆}Z^{≤κ} ⊂ ^{⋆}Z for the closure of ^{⋆}Z^{κ} ⊂ ^{⋆}Z. For κ ∈ Λ_{A}, let Z^{κ} ⊂ Z be the locally closed substack completing the Cartesian diagram

Z^{κ} →  ^{⋆}Z^{κ}
↓  ↓
Bun_{⟨1⟩} → Bun_{M/M_{S}} .

We write Z^{≤κ} ⊂ Z for the closure of Z^{κ} ⊂ Z.

4. Maps

4.1. The map 𝔯 : _{∞}B̃un_{P} → _{∞}\overline{Bun}_{P}. Let Θ be a pair (θ, 𝔘(θ^{pos})), with θ ∈ Λ_{M/[M,M]}, and θ^{pos} ∈ Λ_{G,P}^{pos} . and 𝔘(θ^{pos}) a decomposition θ^{pos} = ∑_{m} n_{m}θ_{m}^{pos}.

Let _{∞}B̃un_{P}^{Θ} ⊂ _{∞}B̃un_{P} be the inverse image of _{∞}\overline{Bun}_{P}^{Θ} ⊂ _{∞}\overline{Bun}_{P} under the natural map

𝔯 : _{∞}B̃un_{P} → _{∞}\overline{Bun}_{P} .

We would like to describe the fibers of the restriction of 𝔯 to the substack ∞B̃un_{P}^{Θ} ⊂ _{∞}B̃un_{P}. First, we define the Hecke ind-substack

ℋ_{M}^{♭(θ)} ⊂ ℋ_{M}

to be the union of the spherical Hecke substacks

ℋ_{M}^{μ} ⊂ ℋ_{M},

for μ ∈ Λ_{M}^{+} such that r(μ) = θ.

Second, if there exists μ̃^{pos} ∈ Λ̃_{G,P}^{pos} such that r(μ̃^{pos}) = θ^{pos}, we define the Hecke substack

ℋ_{M}^{♭(θ^{pos})} ⊂ ℋ_{M}

to be the union of the spherical Hecke substacks

ℋ_{M}^{μ̃^{pos}} ⊂ ℋ_{M},

for μ̃^{pos} ∈ Λ̃_{G,P}^{pos} such that r(μ̃^{pos}) = θ_{m}^{pos}. Finally, we define the Hecke ind-stack

ℋ_{M,0}^{♭(Θ)} → C_{0}^{Θ}

to be that with fiber over (c, c_{𝔘(θ^{pos})}) ∈ C_{0}^{Θ}, where c_{𝔘(θ^{pos})} = ∑_{m,n} θ_{m}^{pos} · c_{m,n}, the fiber product

ℋ_{M}^{♭(θ)}|_{c} ×_{Bun_{M}} ∏_{Bun_{M}} ℋ_{M}^{♭(θ_{m}^{pos})}|_{c_{m,n}}.

The following is an ind-version of [BG, Proposition 6.2.5], or [BFGM, Proposition 1.9], and we leave the proof to the reader. It is also immediately implied by Proposition 3.3.2.

Proposition 4.1.1. Let Θ be a pair (θ, 𝔘(θ^{pos})), with θ ∈ Λ_{M/[M,M]}, θ^{pos} ∈ Λ_{G,P}^{pos} , and 𝔘(θ^{pos}) a decomposition θ^{pos} = ∑_{m} n_{m}θ_{m}^{pos}. If for all m there exists μ̃_{m}^{pos} ∈ Λ̃_{G,P}^{pos} such that r(μ̃_{m}^{pos}) = θ_{m}^{pos}, then on the level of reduced stacks there is a canonical isomorphism

∞B̃un_{P}^{Θ} ≃ Bun_{P} ×_{Bun_{M}}ℋ_{M,0}^{♭(Θ)}

such that the following diagram commutes

∞B̃un_{P}^{Θ} ≃ Bun_{P} ×_{Bun_{M}}ℋ_{M,0}^{♭(Θ)}
↓  ↓
∞\overline{Bun}_{P}^{Θ} ≃  Bun_{P} ×C_{0}^{Θ}

where the right hand side is the obvious projection. If there is an m such that θ_{m}^{pos} is not equal to r(μ̃^{pos}), for any μ̃^{pos} ∈ Λ̃_{G,P}^{pos} ,

then _{∞}B̃un_{P}^{Θ} is empty.

4.2. The map 𝔭 : \overline{Z}_{can} → Z_{can}. Let X be an irreducible affine horospherical G-variety with generic stabilizer S ⊂ G. Recall that the normalizer of a horospherical subgroup S ⊂ G is a parabolic subgroup P ⊂ G with the same derived group [P, P] = [S, S] and unipotent radical U(P) = U(S). We write M for the Levi quotient P/U(P), M_{S} for the Levi quotient S/U(S), and M_{S}^{0} for the identity component of M_{S}. We write A for the quotient torus P/S, and Λ_{A} for its coweight lattice. Similarly, for the identity component S^{0} ⊂ S, we write A_{0} for the quotient torus P/S^{0}, and Λ_{A_{0}} for its coweight lattice. The natural map M/[M, M] → A_{0} induces a surjection of coweight lattices Λ_{M/[M,M]} → Λ_{A_{0}} which we denote by p. The kernel of p is the coweight lattice Λ_{M_{S}^{0}/[M_{S},M_{S}]}. (Note that the component group of M_{S} is abelian.)

Associated to the canonical affine closure \overline{G/S}, we have a Cartesian diagram of ind-stacks

\overline{Z}_{can} → _{∞}\overline{Bun}_{P}
𝔭 ↓  ↓ 𝔭
Z_{can} → ^{⋆}Z_{can}

We would like to describe some properties of the vertical maps.

Proposition 4.2.1. The map 𝔭 : _{∞}\overline{Bun}_{P} → ^{⋆}Z_{can} is ind-finite. For θ ∈ Λ_{M/[M,M]}, its restriction to _{∞}\overline{Bun}_{P}^{θ} is an embedding with image ^{⋆}Z_{can}^{p(θ)},

and its restriction to _{∞}\overline{Bun}_{P}^{≤θ} is finite with image ^{⋆}Z_{can}^{≤p(θ)}.

Proof. For a point (c, 𝒫_{G}, 𝒫_{M/[M,M]}, \overline{σ}) ∈ _{∞}\overline{Bun}_{P}, we write (c, 𝒫_{G}, 𝒫_{M/M_{S}}, σ) ∈ ^{⋆}Z_{can} for its image under 𝔭. Observe that for θ ∈ Λ_{M/[M,M]}, the point (c, 𝒫_{G}, 𝒫_{M/[M,M]}(θ · c), \overline{σ}) ∈ _{∞}\overline{Bun}_{P} maps to (c, 𝒫_{G}, 𝒫_{M/M_{S}}(p(θ) · c), σ) ∈ ^{⋆}Z_{can} under 𝔭. Therefore to prove the proposition, it suffices to show that the restriction of 𝔭 to the canonical embedding Bun_{P} ⊂ _{∞}\overline{Bun}_{P} is an embedding with image the canonical embedding Bun_{P} ⊂ ^{⋆}Z_{can}, and its restriction to _{∞}\overline{Bun}_{P}^{≤0} is a finite map with image ^{⋆}Z_{can}^{≤0}. The first assertion is immediate from the definitions. To prove the second, recall that by [BG, Proposition 1.3.6], _{∞}\overline{Bun}_{P} is proper over Bun_{G}, and so the map 𝔭 is proper since it respects the projection to Bun_{G}. Therefore it suffices to check that the fibers over closed points of the restriction of 𝔭 to _{∞}\overline{Bun}_{P}^{≤0} are finite. Let Θ be a pair (0, 𝔘(θ^{pos})), with θ^{pos} ∈ Λ_{G,P}^{pos} . The stack _{∞}\overline{Bun}_{P}^{Θ} classifies data

(c, 𝒫_{P} , c_{Θ}, 𝒫_{M/[M,M]})

together with an isomorphism

α : 𝒫_{P} ×^{P} P/[P, P] ≃ 𝒫_{M/[M,M]}(c_{Θ}).

The fiber of 𝔭 through such a point classifies data

(𝒫_{P} , c′_{Θ′}, 𝒫′_{M/[M,M]})

together with an isomorphism

α′ : 𝒫_{P} ×^{P} P/[P, P] ≃ 𝒫′_{M/[M,M]}(c′_{Θ′})

such that the labelling c_{Φ} = c_{Θ} − c′_{Θ′} takes values in Λ_{M_{S}^{0}/[M_{S},M_{S}]}. Therefore we need only check that for θ^{pos} ∈ Λ_{G,P}^{pos} , there are only a finite number of ϕ ∈ Λ_{M_{S}^{0}/[M_{S},M_{S}]} such that θ^{pos} + ϕ ∈ Λ_{G,P}^{pos} . By Proposition 2.1.3, the lattice Λ_{M_{S}^{0}/[M_{S},M_{S}]} intersects the semigroup Λ_{G,P}^{pos} only at 0. Since Λ_{G,P}^{pos} is finitelygenerated, this implies that for θ^{pos} ∈ Λ_{M/[M,M]}, the coset θ^{pos} + Λ_{M_{S}^{0}/[M_{S},M_{S}]} intersects Λ_{G,P}^{pos} in a finite set. □

Corollary 4.2.2. The map 𝔭 : \overline{Z}_{can} → Z_{can} is ind-finite. For θ ∈ Λ_{M/[M,M]}, its restriction to \overline{Z}_{can}^{θ} is an embedding with image Z_{can}^{p(θ)},

and its restriction to \overline{Z}_{can}^{≤θ} is finite with image Z_{can}^{≤p(θ)}.

4.3. The map 𝔰 : Z_{can} → Z. Let X be an affine horospherical variety with dense G-orbit X̊ ⊂ X and generic stabilizer S ⊂ G. Associated to the natural map \overline{G/S} → X, we have a Cartesian diagram of ind-stacks

Z_{can} → ^{⋆}Z_{can}
𝔰 ↓  ↓ 𝔰
Z → ^{⋆}Z.

We would like to describe some properties of the vertical maps.

Proposition 4.3.1. The map 𝔰 : ^{⋆}Z_{can} → ^{⋆}Z is a closed embedding. For κ ∈ Λ_{A}, its restriction to ^{⋆}Z_{can}^{κ} is an embedding with image ^{⋆}Z^{κ}, and its restriction to ^{⋆}Z_{can}^{≤κ} is a closed embedding with image ^{⋆}Z^{≤κ}.

Proof. First note that 𝔰 is injective on scheme-valued points since for (c, 𝒫_{G}, 𝒫_{M/M_{S}}σ) ∈ ^{⋆}Z_{can}, the map

σ : 𝒫_{M/M_{S}}|_{C\c} → 𝒫_{G} ×^{G} \overline{G/S}|_{C\c}

factors

σ|_{C′} : 𝒫_{M/M_{S}}|_{C′} → 𝒫_{G} ×^{G} G/S|_{C′} → 𝒫_{G} ×^{G} \overline{G/S}|_{C′},

for some open curve C′ ⊂ C \ c, and the map \overline{G/S} → X restricted to G/S is an embedding. Now to see 𝔰 is a closed embedding, it suffices to check that 𝔰 satisfies the valuative criterion of properness. Let D = Spec ℂ[[t]] be the disk, and D^{×} = Spec ℂ((t)) the punctured disk. Let f : D → Z be a map with a partial lift F^{×} : D^{×} → Z_{can}. Let 𝒫_{G}^{f} be the D-family of G-bundles defined by f, and let 𝒫_{M/M_{S}}^{f} be the D-family of M/M_{S}-bundles defined by f. We must check that any partial lift

Σ^{×} : 𝒫_{M/M_{S}}^{f}|_{(C\c)×D^{×}} → 𝒫_{G}^{f} ×^{G} \overline{G/S}|_{(C\c)×D^{×}}

of a map

σ : 𝒫_{M/M_{S}}^{f}|_{(C\c)×D} → 𝒫_{G}^{f} ×^{G} X|_{(C\c)×D}

which factors

σ|_{C′×D} : 𝒫_{M/M_{S}}^{f}|_{C′×D} → 𝒫_{G}^{f} ×^{G} G/S|_{C′×D} → 𝒫_{G}^{f} ×^{G} X|_{C′×D},

for some open curve C′ ⊂ C \ c, extends to (C \ c) × D. Since \overline{G/S} → X restricted to G/S is an embedding with image G/S, we may lift σ|_{C′×D} to extend Σ^{×} to C′ × D. But then Σ^{×} extends completely since 𝒫_{M/M_{S}}^{f}|_{(C\c)×D}

is normal and the complement of 𝒫_{M/M_{S}}^{f}|_{C′×D} is of codimension 2.

Finally, for a point (c, 𝒫_{G}, 𝒫_{M/M_{S}}, σ_{can}) ∈ ^{⋆}Z_{can}, we write (c, 𝒫_{G}, 𝒫_{M/M_{S}}, σ) ∈ ⋆Z for its image under 𝔰. Observe that for κ ∈ Λ_{A}, the point (c, 𝒫_{G}, 𝒫_{M/M_{S}}(κ · c), σ_{can}) ∈ ^{⋆}Z_{can} maps to (c, 𝒫_{G}, 𝒫_{M/M_{S}}(κ · c), σ) ∈ ^{⋆}Z under 𝔰. Therefore to complete the proof of the proposition, it suffices to show that the restriction of 𝔰 to the canonical embedding Bun_{S} ×C ⊂ ^{⋆}Z_{can} has image the canonical embedding Bun_{S} ×C ⊂ ^{⋆}Z. This is immediate from the definitions. □

Corollary 4.3.2. The map 𝔰 : Z_{can} → Z is a closed embedding. For κ ∈ Λ_{A}, its restriction to Z_{can}^{κ} is an embedding with image Z^{κ}, and its restriction to Z_{can}^{≤κ} is a closed embedding with image Z^{≤κ}.

5. Convolution

Let X be an affine horospherical G-variety with dense G-orbit X̊ ⊂ X and generic stabilizer S ⊂ G. The following diagram summarizes the ind-stacks and maps under consideration

∞B̃un_{P} →^{𝔯} _{∞}\overline{Bun}_{P} →^{𝔭} ^{⋆}Z_{can}
↑ 𝔨  ↑ 𝔨
\overline{Z}_{can}  →^{𝔭} Z_{can} →^{𝔰} Z.

Each of the ind-stacks of the diagram projects to C × Bun_{G}, and the maps of the diagram commute with the projections. Let 𝒵 be any one of the ind-stacks from the diagram, and form the diagram

𝒵 ←^{h\overleftarrow{G}} ℋ_{G} ×_{Bun_{G}×C} 𝒵 →^{h\overrightarrow{G}} 𝒵
↓  ↓  ↓
Bun_{G} ←^{h\overleftarrow{G}}  ℋ_{G}  →^{h\overrightarrow{G}} Bun_{G}

in which each square is Cartesian. For λ ∈ Λ_{G}^{+}, we define the convolution functor

H_{G}^{λ} : Sh(𝒵) → Sh(𝒵)

on an object ℱ ∈ Sh(𝒵) to be

H_{G}^{λ}(ℱ) = h\overleftarrow{G}_{!}(𝒜_{G}^{λ} ˜⊠ℱ)^{r}

where (𝒜_{G}^{λ} ˜⊠ℱ)^{r} is the twisted product defined with respect to h\overrightarrow{G}, and 𝒜_{G}^{λ} is the simple spherical sheaf on the fibers of h\overrightarrow{G} corresponding to λ. (See Section 1.4 for more on the twisted product and spherical sheaf.)

5.1. Convolution on _{∞}B̃un_{P}. Recall that for a reductive group H, and λ ∈ Λ_{H}^{+}, we write V λ ; Ȟ for the irreducible representation of the dual group Ȟ of highest weight λ. We shall deduce our results from the following.

Theorem 5.1.1. [BG, Theorem 4.1.5]. For λ ∈ Λ_{G}^{+}, there is a canonical isomorphism

H_{G}^{λ}(IC_{_{∞}B̃un_{P}}^{≤0} ) ≃ ∑_{μ∈Λ_{M}^{+}} IC_{_{∞}B̃un_{P}}^{≤μ} ⊗ HomM̌(V μ ; M̌, V λ ; Ǧ).

5.2. Convolution on _{∞}\overline{Bun}_{P}. Recall that r : Λ_{M} → Λ_{M/[M,M]} denotes the natural projection, 2ρ̌_{M} the sum of the positive roots of M, and ⟨2ρ̌_{M}, μ⟩ the natural pairing, for μ ∈ Λ_{M}.

Theorem 5.2.1. For λ ∈ Λ_{G}^{+}, there is an isomorphism

H_{G}^{λ}(IC_{_{∞}\overline{Bun}_{P}}^{≤0} ) ≃ ∑_{θ∈Λ_{M/[M,M]}} ∑_{μ∈Λ_{M} ,r(μ)=θ} IC_{_{∞}\overline{Bun}_{P}}^{≤θ} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)[⟨2ρ̌_{M}, μ⟩].

Proof. Step 1. For the projection

𝔯 : _{∞}B̃un_{P} → _{∞}\overline{Bun}_{P} ,

we clearly have

(1)  H_{G}^{λ}(𝔯_{!} IC_{_{∞}B̃un_{P}}^{≤0} ) ≃ 𝔯_{!}H_{G}^{λ}(IC_{_{∞}B̃un_{P}}^{≤0} ).

Let us first analyze the left hand side of equation 1. We may write the pushforward 𝔯_{!} IC_{_{∞}B̃un_{P}}^{≤0} in the form

𝔯_{!} IC_{_{∞}B̃un_{P}}^{≤0} ≃ IC_{_{∞}\overline{Bun}_{P}}^{≤0} ⊕ℐ^{≤0}

where ℐ^{≤0} ∈ Sh(_{∞}\overline{Bun}_{P}) is isomorphic to a direct sum of shifts of sheaves of the form

IC_{_{∞}\overline{Bun}_{P}}^{≤Θ} , for pairs Θ = (0, 𝔘(θ^{pos})), with θ^{pos} ∈ Λ_{G,P}^{pos} \ \{0\}.

The asserted form of ℐ^{≤0} follows from the Decomposition Theorem, the fact that the restrictions of IC_{_{∞}B̃un_{P}}^{≤0} to the strata of _{∞}B̃un_{P} are constant [BFGM, Theorem 1.12], and the structure of the map 𝔯 described in Proposition 4.1.1. For any η^{pos} ∈ Λ_{G,P}^{pos} \ {0}, and decomposition 𝔘(η^{pos}), we have the finite map

τ_{𝔘(η^{pos})} : C^{𝔘(η^{pos})} ×_{∞}\overline{Bun}_{P} → _{∞}\overline{Bun}_{P}

defined by

τ_{𝔘(η^{pos})}(∑_{m,n} η_{m}^{pos} · c_{m,n}, (c, 𝒫_{G}, 𝒫_{M/[M,M]}, σ))

= (c, 𝒫_{G}, 𝒫_{M/[M,M]}(− ∑_{m,n} η_{m}^{pos} · c_{m,n}), σ).

Note that for η ∈ Λ_{M/[M,M]}, and Θ the pair (η, 𝔘(η^{pos})), the restriction of τ_{𝔘(η^{pos})} provides an isomorphism

τ_{𝔘(η^{pos})} : (C^{𝔘(η^{pos})} ×_{∞}\overline{Bun}_{P}^{η})_{0} →^{∼} _{∞}\overline{Bun}_{P}^{Θ}

where the domain completes the Cartesian square

(C^{𝔘(η^{pos})} ×_{∞}\overline{Bun}_{P}^{η})_{0} → C^{𝔘(η^{pos})} ×_{∞}\overline{Bun}_{P}^{η}
↓  ↓
(C^{𝔘(η^{pos})} × C)_{0}  →  C^{𝔘(η^{pos})} × C

where as usual

(C^{𝔘(η^{pos})} × C)_{0} ⊂ C^{𝔘(η^{pos})} × C

denotes the complement to the diagonal divisor. We define the strict full triangulated subcategory of irrelevant sheaves

IrrelSh(_{∞}\overline{Bun}_{P}) ⊂ Sh(_{∞}\overline{Bun}_{P})

to be that generated by sheaves of the form

τ_{𝔘(η^{pos})}_{!}(IC_{C}^{𝔘(η^{pos})} ⊠ℱ)

where η^{pos} runs through Λ_{G,P}^{pos} \ {0}, 𝔘(η^{pos}) runs through decompositions of

η^{pos}, IC_{C}^{𝔘(η^{pos})} denotes the intersection cohomology sheaf of C^{𝔘(η^{pos})}, and ℱ runs through objects of Sh(_{∞}\overline{Bun}_{P}).

Lemma 5.2.2. The sheaf ℐ^{≤0} is irrelevant.

Proof. Let Θ be a pair (θ, 𝔘(θ^{pos})), with θ ∈ Λ_{M/[M,M]}, and θ^{pos} ∈ Λ_{G,P}^{pos} \ {0}. Then we may realize the sheaf IC_{_{∞}\overline{Bun}_{P}}^{≤Θ} as the pushforward

IC_{_{∞}\overline{Bun}_{P}}^{≤Θ} ≃ τ_{𝔘(θ^{pos})}_{!}(IC_{C}^{𝔘(θ^{pos})} ⊠ IC_{_{∞}\overline{Bun}_{P}}^{θ})

To see this, we use the isomorphism

τ_{𝔘(θ^{pos})} : (C^{𝔘(θ^{pos})} ×_{∞}\overline{Bun}_{P}^{θ})_{0} →^{∼} _{∞}\overline{Bun}_{P}^{Θ},

and the fact that τ_{𝔘(θ^{pos})} is finite. □

Lemma 5.2.3. If ℰ is an irrelevant sheaf, then H_{G}^{λ}(ℰ) is an irrelevant sheaf.

Proof. Clearly we have a canonical isomorphism

H_{G}^{λ}(τ_{𝔘(η^{pos})}_{!}(IC_{C}^{𝔘(η^{pos})} ⊠ℱ)) ≃ τ_{𝔘(η^{pos})}_{!}(IC_{C}^{𝔘(η^{pos})} ⊠H_{G}^{λ}(ℱ)).
□

By the preceding lemmas, we may write the left hand side of equation 1 in the form

(2)  H_{G}^{λ}(𝔯_{!} IC_{_{∞}B̃un_{P}}^{≤0} ) ≃ H_{G}^{λ}(IC_{_{∞}\overline{Bun}_{P}}^{≤0} ) ⊕ H_{G}^{λ}(ℐ^{≤0})

where H_{G}^{λ}(ℐ^{≤0}) is an irrelevant sheaf. Let us next analyze the right hand side of equation 1. By Theorem 5.1.1, we have

𝔯_{!}H_{G}^{λ}(IC_{_{∞}B̃un_{P}}^{≤0} ) ≃ ∑_{μ∈Λ_{M}^{+}} 𝔯_{!} IC_{_{∞}B̃un_{P}}^{≤μ} ⊗ HomM̌(V μ ; M̌, V λ ; Ǧ).

Lemma 5.2.4. For μ ∈ Λ_{M}^{+}, we have

𝔯_{!} IC_{_{∞}B̃un_{P}}^{≤μ} ≃ ∑_{ν∈Λ_{M}}(IC_{_{∞}\overline{Bun}_{P}}^{≤r(μ)} ⊕ℐ^{≤μ}) ⊗ HomŤ(V ν ; Ť, V μ ; M̌)[⟨2ρ̌_{M}, ν⟩].

where ℐ^{≤μ} is isomorphic to a direct sum of shifts of sheaves of the form

IC_{_{∞}\overline{Bun}_{P}}^{≤Θ} , for pairs Θ = (θ, 𝔘(θ^{pos})).

Proof. We may form the diagram

∞B̃un_{P} ←^{h\overleftarrow{M}} ℋ_{M} ×_{Bun_{M} ×C} _{∞}B̃un_{P} →^{h\overrightarrow{M}}  _{∞}B̃un_{P}
↓  ↓  ↓
Bun_{G}  ←^{h\overleftarrow{M}}  ℋ_{G}  →^{h\overrightarrow{M}} Bun_{G}

in which each square is Cartesian. We define the convolution functor

H_{M}^{μ} : Sh(_{∞}B̃un_{P}) → Sh(_{∞}B̃un_{P})

on an object ℱ ∈ Sh(_{∞}B̃un_{P}) to be

H_{M}^{μ}(ℱ) = h\overleftarrow{M}_{!}(𝒜_{M}^{μ} ˜⊠ℱ)^{r}

where (𝒜_{M}^{μ} ˜⊠ℱ)^{r} is the twisted product defined with respect to h\overrightarrow{M}, and 𝒜_{M}^{μ} is the simple spherical sheaf on the fibers of h\overrightarrow{M} corresponding to μ. Theorem 4.1.3 of [BG] provides a canonical isomorphism

H_{M}^{μ}(IC_{_{∞}B̃un_{P}}^{≤0} ) ≃ IC_{_{∞}B̃un_{P}}^{≤μ} .

We also have a commutative diagram

∞B̃un_{P}  ←^{h\overleftarrow{M}}  ℋ_{M} ×_{Bun_{M} ×C} _{∞}B̃un_{P}
𝔯 ↓  ↓ 𝔯′
∞\overline{Bun}_{P} ←^{h\overleftarrow{M/[M,M]}} ℋ_{M/[M,M]} ×_{Bun_{M/[M,M]} ×C} _{∞}\overline{Bun}_{P}

where the modification map h\overleftarrow{M/[M,M]} is given by

h\overleftarrow{M/[M,M]}(θ, (c, 𝒫_{G}, 𝒫_{M/[M,M]}, σ)) = (c, 𝒫_{G}, 𝒫_{M/[M,M]}(−θ · c), σ).

We conclude that there is an isomorphism

𝔯_{!} IC_{_{∞}B̃un_{P}}^{≤μ} ≃ h\overleftarrow{M/[M,M]!}𝔯′_{!}(𝒜_{M}^{μ} ˜⊠ IC_{_{∞}B̃un_{P}}^{≤0} )^{r}.

Now the map 𝔯′ factors into the projection of the left hand factor

ℋ_{M} ×_{Bun_{M} ×C} _{∞}B̃un_{P} → ℋ_{M/[M,M]} ×_{Bun_{M/[M,M]} ×C} _{∞}B̃un_{P}

followed by the projection of the right hand factor

ℋ_{M/[M,M]} ×_{Bun_{M/[M,M]} ×C} _{∞}B̃un_{P} →^{𝔯} ℋ_{M/[M,M]} ×_{Bun_{M/[M,M]} ×C} _{∞}\overline{Bun}_{P} .

Thus we have an isomorphism

𝔯′_{!}(𝒜_{M}^{μ} ˜⊠ IC_{_{∞}B̃un_{P}}^{≤0} )^{r} ≃ ∑_{ν∈Λ_{M}}(IC_{_{∞}\overline{Bun}_{P}}^{≤0} ⊕ℐ^{≤0}) ⊗ HomŤ(V ν ; Ť, V μ ; M̌)[⟨2ρ̌_{M}, ν⟩]

where as before

𝔯_{!} IC_{_{∞}B̃un_{P}}^{≤0} ≃ IC_{_{∞}\overline{Bun}_{P}}^{≤0} ⊕ℐ^{≤0}

where ℐ^{≤0} is isomorphic to a direct sum of shifts of sheaves of the form

IC_{_{∞}\overline{Bun}_{P}}^{≤Θ} , for pairs Θ = (0, 𝔘(θ^{pos})), with θ^{pos} ∈ Λ_{G,P}^{pos} \ \{0\}.

Finally, applying the modification h\overleftarrow{M/[M,M]!} with twist r(μ) to the above isomorphism, we obtain an isomorphism

𝔯_{!} IC_{_{∞}B̃un_{P}}^{≤μ} ≃ ∑_{ν∈Λ_{M}}(IC_{_{∞}\overline{Bun}_{P}}^{≤r(μ)} ⊕ℐ^{≤μ}) ⊗ HomŤ(V ν ; Ť, V μ ; M̌)[⟨2ρ̌_{M}, ν⟩].

Here we write ℐ^{≤μ} for the result of applying the modification h\overleftarrow{M/[M,M]!} with twist r(μ) to ℐ^{≤0}. Clearly the modification h\overleftarrow{M/[M,M]!} takes strata to strata so we conclude that ℐ^{≤μ} is isomorphic to a direct sum of shifts of sheaves of the form

IC_{_{∞}\overline{Bun}_{P}}^{≤Θ} , for pairs Θ = (θ, 𝔘(θ^{pos})).
□

Note that the proof actually shows that ℐ^{≤μ} is isomorphic to a direct sum of shifts of sheaves of the form

IC_{_{∞}\overline{Bun}_{P}}^{≤Θ} , for pairs Θ = (0, 𝔘(θ^{pos})), with θ^{pos} ∈ Λ_{G,P}^{pos} \ \{0\},

and so in particular is irrelevant, but we shall have no need for this. Combining the formulas given by Theorem 5.1.1 and the preceding lemma, we may write the right hand side of equation 1 in the form (3)

𝔯_{!}H_{G}^{λ}(IC_{_{∞}B̃un_{P}}^{≤0} ) ≃ ∑_{θ∈Λ_{M/[M,M]}} ∑_{μ∈Λ_{M} ,r(μ)=θ} IC_{_{∞}\overline{Bun}_{P}}^{≤θ} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)[⟨2ρ̌_{M}, μ⟩]⊕𝒥

where 𝒥 is isomorphic to a direct sum of shifts of sheaves of the form

IC_{_{∞}\overline{Bun}_{P}}^{≤Θ} , for pairs Θ = (θ, 𝔘(θ^{pos})).

Finally, comparing the left hand side (equation 2) and the right hand side (equation 3), and noting that IC_{_{∞}\overline{Bun}_{P}}^{≤θ} is not irrelevant, we conclude that

H_{G}^{λ}(IC_{_{∞}\overline{Bun}_{P}}^{≤0} ) ≃ ∑_{θ∈Λ_{M/[M,M]}} ∑_{μ∈Λ_{M} ,r(μ)=θ} IC_{_{∞}\overline{Bun}_{P}}^{≤θ} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)[⟨2ρ̌_{M}, μ⟩]⊕ℳ

where ℳ is is isomorphic to a direct sum of shifts of sheaves of the form

IC_{_{∞}\overline{Bun}_{P}}^{≤Θ} , for pairs Θ = (θ, 𝔘(θ^{pos})).

Step 2. Now we shall show that ℳ is in fact zero. To do this, we shall show that its restriction to each stratum of _{∞}\overline{Bun}_{P} is zero. Let Φ be a pair (ϕ, 𝔘(ϕ^{pos})), with ϕ ∈ Λ_{M/[M,M]}, and ϕ^{pos} ∈ Λ_{G,P}^{pos} . Let

H_{G}^{λ}(IC_{_{∞}\overline{Bun}_{P}}^{≤0} )_{Φ} be the restriction of H_{G}^{λ}(IC_{_{∞}\overline{Bun}_{P}}^{≤0} ) to the stratum_{∞}\overline{Bun}_{P}^{Φ}. For

θ ∈ Λ_{M/[M,M]}, let 𝒜_{Φ}^{θ} be the restriction of IC_{_{∞}\overline{Bun}_{P}}^{≤θ} to the stratum _{∞}\overline{Bun}_{P}^{Φ}, and let ℳ_{Φ} be the restriction of ℳ. Note that by step 1, [BFGM, Theorem 7.3] and Lemma 5.2.5 below, all of the restrictions are locally constant. We shall calculate H_{G}^{λ}(IC_{_{∞}\overline{Bun}_{P}}^{≤0} )_{Φ} in two different ways and compare the results. On the one hand, by Step 1, we have (4)

H_{G}^{λ}(IC_{_{∞}\overline{Bun}_{P}}^{≤0} )_{Φ} ≃ ∑_{θ∈Λ_{M/[M,M]}} ∑_{μ∈Λ_{M} ,r(μ)=θ} 𝒜_{Φ}^{θ} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)[⟨2ρ̌_{M}, μ⟩] ⊕ ℳ_{Φ}

On the other hand, let us return to the definition of the convolution, and consider the diagram

∞\overline{Bun}_{P} ←^{h\overleftarrow{G}} ℋ_{G} ×_{Bun_{G} ×C} _{∞}\overline{Bun}_{P}^{≤0} →^{h\overrightarrow{G}}  _{∞}\overline{Bun}_{P}^{≤0}
↓  ↓  ↓
Bun_{G}  ←^{h\overleftarrow{G}}  ℋ_{G}  →^{h\overrightarrow{G}} Bun_{G}

Recall that by definition

H_{G}^{λ}(IC_{_{∞}\overline{Bun}_{P}}^{≤0} ) = h\overleftarrow{G}_{!}(𝒜_{G}^{λ} ˜⊠ IC_{_{∞}\overline{Bun}_{P}}^{≤0} )^{r}

where (𝒜_{G}^{λ} ˜⊠ IC_{_{∞}\overline{Bun}_{P}}^{≤0} )^{r} is the twisted product defined with respect to h\overrightarrow{G}, and

𝒜_{G}^{λ} is the simple spherical sheaf on the fibers of h\overrightarrow{G} corresponding to λ. To calculate H_{G}^{λ}(IC_{_{∞}\overline{Bun}_{P}}^{≤0} )_{Φ}, consider the inverse image h\overleftarrow{G}^{−1}(_{∞}\overline{Bun}_{P}^{Φ}). Projecting along h\overrightarrow{G}, we may decompose the inverse image into a union of locally closed substacks

h\overleftarrow{G}^{−1}(_{∞}\overline{Bun}_{P}^{Φ}) ≃ ⨆_{ξ∈R_{G,P}^{pos}} 𝒮_{P,ϕ−ξ}^{λ} ×_{Bun_{P}} _{∞}\overline{Bun}_{P}^{(ξ,𝔘(ϕ^{pos}))}.

Projecting each piece back along h\overleftarrow{G}, we arrive at a spectral sequence for H_{G}^{λ}(IC_{_{∞}\overline{Bun}_{P}}^{≤0} )_{Φ} with E_{2} term

∑_{ξ∈R_{G,P}^{pos}} ∑_{μ∈Λ_{M} ,r(μ)=ϕ−ξ} 𝒜_{(ξ,𝔘(ϕ^{pos}))}^{0} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)[⟨2ρ̌_{M}, μ⟩]

In fact, the spectral sequence degenerates here for reasons of parity, but we shall not need this. What we do need is the following cyclicity.

Lemma 5.2.5. Let Ψ be a pair (ψ, 𝔘(ψ^{pos})), with ψ ∈ Λ_{M/[M,M]}, and ψ^{pos} ∈ Λ_{G,P}^{pos} . Let θ ∈ Λ_{M/[M,M]}. Then 𝒜_{(ψ,𝔘(ψ^{pos}))}^{0} ≃ 𝒜_{(ψ+θ,𝔘(ψ^{pos}))}^{θ}.

44 Gaitsgory and Nadler Proof. The modification

(c, 𝒫_{G}, 𝒫_{M/[M,M]}, σ) ↦ (c, 𝒫_{G}, 𝒫_{M/[M,M]}(θ · c), σ).

defines an isomorphism _{∞}\overline{Bun}_{P} →^{∼} _{∞}\overline{Bun}_{P} which restricts to an isomorphism

∞\overline{Bun}_{P}^{(ψ,𝔘(ψ^{pos}))} →^{∼} _{∞}\overline{Bun}_{P}^{(ψ+θ,𝔘(ψ^{pos}))}.

□

We apply the lemma with ψ = ξ, ψ^{pos} = ϕ^{pos}, and make the substitution θ = ϕ − ξ, to write the E_{2} term

(5)  ∑_{ϕ−θ∈R_{G,P}^{pos}} ∑_{μ∈Λ_{M} ,r(μ)=θ} 𝒜_{(ϕ,𝔘(ϕ^{pos}))}^{θ} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)[⟨2ρ̌_{M}, μ⟩]

Comparing our two calculations (equations 4 and 5), we conclude by a dimension count that ℳ_{Φ} must be zero. □ 5.3. Convolution on \overline{Z}_{can}. Theorem 5.3.1. For λ ∈ Λ_{G}^{+}, there is an isomorphism

H_{G}^{λ}(IC_{\overline{Z}_{can}}^{≤0}) ≃ ∑_{θ∈Λ_{M/[M,M]}} ∑_{μ∈Λ_{M} ,r(μ)=θ} IC_{\overline{Z}_{can}}^{≤θ} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)[⟨2ρ̌_{M}, μ⟩].

Proof. By Proposition 3.4.1, for θ ∈ Λ_{M/[M,M]}, we have

𝔨^{∗} IC_{_{∞}\overline{Bun}_{P}}^{≤θ} ≃ IC_{\overline{Z}_{can}}^{≤θ},

Clearly the pullback 𝔨^{∗} commutes with convolution

H_{G}^{λ}(𝔨^{∗} IC_{_{∞}\overline{Bun}_{P}}^{≤θ}) ≃ 𝔨^{∗}H_{G}^{λ}(IC_{_{∞}\overline{Bun}_{P}}^{≤θ}).

Thus by Theorem 5.2.1, we conclude

H_{G}^{λ}(IC_{\overline{Z}_{can}}^{≤0}) ≃ H_{G}^{λ}(𝔨^{∗} IC_{_{∞}\overline{Bun}_{P}}^{≤0} ) ≃ 𝔨^{∗}H_{G}^{λ}(IC_{_{∞}\overline{Bun}_{P}}^{≤0} )

≃ ∑_{θ∈Λ_{M/[M,M]}} ∑_{μ∈Λ_{M} ,r(μ)=θ} 𝔨^{∗} IC_{_{∞}\overline{Bun}_{P}}^{≤θ} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)[⟨2ρ̌_{M}, μ⟩]

≃ ∑_{θ∈Λ_{M/[M,M]}} ∑_{μ∈Λ_{M} ,r(μ)=θ} IC_{\overline{Z}_{can}}^{≤θ} ⊗ HomŤ(V μ ; Ť, V λ ; Ť)[⟨2ρ̌_{M}, μ⟩].

□

Documenta Mathematica 14 (2009) 19–46

5.4. Convolution on Z. Recall the map of coweight lattices

q : Λ_{M} →^{r} Λ_{M/[M,M]} →^{p} Λ_{A_{0}}.

Theorem 5.4.1. For λ ∈ Λ_{G}^{+}, there is an isomorphism

H_{G}^{λ}(IC_{Z}^{≤0}) ≃ ∑_{κ∈Λ_{A_{0}}} ∑_{μ∈Λ_{T} ,q(μ)=κ} IC_{Z}^{≤κ} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)[⟨2ρ̌_{M}, μ⟩].

Proof. By Corollary 4.2.2, for θ ∈ Λ_{M/[M,M]}, we have

𝔭_{!} IC_{\overline{Z}_{can}}^{≤θ} ≃ IC_{Z_{can}}^{≤p(θ)},

By Corollary 4.3.2, for κ ∈ Λ_{A_{0}}, we have

𝔰_{!} IC_{Z_{can}}^{≤κ} ≃ IC_{Z}^{≤κ} .

Clearly the pushforwards 𝔭_{!} and 𝔰_{!} commute with convolution

H_{G}^{λ}(𝔰_{!}𝔭_{!} IC_{\overline{Z}}{≤0 \atop can}) ≃ 𝔰_{!}𝔭_{!}H_{G}^{λ}(IC_{\overline{Z}}{≤0 \atop can}).

Thus by Theorem 5.3.1, we conclude

H_{G}^{λ}(IC_{Z}^{≤0})

≃ H_{G}^{λ}(𝔰_{!}𝔭_{!} IC_{\overline{Z}_{can}}^{≤0})

≃ 𝔰_{!}𝔭_{!}H_{G}^{λ}(IC_{\overline{Z}_{can}}^{≤0})

≃ ∑_{θ∈Λ_{M/[M,M]}} ∑_{μ∈Λ_{M} ,r(μ)=θ} 𝔰_{!}𝔭_{!} IC_{\overline{Z}_{can}}^{≤θ} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)[⟨2ρ̌_{M}, μ⟩]

≃ ∑_{κ∈Λ_{A_{0}}} ∑_{μ∈Λ_{T} ,q(μ)=κ} IC_{Z}^{≤κ} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)[⟨2ρ̌_{M}, μ⟩].

□

6. Complements

For our application [GN], we need a slight modification of our main result. As usual, let X be an affine horospherical G-variety with dense G-orbit X̊ ⊂ X and generic stabilizer S ⊂ G. Let S^{0} be the identity component of S, and let π_{0}(S) be the component group S/S^{0}. For a scheme 𝒮, we write C_{𝒮} for the product 𝒮 × C. For an 𝒮-point (c, 𝒫_{G}, σ) of the ind-stack Z, the section σ defines a reduction of the G-bundle 𝒫_{G} to an S-bundle 𝒫′_{S} over an open subscheme C′_{𝒮} ⊂ C_{𝒮} which is the complement C_{𝒮} \ 𝒟 of a subscheme 𝒟 ⊂ C_{𝒮} which is finite and flat over 𝒮. By induction, the S-bundle 𝒫′_{S} defines a π_{0}(S)-bundle over C′_{𝒮}. We call this the generic π_{0}(S)-bundle associated to the point (c, 𝒫_{G}, σ). We define ′Z ⊂ Z to be the ind-substack whose 𝒮-points (c, 𝒫_{G}, σ) have the property that for every geometric point s ∈ 𝒮, the restriction of the associated generic π_{0}(S)-bundle to {s} × C ⊂ C_{𝒮} is trivial. It is not difficult (see [GN]) to show that ′Z is closed in Z. Observe that we have a short exact sequence

0 → Λ_{A_{0}} → Λ_{A} → S/S^{0} → 0.

Thus for κ ∈ Λ_{A_{0}}, it makes sense to consider the locally closed substack ′Z^{κ} ⊂ ′Z and its closure ′Z^{≤κ} ⊂ ′Z. Observe as well that from the fibration S → G → G/S, we have an exact sequence

π_{1}(G) → π_{1}(X̊) → π_{0}(S).

Thus for λ ∈ Λ_{G}^{+}, we have the convolution functor

H_{G}^{λ} : Sh(′Z) → Sh(′Z).

The same arguments show that our main result holds equally well in this context.

Theorem 6.0.2. For λ ∈ Λ_{G}^{+}, there is an isomorphism

H_{G}^{λ}(IC_{′Z}^{≤0}) ≃ ∑_{κ∈Λ_{A_{0}}} ∑_{μ∈Λ_{T} ,q(μ)=κ} IC_{′Z}^{≤κ} ⊗ HomŤ(V μ ; Ť, V λ ; Ǧ)[⟨2ρ̌_{M}, μ⟩].

References

[BFGM] A. Braverman, M. Finkleberg, D. Gaitsgory, and I. Mirković. Intersection cohomology of Drinfeld’s compactifications, Selecta Math. (N.S.), vol. 8, 2002, 381–418.

[BG] A. Braverman and D. Gaitsgory, Geometric Eisenstein series. Invent. Math., 150 (2), 2002, 287–384.

[GN] D. Gaitsgory and D. Nadler, Spherical varieties and Langlands duality, To appear in Moscow Math. J., volume dedicated to Pierre Deligne.

[Kn] F. Knop, Weylgruppe und Momentabbildung, Invent. Math., vol. 99 (1), 1990, 1–23.

[Pop] V. L. Popov, Contractions of actions of reductive algebraic groups. Mat. Sb. (N.S.), 130(172)(3), 1986, 310–334, 431.

Dennis Gaitsgory

Department of Mathematics Harvard University Cambridge MA 02138 gaitsgde@math.harvard.edu David Nadler Department of Mathematics Northwestern University Evanston IL 60208 nadler@math.northwestern.edu