An Iwahori-Whittaker model for the Satake category

D. ARINKIN AND D. GAITSGORY

Introduction

Date: March 16, 2015.

1

Part I: The Key Calculation

1. Statement of the problem 1.1. The geometric objects.

1.1.1. Recall the stack BunB defined as in [BG1]. Recall that BunB admits a decomposition into locally closed substacks, parameterized by elementsλ∈Λ^pos,

BunB = [

λ∈Λ^pos

Bun^=λ_B .

For each λ∈Λ^pos we let

^λ: Bun^=λ_B ,→BunB

denote the corresponding locally closed embedding.

We have a canonical identification

Bun^=λ_B 'X^λ×BunB.

We let Bun^≤λ_B denote the open substack of Bun_B equal to [

0≤µ≤λ

Bun^=µ_B .

1.1.2. We let

p: Bun_B →Bun_G andq: Bun_B→Bun_T denote the canonical projections, and by

p: BunB →BunG andq: BunB→BunT

their restrictions to BunB ⊂BunB, respectively.

Note that we have a commutative diagram

X^λ×Bun_B −−−−→^∼ Bun^=λ_B

id×q



 y



 y^q◦

λ

X^λ×Bun_T −−−−→ Bun_T, where the bottom horizontal arrow is the map

D,PT 7→PT(−D), whereD∈X^λ is a Λ^pos-colored divisor.

1.1.3. LetPT be aT-bundle. We let BunN,PT denote the fiber product BunB ×

BunT

pt,

where the map BunB→BunT isqand pt→BunT is given byPT.

The group T acts by automorphisms of PT; by functoriality, we obtain a T-action on BunN,PT. The quotient stack BunN,PT/T identifies with

BunB ×

BunT

pt/T; in particular, the resulting map

BunN,PT/T →BunB

is a closed embedding.

We let rdenote the tautological projection Bun_N,PT →Bun_N,PT/T.

The stack Bun_N,PT inherits a decomposition into locally closed substacks with Bun^=λ_N,PT 'X^λ ×

BunT

BunB, where the mapX^λ→BunT is

D7→PT(D).

By a slight abuse of notation, we shall use the same symbol^λ to denote also the maps Bun^=λ_N,P

T →Bun_N,PT and Bun^=λ_N,P

T/T →Bun_N,PT/T, respectively.

Similarly, by a slight abuse of notation we will denote by id×qalso the maps Bun^=λ_N,PT 'X^λ ×

Bun_T

BunB id×q

−→X^λ ×

Bun_T

BunT =X^λ and

(1.1) Bun^=λ_N,PT/T '(X^λ ×

Bun_T

BunB)/T ^id−→^×q(X^λ ×

Bun_T

BunT)/T =X^λ×pt/T, respectively.

1.2. The basic character sheaf.

1.2.1. In what follows we fix one and for all a line bundleω_X¹² equipped with an isomorphsim (ω

1 2

X)^⊗2'ωX.

We letρ(ωX) denote theT-bundle onX induced from the line bundleω

1 2

X by means of the cocharacter 2ρ:Gm→T.

Consider the stack

Bun_N,ρ(ωX). As in [FGV], there exists a canonical map

ev : Bun_N,ρ(ωX)→A¹.

1.2.2. The map ev is constructed as follows. Consider the map B→B/[N, N]ZG'Π

i (Gmn Ga),

wherei runs through the set of vertices of the Dynkin diagram of G. From here we obtain a map of stacks

Bun_N,ρ(ωX)→Π

i

Bun_GmnGa ×

Bun_Gm

pt

,

where pt→Bun_Gm is given by the line bundleωX. Now, the stack Bun_GmnGa ×

Bun_Gm

pt classfies short exact sequences 0→ωX→E→OX →0,

and hence is equipped with a canonocal map toA¹. Finally, we let ev be the composition

(1.2) Bun_N,ρ(ωX)→Π

i

Bun_GmnGa ×

Bun_Gm

pt

→Π

i A^{1 sum}−→A¹. 1.2.3. Set

ψ:= ev^∗(A-Sch),

where A-Sch ∈ D-mod(A¹) is the algebraic-Schreier sheaf (i.e., the exponential D-module, shifted cohomologically by [−1]).

Set

ψ:= (⁰)!(ψ)∈D-mod(Bun_N,ρ(ωX)).

Remark 1.2.4. As was observed in [FGV], the extensionψ ψis clean, i.e., the canonical map

(1.3) (⁰)_!(ψ)→(⁰)_∗(ψ)

is an isomorphism.

1.2.5. Consider the object

r!(ψ)∈D-mod(Bun_N,ρ(ωX)/T).

For each λ∈Λ^pos consider

(^λ)^!◦r!(ψ)∈D-mod(Bun^=λ_N,ρ(ωX)/T).

Consider now the object

(1.4) M^λ:= (id×q)∗◦(^λ)^!◦r!(ψ)∈D-mod(X^λ×pt/T), where id×q: Bun^=λ_N,ρ(ω

X)/T →X^λ×pt/T is as in (1.1).

1.2.6. The goal of Part I of this paper is concerned with the objectM^λ of (1.4). On the one hand, in Theorem 1.3.6, we will calculateM^λmore or less explicitly.

On the other hand, and which is more important, in Theorem 1.4.6, we will relateM^λ to the calculation of a certainConstant Term functor.

The proofs of Theorems 1.3.6 and 1.4.6 will share a common core, given by Theorem 2.2.3.

The latter theorem (like most of the assertions of this kind) is a corollary of Braden’s theorem about hyperbolic restrictions; the deduction will be the subject of Part II of the paper.

In Part III of the paper we will explain how Theorem 1.4.6 fits into the geometric Langlands program.

1.2.7. Example. Let us describe explicitly the objectM⁰, i.e.,M^λforλ= 0. This is easy to do

“by hand”. Namely, we claim thatM⁰∈D-mod(pt/T) identifies withHc(T), where the latter is the direct image with compact supports ofk∈Vect = D-mod(pt) under the map

triv : pt→pt/T.

Indeed, note that in the composition (1.2), the first map is a smooth unipoten gerbe (see [DrGa1] for what this means), so the functors of *-pullback and *-pushforward define mutually inverse equivalences of categories.

Hence, M⁰ identifies with the direct image underA^r/T → pt/T of the direct image with compact supports underA^r→A^r/T of

A-Sch^r∈D-mod(A^r), whererdenotes the semi-simple rank ofG.

Hence, M⁰ identifies with the direct image underA^r/T → pt/T of the direct image with compact supports underA^r→A^r/T of

A-Sch^r∈D-mod(A^r).

By the contraction principle (see [DrGa2]), the operation of direct image underA^r/T →pt/T can be replaced by that of *-pullback under pt/T →A^r/T.

The stated answer forM⁰ follows now by base change from the Cartesian square pt −−−−→ A^r

triv



 y



 y pt/T −−−−→ A^r/T.

1.3. The sheaves Ω^λ.

1.3.1. In this subsection we recall the objects

Ω^λ∈D-mod(X^λ)^♥, λ∈Λ^pos, introduced in [BG2].

The object Ω^λ is characterised by the requirement that its !-fiber at D= Σ

i λi·xi∈X^λ, xi6=xj, identifies with

O

i

(C^·(ˇn))^−λi,

where C^·(ˇn) is the cohomological Chevalley complex of the Lie algebra ˇn, and superscript −λi

means the (−λi)-graded component with respect to the adjoint action of ˇT. The sheaf structure on Ω^λ is given by the structure on C^·(ˇn) of commutative DG algebra.

1.3.2. Examples. For G = SL2 and λ =n ∈ Z^≥0 (so that X^λ is the n-th symmetric power X⁽ⁿ⁾), the sheaf Ω^λ is the (clean) extension of the sign local system onX^◦(n).

The same happens for anyGforλ=n·α_i, whereα_i is a simple coroot.

Consider now G = SL₃ with its two simple roots α₁ and α₂. For λ = α₁+α₂ we have X^λ=X×X, and we have

Ω^α1+α2=j_∗(k_Xk_X)[2], wherej:X×X−∆(X),→X×X.

1.3.3. Although this will not be necessary for the sequel, let us add a few more pieces of information regarding the sheaves Ω^λ.

First, the assigment

λ7→Ω^λ has a factorization property:

(1.5) Ω^λ1+λ2|_(Xλ1×X^λ2)disj 'Ω^λ1Ω^λ2|_(Xλ1×X^λ2)disj, where

(X^λ1×X^λ2)_disj⊂X^λ1×X^λ2 is the open subset corresponding to pairs

(D₁, D₂), supp(D₁)∩supp(D₂) =∅.

Note that the addition map

X^λ1×X^λ2 →X^λ1+λ2

is ´etale when retsricted to (X^λ1×X^λ2)disj, so in the left-hand side of (1.5) we can take either the ! or the *-retsriction.

1.3.4. The isomorphism (1.5) allows to describe the sheaves Ω^λinductively. Indeed, suppose we know Ω^λforλ⁰< λ. Then (1.5) implies that we know the restriction of Ω^λtoX^λ−∆(X),→^j X^λ.

According to [BG2, Proposition 3.2], we have:

Ω^λ=











k_X[1] if λis a simple coroot,

j!∗(Ω^λ|_Xλ−∆(X)) if∃w∈W, λ=ρ−w(ρ) with`(w) = 2,

j_!∗(Ω^λ|_Xλ−∆(X))'H⁰(j_∗(Ω^λ|_Xλ−∆(X))), if∃w∈W, λ=ρ−w(ρ) with`(w)≥3, H⁰(j∗(Ω^λ|_Xλ−∆(X)))'j∗(Ω^λ|_Xλ−∆(X)) if 6 ∃w∈W, λ=ρ−w(ρ),

whereH⁰referes to taking cohomology with respect to the D-module (i.e., perverse) t-structure.

1.3.5. The first main result of Part I of this paper reads:

Theorem 1.3.6. There exists a canonical isomorphism inD-mod(X^λ×pt/T) M^λ'Ω^λ[−2|λ|]H_c(T),

where|λ|is the length ofλ.

1.4. Relation to the Constant Term functor(s).

1.4.1. Recall that the map

Bun_N,ρ(ωX)/T →Bun_B is a closed embedding.

By a slight abuse of notation we shall identify objects of D-mod(Bun_N,ρ(ωX)/T) with the corresponding objects of D-mod(Bun_B).

Note that we have a commutative diagram

Bun^=λ_N,ρ(ωX)/T −−−−→ Bun^=λ_B −−−−→^id×q X^λ×BunT

∼



 y



 y^pXλ×id (X^λ ×

Bun_T

BunB)/T −−−−→ X^λ×pt/T −−−−→^AJ BunT, where AJ is a version of the Abel-Jacobi map,

D7→ρ(ωX)(D), and wherep_Xλ :X^λ→pt.

1.4.2. LetW∈D-mod(Bun_G) denote the direct image ofψ∈D-mod(Bun_N,ρ(ωX)) with compact supports under the forgetful map

Bun_N,ρ(ωX)→Bun_G. Tautologically,

W'p_!◦r_!(ψ).

Sometimes, the objectW∈D-mod(BunG) goes under the samethe first Whittaker coefficient.

1.4.3. Recall theConstant Term functor

CT∗:=q∗◦p^!: D-mod(BunG)→D-mod(BunT), see [DrGa4].

Recall the object M^λ ∈ D-mod(X^λ×pt/T), see Sect. 1.4. We claim that there exists a canonically defined map

(1.6) AJ!(M^λ)→CT∗(W).

Indeed, let us write

AJ!(M^λ) = (p_Xλ×id)!◦(id×q)∗◦(^λ)^!◦r!(ψ) andW=p_!◦r!(ψ), and we claim that there is a natural transformation

(1.7) (p_Xλ×id)!◦(id×q)∗◦(^λ)^!→CT∗◦p_!.

1.4.4. Namely, consider the unit of the adjunction Id→p^!◦p_!, and apply to both sides the functor

(p_Xλ×id)_!◦(id×q)∗◦(^λ)^! : D-mod(Bun_B)→D-mod(Bun_T).

We obtain a map

(1.8) (p_Xλ×id)!◦(id×q)_∗◦(^λ)^!→(p_Xλ×id)!◦(id×q)_∗◦(^λ)^!◦p^!◦p_!'

'(p_Xλ×id)_!◦(id×q)_∗◦ p◦^λ!

◦p_!. So, it is enough to construct a natural transformation

(1.9) (p_Xλ×id)!◦(id×q)_∗◦ p◦^λ!

→CT_∗. However,

p◦^λ=p◦(p_Xλ×id) :X^λ×BunB→BunG, so the expression in (1.9) is isomorphic to

(1.10) (p_Xλ×id)!◦(id×q)∗◦(p_Xλ×id)^!◦p^!. Further, from the Cartesian diagram

X^λ×BunB

p_Xλ×id

−−−−−→ BunB id×q



 y



 y^q X^λ×BunT

p_Xλ×id

−−−−−→ BunT

we obtain a natural transformation (in fact, an isomorphism, since the horizontal maps are proper)

(p_Xλ×id)!◦(id×q)∗→q∗◦(p_Xλ×id)!. Hence, the expression in (1.10) maps (in fact, isomorphically) to (1.11) q_∗◦(p_Xλ×id)!◦(p_Xλ×id)^!◦p^!.

Finally, applying the co-unit of the adjunction (p_Xλ×id)!◦(p_Xλ×id)^! →Id, we obtain that the expression in (1.11) maps to

q∗◦p^!= CT∗, as desired.

1.4.5. The main result of Part I of this paper is:

Theorem 1.4.6. The map (1.6)is an isomorphism.

2. Calculation via the Zastava spaces 2.1. Zastava spaces: recollections.

2.1.1. LetZ_P^λ

T denote the Zastava space corresponding toPT ∈BunT. I.e., Z_P^λ

T is the open substack

BunN,PT ×

BunG

Bun_B⁻_,λ+deg(PT)

^gen.trans

⊂BunN,PT ×

BunG

Bun_B⁻_,λ+deg(PT),

corresponding toB- andB⁻-reductions that are transversal at the generic point of the curve.

In the above formula Bun_B−,λ+deg(PT)is the connected component of Bun_B− corresponding toB-bundles, for which the inducedT-bundle has degreeλ+ deg(PT).

Remark 2.1.2. A basic feature ofZ_P^λ

T is that it is actually a (quasi-projective) scheme.

2.1.3. We let

0p⁻:Z_P^λ

T →Bun_N,PT denote the tautological projection.

For 0≤µ≤λ, we let⁰^µ denote the locally closed embedding Z_P^λ,=µ

T :=Z_P^λ

T ×

Bun_N,PT

Bun^=µ_N,P

T ,→Z_P^λ

T.

In particular, forµ= 0 we let

◦

Z^λ_P

T ⊂Z_P^λ

T

denote the corresponding open subscheme.

2.1.4. According to [BFGM], there exists a canonical projection π^λ:Z_P^λ_T →X^λ

that makes the diagram

Z_P^λ

T −−−−→ Bun_B−,λ+deg(PT) π^λ



 y



 y X^λ −−−−→ BunT

commute, where the bottom horizontal arrow is the mapD7→PT(D).

It is known that the map π^λ induces an isomorphism between Z_P^λ,=λ

T and X^λ. Hence, the map⁰^λ provides a section of the projectionπ^λ. We shall also use the notation

s^λ:=⁰^λ. The composed map

X^λs

λ

'Z_P^λ,=λ

T 0p⁻

−→Bun^=λ_N,PT =X^λ ×

BunT

BunB

equals the mapι^λ.

2.2. An adaptation of Braden’s theorem.

2.2.1. For anyPT consider the diagram

(2.1)

X^λ×pt/T ^s

λ

−−−−→ Z_P^λ

T/T ^π

λ

−−−−→ X^λ×pt/T

ι^λ



 y



 y

0p⁻

(X^λ ×

Bun_T

Bun_B)/T ^

λ

−−−−→ Bun_N,PT/T

id×q



 y X^λ×pt/T,

in which the inner square is Cartesian and the composite maps are both equal to id_Xλ×pt/T. By base change, we have an isomorphism of functors

(2.2) (id×q)∗◦(^λ)^!◦(⁰p⁻)_∗◦(π^λ)^!'Id_D-mod(Xλ×pt/T), from which we obtain a natural transformation of functors

(2.3) (id×q)_∗◦(^λ)^! →(π^λ)!◦(⁰p⁻)^∗, D-mod(BunN,PT/T)→D-mod(X^λ×pt/T).

2.2.2. We will apply Braden’s theorem (see [DrGa3]) to prove:

Theorem 2.2.3. The natural transformation (2.3)is an isomorphism.

Theorem 2.2.3 will be proved in Sect. 5.1.

2.3. Proof of Theorem 1.3.6. In this subsection we will show how Theorem 2.2.3 implies Theorem 1.3.6.

2.3.1. TakePT =ρ(ω_X), and consider the corresponding Zastava spaceZ_ρ(ω^λ

X). Recall the map

0p⁻:Z_ρ(ω^λ

X)→Bun_N,ρ(ωX).

The key ingredient that connects M^λ with Ω^λ is provided by the followiing theorem (see [Ras]):

Theorem 2.3.2. There exists a canonical isomorphism π_!^λ◦(⁰p⁻)^∗(ψ)'Ω^λ[−2|λ|].

2.3.3. Thus, in order to prove Theorem 1.3.6, we need to construct a canonical isomorphism (2.4) (id×q)∗◦(^λ)^!◦r!(ψ)'π_!^λ◦(⁰p⁻)^∗(ψ)Hc(T)

of objects of D-mod(X^λ×pt/T).

Let us apply the natural isomorphism (2.3) to

r!(ψ)∈D-mod(Bun_N,ρ(ωX)/T).

We obtain that the left-hand side identifies with (π^λ)_!◦(⁰p⁻)^∗◦r_!(ψ).

Applying base change along the lower square in the diagram X^λ −−−−−→^id×triv X^λ×pt/T

π^λ

x





x



^π

λ

Z_P^λ

T

0r

−−−−→ Z_P^λ

T/T

0p⁻



 y



 y

0p⁻

Bun_N,ρ(ωX) −−−−→^r Bun_N,ρ(ωX)/T, we rewrite

(π^λ)!◦(⁰p⁻)^∗◦r!(ψ)'(π^λ)!◦(⁰r)!◦(⁰p⁻)^∗(ψ)'(id×triv)!◦(π^λ)!◦(⁰p⁻)^∗(ψ)'

'(π^λ)!◦(⁰p⁻)^∗(ψ)Hc(T), as required.

2.4. A reduction step towards the proof of Theorem 2.2.3. As was mentioned earlier, Theorem 2.2.3 will be deduced from Braden’s theorem. We would like to apply Braden’s theorem to the algebraicstack BunN,PT. The problem is that we cannot quite do this, because Braden’s theorem is only known for algebraicspaces, and not general algebraic stacks.

In this subsection we will show that the assertion of Theorem 2.2.3 can be formally deduced from the case when the pair (PT, λ) is such that Bun^≤λ_N,P

T is an algebraic space, in which case Braden’s theorem can be applied.

That said, we should say that, given Theorem 5.1.3 established in Part II of the paper, we can reprove an analog of Braden’s theorem specifically for the stack Bun_N,PT, using the method of [DrGa3]. I.e., the reduction to the case of algebraic spaces is not really necessary.

2.4.1. Let us be given a diagram of algebraic stacks

(2.5)

Y⁰ −−−−→^ι− Y⁻ −−−−→^q− Y⁰

ι⁺



 y



 y^p

−

Y⁺ −−−−→^p+ Y

q⁺



 y Y⁰, where

q⁺◦ι⁺= id_Y⁻ =q⁻◦ι⁻. and where the map

Y⁰→Y⁺×

YY⁻ is an open embedding.

Then as in (2.2), we obtain a natural transformation

q⁺_∗ ◦(p⁺)^!◦(p⁻)_∗◦(q⁻)^!→Id_D-mod(Y0), from which we obtain a natural transformation

(2.6) (q⁺)_∗◦(p⁺)^!→(q⁻)_!◦(p⁻)^∗.

2.4.2. We wish to know when for a given

F∈D-mod(Y)

the resulting map (q⁺)_∗◦(p⁺)^!(F)→(q⁺)_∗◦(p⁺)^!(F) is an isomorphism.

Suppose that we are given a similar diagram

(2.7)

eY⁰ −−−−→^eι− Ye⁻ −−−−→^eq− eY⁰

eι⁺



 y



 y^ep

−

Ye⁺ −−−−→^ep+ Ye

eq⁺



 y Y⁰.

2.4.3. Suppose that we are given a map from the diagram (2.7) to the diagram (2.5), such that:

• The mapsφ⁻:eY⁻→Y⁻ andφ⁰:eY⁰→Y⁰ are isomorphisms.;

• The mapsφ:eY→Yandφ⁺:eY⁺→Y⁺ are smooth;

• The diagram

eY⁺ −−−−→^ep+ eY

φ⁺



 y



 y^φ Y⁺ −−−−→^p+ Y is Cartesian.

The following is an easy particular case of [DrGa4, Theorem 4.3.4]

Lemma 2.4.4. Let F∈D-mod(Y)be such that for Fe:=φ^∗(F)the map (eq⁺)_∗◦(ep⁺)^!(eF)→(eq⁺)_∗◦(ep⁺)^!(eF) is an isomorphism. Assume also that the maps

(2.8) (q⁺)_∗◦(p⁺)^!(F)→(q⁺)_∗◦(ι⁺)_∗◦(ι⁺)^∗◦(p⁺)^!(F)'(ι⁺)^∗◦(p⁺)^!(F) and

(2.9) (eq⁺)_∗◦(ep⁺)^!(eF)→(eq⁺)_∗◦(eι⁺)_∗◦(eι⁺)^∗◦(ep⁺)^!(eF)'(eι⁺)^∗◦(ep⁺)^!(eF) are isomorphisms. Then the map

(q⁺)_∗◦(p⁺)^!(F)→(q⁺)_∗◦(p⁺)^!(F) is an isomorphism.

Proof. Diagram chase shows that we have the following commutative diagram of natural trans- formations

(φ⁰)^∗◦(q⁺)∗◦(p⁺)^{! (φ}

0)^∗(2.6)

−−−−−−→ (φ⁰)^∗◦(q⁻)!◦(p⁻)^∗



 y



 y

(eq⁺)_∗◦(φ⁺)^∗◦(p⁺)^! (eq⁻)_!◦(φ⁻)^∗◦(p⁻)^∗



 y



 y^∼ (eq⁺)∗◦(ep⁺)^!◦φ^{∗ (2.6)(φ}

∗)

−−−−−−→ (eq⁻)!◦(ep⁻)^∗◦φ^∗.

We need to show that the top horizontal arrow in this diagram is an isomorphism when evaluated onF. We shall do so by showing that all other maps are isomorphisms.

The assumption on the map of diagrams implies that the lower left vertical arrow and the upper right vertical arrows are isomorphisms. The bottom horizontal arrow is an isomorphism by assumption. Hence, it remains to show that the upper left vertical arrow is an isomorphism.

However, this follows from the commutative diagram

(φ⁰)^∗◦(q⁺)_∗◦(p⁺)^! −−−−→ (φ⁰)^∗◦(ι⁺)^∗◦(p⁺)^!



 y



 y^∼

(eq⁺)_∗◦(φ⁺)^∗◦(p⁺)^! −−−−→ (ι⁺)^∗◦(φ⁺)^∗◦(p⁺)^! and the assumption that the maps (2.8) and (2.9) are isomorphisms.

2.4.5. Let us first take the diagram (2.5) to be (2.1). To prove Theorem 2.2.3 we have to show that the corresponding natural transformation (2.6) is an isomorphism.

For a point x∈X , let

Bun^good_N,PT^x ⊂BunN,PT

be the open substack, where we do not allow the generalized B-reduction to degenerate at x∈X.

Denote also

Bun^≤λ,good_N,PT ^x and Bun^=λ,good_N,PT ^x

the corresponding open substacks of Bun^≤λ_N,PT and Bun^=λ_N,PT, respectively. We have Bun^=λ,good_N,P ^x

T '(X−x)^λ ×

Bun_T

Bun_B.

It is enough to show that the natural transformation (2.6) is an isomorphism for the diagram

(2.10)

(X−x)^λ×pt/T −−−−→ Z_P^λ

T/T ×

X^λ

(X−x)^λ −−−−→ (X−x)^λ



 y



 y Bun^=λ,good_N,P ^x

T /T −−−−→ Bun^≤λ,good_N,P ^x

T /T



 y (X−x)^λ

for anyx. We shall do so by applying Lemma 2.4.4.

Note that the natural transformation (2.8) is an isomorphism, by the contraction principle, see [DrGa3].

2.4.6. Letµbe a dominant coweight, and setP⁰T :=PT(−µ·x). We claim that there exists a canonically defined commutative (in fact Cartesian) diagram

Bun^=λ,good_N,P0 ^x

T −−−−→ Bun^good_N,P0^x T



 y



 y Bun^=λ,good_N,PT ^x −−−−→ Bun^good_N,PT^x, in which the vertical arrows are smooth.

Indeed, let Bun_B,PT(Dx) (resp., Bun_B,P⁰

T(Dx)) denote the stack ofB-bundles on the formal disc aroundxinX, equipped with an identification of the inducedT-bundle withPT|_Dx(resp., P⁰T|_Dx).

We have canonically defined maps Bun^good_N,P^x

T →Bun_B,PT(Dx) and Bun^good_N,P0^x

T →Bun_B,P⁰

T(Dx).

We also have a smooth map

BunB,P⁰T(Dx)→BunB,PT(Dx), and a Cartesian square

Bun^good_N,P0^x

T −−−−→ Bun_B,P⁰

T(Dx)



 y



 y Bun^good_N,PT^x −−−−→ BunB,PT(Dx).

2.4.7. Note also that there is a canonical isomorphism Z_P^λ_T ×

X^λ

(X−x)^λ'Z_P^λ0 T ×

X^λ

(X−x)^λ (the Zastava space is “local” inX).

By construction, the diagram Bun^good_N,P0^x

T 0p⁻

←−−−− Z_P^λ0 T ×

X^λ

(X−x)^λ



 y



 y Bun^good_N,P^x

T 0p⁻

←−−−− Z_P^λ

T ×

X^λ

(X−x)^λ commutes as well.

Hence, we obtain a map from the diagram

(2.11)

(X−x)^λ×pt/T −−−−→ Z_P^λ0 T

/T ×

X^λ

(X−x)^λ −−−−→ (X−x)^λ



 y



 y Bun^=λ,good_N,P0 ^x

T /T −−−−→ Bun^≤λ,good_N,P0 ^x T /T



 y (X−x)^λ to that in (2.11).

2.4.8. Applying Lemma 2.4.4, we obtain that the assertion of Theorem 2.2.3 for a given pair PT and λfollows from the validity of Theorem 2.2.3 forP⁰T :=PT(−µ·x) with µ∈Λ⁺, and the sameλfor allx∈X.

Chooseµso that deg(PT) +λ−µis anti-dominant and regular. Note that in this case points of Bun^≤λ_N,P0

T admit no non-trivial automorphisms, so the open substack Bun^≤λ_N,P0

T is an algebraic space.

Hence, we obtain that assertion of Theorem 2.2.3 for a givenλ follows from the case when PT is such that Bun^≤λ_N,P

T is an algebraic space.

3. Proof of Theorem 1.4.6 3.1. Compatibility between diagrams.

3.1.1. Let us be given be a pair of diagrams (2.5) and (2.7) and a map between them. Then the construction in Sect. 1.4.4 gives rise to a natural transformation

(3.1) (φ⁰)!◦(eq⁺)_∗◦(ep⁺)^!→(q⁺)_∗◦(p⁺)^!◦φ!. Let us make the following assumptions:

• The diagram

(3.2)

eY⁰ −−−−→^eι+ eY⁺

φ⁰



 y



 y^φ

+

Y⁰ −−−−→^ι+ Y⁺ is Cartesian;

• The mapYe⁻ →eY×

YY⁻ is an open embedding.

In particular, we obtain a natural transformation

(3.3) (φ⁻)!◦(ep⁻)^∗→(p⁻)^∗◦φ!.

Furthermore, diagram chase shows that the following diagram of natural transformations is commutative:

(3.4)

(φ⁰)!◦(eq⁺)_∗◦(ep⁺)^! −−−−→^(3.1) (q⁺)_∗◦(p⁺)^!◦φ! (φ⁰)!(2.6)



 y

(φ⁰)_!◦(eq⁻)_!◦(ep⁻)^∗ 

 y^(2.6)(φ!)

∼



 y

(q⁻)!◦(φ⁻)!◦(ep⁻)^{∗ (q}

−)_!(3.3)

−−−−−−→ (q⁻)!◦(p⁻)^∗◦φ!.

3.1.2. We apply the commutative diagram (3.4) in the following situation. We take (2.7) to be the diagram

X^λ×pt/T ^s

λ

−−−−→ Z_P^λ

T/T ^π

λ

−−−−→ X^λ×pt/T

ι^λ



 y



 y

0p⁻

(X^λ ×

Bun_T

BunB)/T ^

λ

−−−−→ BunN,PT/T

id×q



 y X^λ×pt/T, of (2.1).

We take the diagram (2.5) to be the diagram BunT −−−−→ Bun_B−

q⁻

−−−−→ BunT



 y



 y^p

−

BunB

−−−−→p BunG q



 y BunT.

We evaluate the functors in (3.4) on the object

r!(ψ)∈D-mod(BunB).

The statement of Theorem 1.4.6 says that the top horizontal arrow in the resulting diagram (3.4) is an isomorphism. We shall do so by showing that all other arrows are isomorphisms.

3.1.3. The fact that the upper left vertical arrow is an isomorphism is the content of Theo- rem 2.2.3. The fact that the right vertical arrow is an isomorphism follows from the fact that the natural transformation

q_∗◦p^! →(q⁻)!◦(p⁻)^∗ is an isomorphism, see [DrGa4].

3.1.4. Thus, it remains to show that the natural transformation (q⁻)_!◦(φ⁻)_!◦(ep⁻)^∗→(q⁻)_!◦(φ⁻)_!◦(ep⁻)^∗, induced by (3.3), yields an isomorphism when evaluated on the object

r_!(ψ)∈D-mod(Bun_N,PT/T).

Concretely, we need to show the following. Consider the open embedding

Bun_N,ρ(ωX) ×

Bun_G

Bun_B−

^gen.trans.

,→j

Bun_N,ρ(ωX) ×

Bun_G

Bun_B−

.

We need to show that the map

j!◦j^∗◦(⁰p⁻)^∗(ψ)→(⁰p⁻)^∗(ψ)

induces an isomorphism after taking the direct image with compact supports along the projec- tion

BunN,ρ(ω_X) ×

BunG

BunB⁻ →BunB⁻ q⁻

−→BunT.

3.1.5. The stack BunN,ρ(ω_X) ×

BunG

Bun_B⁻ admits a decomposition into locally closed pieces in- dexed by the Weyl group:

Bun_N,ρ(ωX) ×

BunG

Bun_B⁻

=[

w

Bun_N,ρ(ωX) ×

BunG

Bun_B⁻ w

,

where the open piece

Bun_N,ρ(ωX) ×

BunG

Bun_B⁻

^gen.trans.

corresponds tow= 1.

We will show that for eachw6= 1, the direct image with compact supports under the map

(3.5) Z^w:=

Bun_N,ρ(ωX) ×

BunG

Bun_B^{− w}

→Bun_B^{− q}

−

−→BunT

of the *-restriction ofψ, vanishes.

3.2. Calculation for w6= 1.

3.2.1. The stackZ^w maps to

X^Λpos:= G

µ∈Λ^pos

X^µ,

see [BG2, Sect. 10.9]. Let us denote this map byπ^w.

The map (3.5) is the composition of the mapπ^w, followed by the Abel-Jacobi map X^Λpos →BunT, D7→ρ(ωX)(w(D)).

We will show that the object

(π^w)!(ψ|Z^w)∈D-mod(X^Λpos) vanishes.

For that it suffices to show that (π^w)!(ψ|Z^w) vanishes, when restricted to (X−x)^Λpos for any x∈X.

3.2.2. LetB^w denote the subgroupB∩Adw(B⁻). It is equipped with a canonical projection toT. LetN^wdenote the kernel of this projection, i.e.,N^w=N∩Adw(B⁻).

For a test-schemeSand a mapz:S→Z^w letDdenote the resulting mapS→X^Λpos. The pointz corresponds to aG-bundlePG onS×X, equipped with a reductionαtoB (for which the inducedT-bundle isω(ρ)), and a reductionβ toB⁻, which are in positionwat the generic point ofX.

By the construction of the map π^w (see [BG2, Sect. 10.9]), the data of (PG, α, β), when restricted to the open subset

U :=S×X−Graph_D,

is induced from a uniquely definedB^w-bundlePB^w, for which the inducedT-bundle is identified withρ(ω_X).

Let us denote

Z^w,goodx :=Z^w ×

X^Λpos

(X−x)^Λpos,

and let us choose a trivialization of theT-bundleρ(ωX) over the formal disc Dx around xin X.

Thus, we obtain a map of stacks

Z^w,goodx→Bun_Nw(Dx)'pt/N^w(Dx),

where Dx denotes the formal disc around x, and where N^w(Dx) denotes the group-scheme classifying mapsDx→N^w.

3.2.3. Consider the fiber product

Z^w,levelx:=Z^w,goodx ×

pt/N^w(Dx)

pt.

This is the moduli stack of (PG, α, β, ), where (PG, α, β) are as above, and is the datum of trivialization ofPN^w over Dx. One can show thatZ^w,levelx is in fact a scheme (of infinite type).

The stack (scheme)Z^w,levelx is acted on by the group ind-schemeN^w(D^×x), whereN^w(D^×x) classifies maps from the formal punctured discD^×x toN^w. This action preserves the map

Z^w,levelx→(X−x)^Λpos,

For any group-subscheme N⁰⊂N:=N^w(D^×x), we can find a group-subschemeN⁰⁰⊂N0:=

N^w(Dx) of finite codimension, which is normal inN⁰. Thus, we obtain an action of the (finite- dimensional) unipotent groupN⁰/N⁰⁰on the stack (of finite type)

Z^w,levelx/N⁰⁰. This action preserves the projection

(3.6) Z^w,levelx/N⁰⁰→Z^w,levelx/N0=Z^w,levelx→(X−x)^Λpos. 3.2.4. Consider the canonical projection

N →G^ra. Consider the map

N(D^×x)→G^ra(D^×x)−→^res G^ra

−→sum Ga, where the map

res :G^ra(D^×x)→G^ra

is defined using our chosen trivializationρ(ωX)|_Dx. Sincew6= 1, the composition

N=N^w(D^×x),→N(D^×x)→Ga

is non-zero.

LetN⁰⊂N^w(D^×x) be such that the composition N⁰ →Ga(D^×x)→Ga

is non-zero. Choose a subgroupN⁰⁰ ⊂N₀ as above. Thus, we obtain a non-trivial homomor- phism

(3.7) N⁰/N⁰⁰→Ga.

3.2.5. Since N0/N⁰⁰ is unipotent, it is enough to show that the direct image with compact supports under the map (3.6) of

(3.8) ψ|_Zw,levelx/N⁰⁰∈D-mod(Z^w,levelx/N⁰⁰) vanishes.

However, this follows from the fact that the object (3.8) is equivariant against the pull-back of

A-Sch∈D-mod(Ga)

under anon-trivial homomorphismN⁰/N⁰⁰→Ga, namely, the homomorphism (3.7).

Part II: Geometry ofG^m-actions on moduli problems

4. Gm-actions on classifying stacks 4.1. Attractors and repellers.

4.1.1. LetYbe an arbitrary prestack (i.e., a contravariant functor from the category of schemes to that of groupoids), equipped with an action ofGm.

We let Fixed(Y), Attr(Y) and Repel(Y) denote the prestacks given by Hom(S,Fixed(Y)) = Hom^Gm(S,Y), and

Hom(S,Attr(Y)) = Hom^Gm(S×A¹,Y), Hom(S,Repel(Y)) = Hom^Gm(S×A¹,Y), respectively, where in the case of attractors the action ofGmonA¹ is the standard one, and in the case of repellers its inverse.

4.1.2. We have the natural forgetful map Fixed(Y)→Y, as well as the maps Attr(Y)→Yand Repel(Y)→Y,

given by evaluation at 1∈A¹, and the maps

Fixed(Y)→Attr(Y) and Fixed(Y)→Repel(Y), given by the projectionA¹→pt.

The latter maps admit respective left inverses

Attr(Y)→Fixed(Y) and Repel(Y)→Fixed(Y), given by evaluation at 0∈A¹.

4.1.3. It is shown in [Dr] that if Y is a (resp., separated) scheme of finite type, then so are Fixed(Y), Attr(Y) and Repel(Y).

IfYis a scheme, on which theGm-action is locally linear (see [DrGa3] for what this means), then the above representability is much easier to establish.

WhenYis an affine scheme, the maps

Fixed(Y)→Attr(Y)→Y are all closed embeddings.

4.1.4. We have the following general observation:

Proposition 4.1.5. LetN be a unipotent group, acted on byGmwith strictly positive eigevalues on the Lie algebra. Consider the resultingGm-action on the stackpt/N. Then:

(a) Fixed(pt/N)'pt.

(b)The forgetful mapAttr(pt/N)→pt/N is an isomorphism.

(c)The maps Fixed(pt/N)→Repel(pt/N)→Fixed(pt/N) are isomorphisms.

This proposition implies that we have the following commutative diagrams pt −−−−→ Fixed(pt/N)



 y



 y pt/N −−−−→ Attr(pt/N)



 y



 y pt −−−−→ Fixed(pt/N) and

pt −−−−→ Fixed(pt/N)



 y



 y pt −−−−→ Repel(pt/N)



 y



 y pt −−−−→ Fixed(pt/N) with horizontal arrows being isomorphisms.

Proof. Filtering N, we can assume thatN =Ga with Gmacting by the n-th power power of the standard character, wheren >0.

LetSbe an arbitrary test-scheme, and letEbe anGa-bundle onS(resp.,S×A¹) is equipped with a structure ofGm-equivariance with respect to the aboveGm-action on pt/Ga (resp., and the correspondingGm-action onA¹).

With no restriction of generality we can assume thatS is affine. Then Eis non-canonically trivial. LetV denote theGm-module of automorphisms ofE. In case (a) we have

V = Γ(S,OS),

with the standard action ofGm. In cases (b) and (c), we have V = Γ(S,OS)⊗k[t],

whereGmacts by then-th power of standard character on Γ(S,OS) and wherethas degree−1 (resp., 1).

Note that E admits a Gm-equivariant trivialization: indeed an osbtruction would be an element ofH¹(Gm, V), whereas this group is 0.

Now, points (a) and (c) of the lemma follow from the fact that in these casesH⁰(Gm, V) = 0.

Point (b) follows from the fact that in this case evaluation at 1∈A¹ defines an isomorphism H⁰(Gm, V)→Γ(S,OS).

4.2. A digression: some maps related to parabolic subgroups. This subsection will contain some tautological manipulations and notation, which will be used in Sect. 4.3.

4.2.1. In what follows we will use the following observation:

For a parabolicP with Levi quotientM there exists a canonically defined map¹

(4.1) pt/M →pt/P,

corresponding tosplit P-bundles, right inverse to the tautological projection pt/P →pt/M.

Indeed, since the different splittings of the projection P → M are uniquely conjugate by means of the of theunipotent radical of P, they give rise to canonically isomorphic splittings of the map pt/P →pt/M.

4.2.2. Letγ:G^m→Gbe a homomorphism. LetMγ be the centralizer ofγ, which is the same as Fixed(G), whereG^macts onGby conjugation viaγ. LetPγ be the subgroup Attr(G).

It is easy to see thatPγ is a parabolic subgroup, and the maps Mγ →Pγ →Mγ

realizeM_γ as both the Levi subgroup and the Levi quotient group of P_γ.

4.2.3. Note that the homomorphismγ:Gm→Z(Mγ) defines an action ofGmon the identity automorphism of pt/Mγ. Hence, we obtain a map

(4.2) pt/M_γ →Maps(pt/Gm,pt/M_γ),

where for two prestacksY1andY2 we let Maps(Y1,Y2) denote the prestack Hom(S,Maps(Y1,Y2)) = Hom(S×Y1,Y2).

Consider the fiber product

Maps(pt/Gm,pt/Pγ) ×

Maps(pt/Gm,pt/M_γ)

pt/Mγ, where pt/Mγ →Maps(pt/Gm,pt/Mγ) is the map (4.2).

Lemma 4.2.4. The map

Maps(pt/Gm,pt/Pγ) ×

Maps(pt/Gm,pt/Mγ)

pt/Mγ →

→Maps(pt/G^m,pt/Mγ) ×

Maps(pt/Gm,pt/Mγ)

pt/Mγ = pt/Mγ

is an isomorphism.

Proof. Follows from Proposition 4.1.5(a).

1We learned this from J. Lurie.