Introduction - Harvard Universitypeople.math.harvard.edu/~gaitsgde/GL/WhatActs.pdf · In the classical theory, the analogue of the term main is a certain intertwining operator (acting

WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES

D. GAITSGORY

1. Introduction

1.1. The Functor. Geometric Eisenstein series is the functor Eis! : D(BunT ) → D(BunG)defined by

F 7→ p! q∗(F),

(up to a cohomological shift), where

BunG BunT

BunB

p

q

The question that we are concerned with in this paper is the description ofHomD(BunG)(Eis!(F1),Eis!(F2)) in terms of F1,F2.

In some sense, the answer is tautological. Let CT∗ denote the right adjoint to Eis!. Thecomposition Φ := CT∗ Eis! is a monad acting on D(BunT ), and we have:

HomD(BunG)(Eis!(F1),Eis!(F2)) ' HomD(BunT )(F1,Φ(F2)).

So, what we are after is to have a more detailed understanding of the monad Φ.

1.2. The Monad.

1.2.1. Parallel to what happens in the classical theory of automorphic functions, the functor Φadmits a canonical filtration by functors numbered by the Weyl group W (viewed as a posetwith respect to the Bruhat order); we denote the subquotient functor corresponding to anelement w ∈ W by Φw. The term Φmain := Φ1 happens to be the most interesting; in factΦmain is itself a monad, and the canonical map Φmain → Φ is a homomorphism. (By contrast,the term Φw0

is the simplest: it’s given by the action of w0 on BunT .)

In the classical theory, the analogue of the term Φmain is a certain intertwining operator(acting on automorphic functions on the abelian group T ), and it decomposes as a product oflocal intertwining operators.

We can now define our goal more precisely as follows: we’d like to describe

Φmain : D(BunT )→ D(BunT )

in terms that are local with respect to the curve X. The latter phrase can be given a precisemeaning as follows:

Date: July 4, 2011.

1

2 D. GAITSGORY

Recall that for a reductive group M and x ∈ X one can attach the Satake category at x,denoted SatM,x that acts on D(BunM ) by Hecke functors. The assignment

x 7→ SatM,x

forms what is called a factorization (a.k.a. chiral) category over X, equipped with a compatiblemonoidal structure. We denote this category simply by SatM . Now, if E ∈ SatM is a factor-ization (a.k.a. chiral) algebra equipped with a compatible associative algebra structure in thiscategory, the chiral homology Hch(X,E) acts as a monad on D(BunM ).

So, our first goal can be stated as follows:

Goal 1a: Describe explicitly the factorization algebra E ∈ SatT , such that Φmain ' Hch(X,E).

1.2.2. By the definition of the functor Eis!, its construction uses the stack BunB . However,BunB is a little ”stupid” as a stack: it splits into connected components numbered by elementsof Λ–the coroot lattice of T .

However, BunB admits a relative compactification

BunB

BunG BunT

BunB

p

q

//

with p = p and q = q .The stack BunB is stratified by locally closed substacks of the form

ιλ : Xλ × BunB → BunB ,

where Xλ is the space of configuration on Λpos-colored divisors of total degree λ. (Here Λpos ⊂ Λis the semi-group spanned by positive simple roots.)

Our second goal can be stated as follows:

Goal 1b: Describe the factorization algebra E in terms adjunction of the strata in BunB .

1.3. The ”space” of rational reductions to B.

1.3.1. The situation with BunB can be pushed even further. In 2004 Drinfeld proposed thatthere should exist a stack BunratB ”of G-bundles equipped with a rational reduction to B”. Thisstack is supposed to glue together the connected components BunµB of BunB for µ projectingto the same element of π1(G). In other words, the should exist a projection

BunB → BunratB

that collapses each stratum Xλ × BunµB to just BunµB .

However, BunB does not exist as an algebraic stack (the strata that need to collapse havewrong self-intersections). Nonetheless we will achieve:

Goal 2a: Construct a category D(BunratB ), equipped with a ”direct image” functor Av :D(BunB)→ D(BunratB ) and a functor prat! : D(BunratB )→ D(BunG) so that p! ' prat! Av.

1.3.2. As we shall see, the monad Φmain will have a natural interpretation in terms of thecategory D(BunratB ). Namely, we’ll achieve

Goal 2b: Interpret the monad Φmain in terms of Homs in the category D(BunratB ).

WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES 3

1.4. Compactified Eisenstein series. The existence of the compactification BunB has leadthe authors of [BG1] and [FFKM] to consider another functor D(BunT ) → D(BunG), namelythat of compactified Eisenstein series:

Eis!∗(F) := p!(q∗(F)⊗ ICBunB

),

where ICBunBis the intersection cohomology sheaf on BunB .

Our next goal can be stated as follows:

Goal 3: From the action of the monad Φmain on Eis!, find what monad acts on the functorEis!∗.

1.5. The Langlands dual picture.

1.5.1. Let us recall now that the functor Eis! is supposed to fit into the Geometric Langlandspicture

(1.1)

QCohN(LocSysG)LG−−−−→ D(BunG)

Eisspec

x Eis! −ρ(ωX)

xQCoh(LocSysT )

LT−−−−→ D(BunT ).

Here QCohN(LocSysG) is a certain modification of the category QCoh(LocSysG), describedin [Sum].

The horizontal arrow LG is the conjectural Langlands transform. The horizontal arrow LT isthe Langlands transform for T , which since T is abelian, is the Fourier-Mukai transform. Thefunctor Eisspec is the spectral Eisenstein series functor defined using the diagram

LocSysG LocSysT

LocSysB

pspec

qspec

byEisspec := pspec∗ q∗spec.

Finally, −ρ(ωX) is the functor of shift by −ρ(ωX) ∈ BunT acting on D(BunT ).

1.5.2. Our main goal can be stated as follows:

Goal 4a: Give an interpretation of the monad Φmain and of the chiral algebra E on the Langlandsdual side as a monad acting on QCoh(LocSysT ).

1.5.3. Note that the map of stacks pt /B → pt /T admits a canonical section, and hence sodoes the map of stacks

e : LocSysT → LocSysB .

We’ll show that if the diagram (1.2) takes place, then so does the diagram

(1.2)

QCohN(LocSysG)LG−−−−→ D(BunG)

pspec∗e∗x Eis!∗ −ρ(ωX)

xQCoh(LocSysT )

LT−−−−→ D(BunT ).

4 D. GAITSGORY

Our final goal is:

Goal 4b: Give an interpretation of the monad from Goal 3 on the Langlands dual side as amonad acting on QCoh(LocSysT ).

We observe that our Goal 4b is a categorical upgrade of the following remarkable fact dis-covered in [FFKM]: namely, that the object Eis!∗(CBunT ) carries an action of the Langlandsdual Lie algebra g.

1.6. Structure of the paper. This paper is divided into three parts, according to the levelof sophistication at which we employ the machinery of factorization algebras and categories.

1.6.1. Part I is largely preparatory and is meant to set the scene for the more involved butsimilar in spirit manipulation in the later sections.

In particular, instead of the factorization category SatT , which is needed for the description ofthe monad Φmain, we use its more simply-minded version that deals with factorization algebrasof Λpos-vector space.

Although this restricted framework will not allow us to get to Φmain or our goals that haveto do with the Langlands dual picture, we will be able to observe some interesting phenomena,e.g. why the functor Eis! factors on the Langlands dual side through a functor q∗spec.

We should also remark that Part I is to a large extent a restatement of the results from[BG2] in the language of factorization algebras.

1.6.2. In Part II we’ll achieve some of the goals stated above.

First, we’ll show what the monad Φmain has to do with the stacks BunB . Secondly, we’lldefine the category D(BunratB ) with the expected properties.

1.6.3. Finally, in Part III we’ll state a certain local conjecture that will relate the monad Φmainto the factorization category SatT , and assuming this conjecture, we’ll be able to make a contactwith the Langlands dual picture.

We note that the results ”proved” in Part III heavily result on the yet unpublished theoryof factorization categories.


Part I

2. Factorization algebras in a simplified context

This section introduces a language of factorization algebras graded by a semi-lattice Λpos. 1

This is nothing but a particular case of the set-up of factorization algebras from [CHA], withthe difference that the presence of Λ allows to replace the Ran space by genuine schemes.

2.1. The graded Ran space.

2.1.1. For λ ∈ Λpos, let Xλ denote the corresponding partially symmetrized power of X. Forλ = λ1 + λ2 let

addλ1,λ2: Xλ1 ×Xλ2 → Xλ

denote the canonical map.

For Fi ∈ D(Xλi), with λ = λ1 + λ2 we’ll denote by F1 ? F2 the object of D(Xλ) equal to

(addλ1,λ2)∗(F1 F2).

2.1.2. We regard the disjoint union ∪λXλ is a Λpos-graded version of the Ran space Ran(X)

and denote it by Ran(X,Λpos).

The category

D(Ran(X,Λpos)) := λ 7→ Fλ ∈ D(Xλ)has a natural monoidal structure with respect to ?: for two families Fλ1 and Fλ2 the valueof their tensor product on Xλ is

⊕λ=λ1+λ2

Fλ11 ? Fλ2

2 .

This monoidal structure is naturally symmetric.

2.1.3. For λ ∈ Λpos consider the natural closed embedding ∆λ : X → Xλ.

The functor ∆∗ := ∆λ∗ makes the category D(X)Λpos of Λpos-graded objects of D(X) a full

subcategory of D(Ran(X,Λpos)). This embedding has a right adjoint given by ∆! := ∆λ!.

The category D(X)Λpos has a natural symmetric monoidal structure given by!⊗. The functor

∆! is strictly monoidal.

By adjunction, for F1, ...,Fn, F ∈ D(X)Λpos we have a natural map

(2.1) HomD(X)Λpos (F,F1

!⊗ ...

!⊗ Fn)→ HomD(Ran(X,Λpos))(∆∗(F),∆∗(F1) ? ... ?∆∗(Fn)),

and these maps are compatible with iterated tensor products. However, we have the followingstraightforward assertion:

Lemma 2.1.4. The maps (2.1) are isomorphisms.

2.1.5. One can speak about associative, commutative and Lie algebras and co-algebras in eitherD(X)Λpos orD(Ran(X,Λpos)). The functor ∆! maps such objects inD(Ran(X,Λpos)) to objectsof a similar nature in D(X)Λpos .

In addition, Lemma 2.1.4 implies that for an object M ∈ D(X)Λpos , a structure of co-algebraof any kind on it is equivalent to one on ∆∗(M).

1Unless specified otherwise, we consider Λpos without the 0 element.

6 D. GAITSGORY

2.1.6. The usual Koszul duality defines equivalences

KDA→cA : Assoc.Alg→ co-Assoc. co-Alg,

KDcC→L : co-Com. co-Alg→ Lie alg,

andKDC→cL : Com.Alg→ Lie co-alg

in either context.

NB: We’ll use the subscripts ? or!⊗ to indicate which category we’re in when an ambiguity is

likely to occur.

Explicitly, KDA→cA attaches to an associative augmented algebra A the co-associative co-algebra Bar(A) that computes Tor’s of the augmentation module with itself over A. The inversefunctor sends a co-associative co-algebra A∨ to coBar(A∨), where the latter computes Exts’sof the augmentation module with itself over A∨.

The inverse to KDcC→L sends a Lie algebra L to its homological Chevalley complex C·(L).The inverse to KDC→cL sends a Lie co-algebra L∨ to its homological Chevalley complex C·(L).

2.2. Factorization algebras.

2.2.1. By definition, a Λpos-factorization algebra is the same as a chiral algebra on X, gradedby Λpos. The corresponding functor is given by

A 7→ C(A),

where C stands for the chiral Chevalley-Cousin complex, following the conventions from [CHA].

2.2.2. Explicitly, we can think of a Λpos-graded factorization algebra A as follows. To eachλ ∈ Λpos we assign Aλ ∈ D(Xλ) and whenever λ = λ1 + λ2, we have an isomorphism

(2.2) Aλ|(Xλ1×Xλ2 )disj ' Aλ1 Aλ2 |(Xλ1×Xλ2 )disj ,

satisfying the natural compatibilities.

NB: The restriction of the map addλ1,λ2to the open subscheme (Xλ1 ×Xλ2)disj ⊂ Xλ1 ×Xλ2

is etale, so the notation F 7→ F|(Xλ1×Xλ2 )disj is unambiguous.

2.2.3. Whenever we have two factorization algebras A1 and A2 their ?-product A1 ? A2 has anatural factorization algebra structure.

2.3. Commutative factorization algebras.

2.3.1. We shall say that A is commutative if the above isomorphisms have been extended tomaps

Aλ1 Aλ2 → add!λ1,λ2

(Aλ),

which also satisfy the natural compatibilities.

By adjunction, for a commutative A, we obtain the maps

Aλ1 ?Aλ2 → Aλ1+λ2 ,

which make A into a commutative algebra in D(Ran(X,Λpos)) with respect to ?.

Note that commutative factorization algebras form a full subcategory among commutativealgebras in D(Ran(X,Λpos)). Indeed, for a commutative algebra A := Aλ ∈ D(Ran(X,Λpos))we have the maps

Aλ1 Aλ2 |(Xλ1×Xλ2 )disj → Aλ|(Xλ1×Xλ2 )disj

and A is a commutative factorization algebra if and only if these maps are isomorphisms.


2.3.2. As in [CHA], one can prove the following:

Proposition 2.3.3. The assignment

A 7→ ∆!(A)

establishes an equivalence between the category of commutative factorization algebras and thatof commutative algebras in D(X)Λpos .

NB: Note that the chiral algebra corresponding to such A is ∆!(A)[−1].

For a commutative algebra A in D(X)Λpos , we’ll denote by ARan the corresponding commu-tative factorization algebra.

2.3.4. The following proposition is due to [JNKF].

Proposition 2.3.5. The following diagram of functors is commutative:

Lie co-alg(D(X)Λpos ,!⊗)

KDcL→C(D(X)Λpos ,!⊗)−−−−−−−−−−−−−−−→ Com.Alg(D(X)Λpos ,

!⊗)

∆∗

y yA 7→ARan

Lie co-alg(D(Ran(X,Λpos), ?)KDcL→C(D(Ran(X,Λpos),?)−−−−−−−−−−−−−−−−−−→ Com.Alg(D(Ran(X,Λpos), ?).

The above proposition reads as follows: for a Lie co-algebra L∨ ∈ D(X)Λpos with respect to

the!⊗ tensor structure, consider the following two procedures:

• Consider the commutative algebra C·(L∨) ∈ D(X)Λpos with respect to the!⊗ tensor

structure, and apply to it the equivalence of Proposition 2.3.3, i.e., C·(L∨)Ran.• Consider the Lie co-algebra ∆∗(L

∨) ∈ D(Ran(X,Λpos)), and consider its cohomologicalChevalley complex C·(∆∗(L

∨)) with respect to the ? tensor structure.

We’ll denote the resulting commutative factorization algebra by Ω(L), i.e.,

C·(L∨)Ran ' Ω(L) ' C·(∆∗(L∨)).

It follows from the construction that if L∨ is such that L∨[−1] is a D-module (belongs tothe heart of the t-structure), then so does Ω(L).

2.3.6. The following construction will be used in the sequel. Let L∨ be as above. Consider∆∗(L

∨) as a Lie co-algebra in D(Ran(X,Λpos)) with respect to ?, and consider its universalco-enveloping co-algebra in D(Ran(X,Λpos)); denote the resulting object of D(Ran(X,Λpos))by U∨(L)Ran. 2

By construction, U∨(L)Ran is a commutative algebra in D(Ran(X,Λpos)) with a compatibleco-associative co-algebra structure, i.e., it’s a commutative Hopf algebra.

In addition, U∨(L) has the following properties:

Lemma 2.3.7.

(1) As a commutative algebra in D(Ran(X,Λpos)), U∨(L)Ran is factorizable.

(2) The Hopf algebra ∆!(U∨(L)Ran) ∈ D(X)Λpos identifies with the universal co-enveloping

co-algebra U∨(L) of L∨, considered as a Lie co-algebra with respect to the!⊗ tensor structure.

2We emphasize that U∨(L) is not the Verdier dual of the chiral universal envelope of the Verdier dual L ofL∨, assuming it was dualizable. As we shall see, U∨(L) is that for the loop object of L[−1].

8 D. GAITSGORY

Note also that the universal co-enveloping co-algebra U∨(L)Ran of a Lie co-algebra L∨ (inany tensor category) has the following additional interpretations:

2.3.8. For a Lie co-algebra L∨ ∈ D(X)Λpos consider its suspension L∨[1]. This is a co-algebraobject in the category of Lie co-algebras in D(X)Λpos . Then we have:

U∨(L)Ran ' Ω(L∨[1]),

as commutative factorization algebras in D(Ran(X,Λpos)) equipped with a co-algebra structure.

2.3.9. Another (in a sense, tautologically equivalent) interpretation is as follows: for a Lie co-algebra L∨, consider C·(L∨) as an associative algebra. Consider the co-associative co-algebraBar (C·(L∨)). The commutative algebra structure on C·(L∨) makes Bar (C·(L∨)) into a com-mutative Hopf algebra. We have a canonical isomorphism

U∨(L) ' Bar (C·(L∨)) ,

as commutative Hopf algebras. So, we have:

U∨(L)Ran ' Bar(Ω(L)),

as commutative factorization algebras in D(Ran(X,Λpos)) equipped with a co-algebra structure.

2.4. Co-commutative factorization algebras.

2.4.1. We say that a Λpos-factorization algebra is co-commutative if the factorization isomor-phisms (2.2) come by adjunction from maps

Aλ1+λ2 → Aλ1 ?Aλ2 ,

which make A into a co-commutative co-algebra in D(Ran(X,Λpos)).

As in the case of commutative factorization algebras, the category of co-commutative fac-torization algebras is a full subcategory in the category of co-commutative co-algebras inD(Ran(X,Λpos)) with respect to the ? tensor structure.

NB: We refrain from formulating the structure of co-commutative factorization algebra as amap on Xλ1 ×Xλ2 because the latter would involve the functor add∗λ1,λ2

, which is not definedfor all D-modules.

2.4.2. The following is parallel to Proposition 2.3.5 combined with Proposition 2.3.3:

Proposition 2.4.3.

(1) For a Lie-* algebra L in D(X)Λpos , the co-commutative co-algebra in D(Ran(X,Λpos)) withrespect to the ? tensor structure, given by C·(∆∗(L)) is factorizable.

(2) The above assignment L 7→ C·(∆∗(L)) is an equivalence between the category of Lie-*algebras in D(X)Λpos and co-commutative factorization algebras.

For a Lie-* algebra L in D(X)Λpos let us denote the resulting co-commutative factorizationalgebra by Υ(L).

It follows from the construction that it L is such that L[1] is a D-module (i.e., belongs tothe heart of the t-structure), then so does Υ(L).

Let U(L) denote the chiral universal enveloping algebra of L; this is a Λpos-graded chiralalgebra on X. By construction, we have:

C(U(L)) ' Υ(L).


2.4.4. The following is parallel to Sect. 2.3.6. For a Lie-* algebra L in D(X)Λpos , consider ∆∗(L)as a Lie algebra in D(Ran(X,Λpos)) with respect to the ? tensor structure, and let U(L)Ran

denote its universal enveloping algebra. This is a co-commutative factorization algebra inD(Ran(X,Λpos)) with a compatible associative algebra structure with respect to ?.

We can also think of U(L)Ran as coBar(Υ(L)), with its natural structure of co-commutativeHopf algebra. In addition, we have a canonical isomorphism

Υ(L[−1]) ' U(L)Ran,

as co-commutative Hopf algebras.

2.5. Verdier duality.

2.5.1. Parallel to the above discussion, we can consider the semi-group Λneg. For a compactobject F ∈ D(Xλ), we’ll think of its Verdier dual D(F) as an object of D(X−λ).

We shall say that an object F = Fλ of D(Ran(X,Λpos)) is locally compact if each of itscomponents Fλ is compact as an object of D(Xλ).

Thus, we obtain that Verdier duality defines a contravariant equivalence between the sub-categories D(Ran(X,Λpos)) and D(Ran(X,Λneg)), consisting of locally compact objects.

2.5.2. The ? tensor product sends (locally) compact objects to (locally) compact ones, andsatisfies:

D(F1 ? F2) ' D(F1) ? D(F2).

In particular, Verdier duality defines anti-equivalences between the categories of locally com-pact associative/commutative/Lie algebras in D(Ran(X,Λpos)) with respect to the ? tensorstructure and the corresponding co-algebras in D(Ran(X,Λneg)).

Moreover, Koszul duaility in any of the contexts: associative/commutative/Lie sends locallycompact objects to locally compact ones, and we have:

D KD ' KD D.

2.5.3. Let A be a factorization algebra which is locally compact. It is clear that D(A) has anatural structure of factorization algebra.

The following is evident from the definitions:

Lemma 2.5.4. Let A be a factorization algebra which is locally compact. Then the structureon it of commutative/co-commutative factorization algebra is equivalent to the structure of co-commutative/commutative factorization algebra on D(A).

2.5.5. Let L be a Lie-* algebra in D(X)Λpos , which is compact in every degree. Then L∨ :=D(L) ∈ D(X)Λneg is also compact in every degree, and has a natural structure of Lie co-algebra

with respect to the!⊗ tensor structure.

From Sect. 2.5.2, we obtain that the objects Υ(L), Ω(L), U(L)Ran and U∨(L)Ran are alllocally compact. Moreover,

Lemma 2.5.6. We have canonical isomorphisms D(Υ(L)) ' Ω(L) as commutative algebras,and D(U(L)) ' U∨(L) as commutative Hopf algebras in the ? tensor structure.

10 D. GAITSGORY

3. Eisenstein series, take I (a summary of [BG2])

3.1. Action of Ran(X,Λpos)) on BunT .

3.1.1. We assume now that X is compact and that Λpos maps to the lattice Λ of coweights ofsome torus T . Consider the category D(BunT ).

We denote the action of D(Ran(X,Λpos)) as a monoidal category on D(BunT ) by

(F := Fλ, S) 7→ F ? S := ⊕λ

(πλ × idBunT )∗

(πλ!(Fλ)

!⊗mult!

λ(S)

),

where multλ is the natural map

Xλ × BunTAJλ× id−→ BunT ×BunT

mult−→ BunT ,

where

AJλ : Xλ → BunT

is the Abel-Jacobi map.

3.1.2. Similarly, we define an action of D(Ran(X,Λneg)). Since the maps multλ are smoothand proper, we have:

Lemma 3.1.3. For a compact object F ∈ D(Ran(X,Λpos)), the functors

S 7→ F ? S and S′ 7→ D(F) ? S′

are both left and right adjoint of one another.

3.2. Action on Drinfeld’s compactifications.

3.2.1. For λ ∈ Λpos denote by

ιλ : Xλ × BunB → BunB

the corresponding map obtained by ”adding zeros”. This is a finite map. We let ιλ denote itsrestriction to the open substack Xλ × BunB .

The above procedure defines an action of D(Ran(X,Λpos)) as a monoidal category onD(BunB) by

(F := Fλ, S) 7→ F ? S := ⊕λιλ∗(F

λ S).

3.2.2. Let p, q denote the projections:

BunGp←− BunB

q−→ BunT .

We define the functor of Eisenstein series

Eis : D(BunB)×D(BunT )→ D(BunG)

by

T,F 7→ p∗

(T

!⊗ q!(F)

).

The following is a diagram chase:

Proposition 3.2.3. For S ∈ D(Ran(X,Λpos)) there exists a canonical isomorphism:

Eis(S ? T,F) ' Eis(T, S ? F).

3.3. A factorization algebra attached to n−.


3.3.1. Consider the Lie algebra n−; consider the constant Lie-* algebra n−X := n−⊗CX , which is

graded by Λneg, and its Verdier dual (n−X)∨ ' (n−)∨⊗CX [2] which is graded by Λpos. Consider

the corresponding commutative algebra Ω(n−X) in D(Ran(X,Λpos)) with respect to ?.

By Sect. 2.3.4, Ωλ(n−X) is a D-module (i.e., belongs to the heart of the t-structure) for everyλ.

3.3.2. Let denote the open embedding BunB → BunB , and consider the object

!(ICBunB ) ∈ D(BunB).

NB: Since BunB is smooth, ICBunB is isomorphic to CBunB [dim(BunB)], where we apply thecohomological shift by the corresponding amount on each connected component.

The following has been established in [BG2]:

Theorem 3.3.3. The object !(ICBunB ) ∈ D(BunB) is naturally a Ω(n−X)-module, with respect

to the above action of D(Ran(X,Λpos)) on D(BunB). Moreover, the map

Ωλ(n−X) ICBunB → ιλ! !(ICBunB ),

arising by adjunction from

ιλ!

(Ωλ(n−X) ICBunB

)' ιλ!

(Ωλ(n−X) !(ICBunB )

)→ !(CBunB ),

identifies

Ωλ(n−X) ICBunB ' H0(ιλ!(!(ICBunB ))

).

3.3.4. Let Eis! denote the functor D(BunT )→ D(BunG) defined as Eis(!(ICBunB ),−), i.e.,

Eis!(F) = p!

(ICBunB

!⊗q!(F)

).

As a corollary of Theorem 3.3.3 we obtain:

Corollary 3.3.5. There exists a natural transformation

Eis!(Ω(n−X) ? F)→ Eis!(F),

compatible with the algebra structure on Ω(n−X).

3.4. Intermediate Eisenstein series.

3.4.1. Let Ω(n−X)-mod(D(BunT )) denote the category of Ω(n−X)-modules in D(BunT ). We havethe natural forgetful functor

Ω(n−X)-mod(D(BunT ))→ D(BunT ),

which admits a left adjoint, denoted indΩ(n−X),

F 7→ Ω(n−X) ? F.

Corollary 3.3.5 implies:

Corollary 3.4.2. The functor Eis! canonically extends to a functor

Eisint! : Ω(n−X)-mod(D(BunT ))→ D(BunG),

so that Eis! ' Eisint! indΩ(n−X).

12 D. GAITSGORY

3.4.3. Let’s give an interpretation of the category Ω(n−X)-mod(D(BunT )) in terms of geometricLanglands.

Recall that we have an equivalence (Fourier-Mukai transform):

ΦT : D(BunT ) ' QCoh(LocSysT ).

The following results from deformation theory:

Proposition 3.4.4. There exists a canonical equivalence

ΦB : Ω(n−X)-mod(D(BunT )) ' QCoh(LocSysB),

making the diagrams

Ω(n−X)-mod(D(BunT ))ΦB−−−−→ QCoh(LocSysB)y qspec∗

yD(BunT )

ΦT−−−−→ QCoh(LocSysT ),

(where the left vertical arrow is the forgetful functor) and

Ω(n−X)-mod(D(BunT ))ΦB−−−−→ QCoh(LocSysB)

indΩ(n−X

)

x q∗spec

xD(BunT )

ΦT−−−−→ QCoh(LocSysT )

commute.

3.4.5. The above achieves the stated goal 2 mentioned in the introduction:

The functor Eisint! ΦB can be interpreted as a functor

QCoh(LocSysB)→ D(BunG),

which is the thought-for ΨG pspec∗.The composition

Eisint! ΦB q∗spec : QCoh(LocSysT )→ D(BunG)

identifies with Eis! ΦT and is supposed to be isomorphic to the composition

QCoh(LocSysT )Eisspec−→ QCohN(LocSysG)

ΦG−→ D(BunG).

3.5. Compactified Eisenstein series.

3.5.1. Let’s recall the second main result of [BG2]. Consider the Bar-construction

(3.1) Bar(Ω(n−X), !(ICBunB )

)∈ D(BunB).

Theorem 3.5.2. There exists a canonical isomorphism in D(BunB):

Bar(Ω(n−X), !(ICBunB )

)' ICBunB

.

3.5.3. LettrivΩ(n−X) : D(BunT )→ Ω(n−X)-mod(D(BunT ))

be the functor that associates to F ∈ D(BunT ) the same object endowed with the trivial actionof Ω(n−X). From Theorem 3.5.2 we obtain:

Corollary 3.5.4. There exists a canonical isomorphism of functors D(BunT )→ D(BunG):

Eis!∗ ' Eisint! trivΩ(n−X).


3.5.5. By Sect. 2.3.6, the object (3.1) acquires a natural co-action of the co-associative co-algebra U∨(n−X) (again, with respect to the above action of D(Ran(X,Λpos)) on D(BunB)). Asa corollary of Theorem 3.5.2, we obtain:

Corollary 3.5.6. The object ICBunB∈ D(BunB) is naturally a U∨(n−X)-comodule, or, equiva-

lently, a co-module for the Lie co-algebra (n−X)∨ ∈ D(Ran(X,Λpos)).

3.5.7. We define the functor of compactified Eisenstein series Eis!∗ : D(BunT ) → D(BunG) asEis(ICBunB

,−), i.e.,

Eis!(F) = p∗

(ICBunB

!⊗q!(F)

).

From Corollary 3.5.6, we obtain:


Eis!∗(F)→ Eis!∗(U∨(n−X) ? F)

compatible with the co-algebra structure on U∨(n−X).

From Lemma 3.1.3, we obtain:


Eis!∗(U(n−X) ? F)→ Eis!(F)

compatible with the algebra structure on U(n−X).

3.5.10. Let(n−X)∨-comod(D(BunT )) = n−X -mod(D(BunT ))

denote the category of (n−X)∨-comodules or (which is equivalent by Lemma 3.1.3) of n−X -modulesin D(BunT ). Let

indn−X : D(BunT )→ n−X -mod(D(BunT ))

denote the left adjoint to the forgetful functor. From Corollary 3.5.9 we obtain:

Corollary 3.5.11. The functor Eis!∗ canonically extends to a functor

Eisint!∗ : n−X -mod(D(BunT ))→ D(BunG),

so that Eis!∗ ' Eisint!∗ indn−X .

4. Relationship between two kinds of Eisenstein series and ε-factors

4.1. The renormalized categories.

4.1.1. Let A be an associative algebra in D(Ran(X,Λpos)), which is locally compact as anobject of this category. Let A∨ be the Verdier dual co-algebra in D(Ran(X,Λneg)).

Note that the forgetful functor

A-mod(D(BunT ))→ D(BunT )

does not send compact objects to compact ones.

We define the renormalized version of category A-mod(D(BunT )), denoted

A-mod(D(BunT ))ren

to be the ind-completion of the full subcategory of A-mod(D(BunT )) consisting of objects whoseimage under the forgetful functor in D(BunT ) is compact.

14 D. GAITSGORY

We have a tautological functor

ΞA : A-mod(D(BunT ))ren → A-mod(D(BunT )).

4.1.2. By construction, the forgetful functor

ΞA : A-mod(D(BunT ))ren → D(BunT )

sends compact objects to compact ones. Hence, it admits a right adjoint that commutes withdirect sums, which we will denote by coindA. Explicitly,

coindA(F) ' A∨ ? F,

where we regard A∨ := D(A) as an A-module.

In addition to the above functor ΞA, we have a functor

ΨA : A-mod(D(BunT ))→ A-mod(D(BunT ))ren

defined so that

ΨA indA ' coindA.

4.1.3. Let trivA denote the functor

D(BunT )→ A-mod(D(BunT ))

that attaches to an object of D(BunT ) the same object endowed with the trivial action of A.

This functor factors naturally as ΞA trivA,ren. The resulting functor

trivA,ren : D(BunT )→ A-mod(D(BunT ))ren

has the property that it sends compact objects to compact ones. Hence, it admits a rightadjoint that commutes with direct sums. We’ll denote this right adjoint by invA.

4.2. Koszul dualities.

4.2.1. For A ∈ D(Ran(X,Λpos)) as above, let B∨ ∈ D(Ran(X,Λpos)) be the Koszul dualco-algebra. By the local compactness assumption, B∨ is also locally compact. Let B ∈D(Ran(X,Λneg)) be its Verdier dual algebra.

We have the following Koszul duality type result:

Proposition 4.2.2.

(1) The functor

invA : A-mod(D(BunT ))ren → D(BunT )

canonically factors through a functor

invAenh : A-mod(D(BunT ))ren → B-mod(D(BunT )),

followed by the forgetful functor.

(2) The functor invAenh is an equivalence. Its inverse is the functor

coinvBenh : B-mod(D(BunT ))→ A-mod(D(BunT ))ren,

whose composition with the forgetful functor is the functor

coinvB : B-mod(D(BunT ))→ D(BunT ),

left adjoint to trivB.


(3) The following diagram of functors is commutative:

(4.1)

A-mod(D(BunT )) −−−−→ B-mod(D(BunT ))ren

ΞA

x ΨB

xA-mod(D(BunT ))ren −−−−→ B-mod(D(BunT )).

4.2.3. Let’s apply the above discussion to A being Ω(n−X). In this case B = U(n−X). We obtain:

Corollary 4.2.4.

(1) The functor

invΩ(n−X) : Ω(n−X)-mod(D(BunT ))ren → D(BunT )


invΩ(n−X)

enh : Ω(n−X)-mod(D(BunT ))ren → n−X -mod(D(BunT )),


(2) The functor invΩ(n−X)

enh is an equivalence. Its inverse is the functor

coinvn−Xenh : n−X -mod(D(BunT ))→ Ω(n−X)-mod(D(BunT ))ren,


coinvn−X : n−X -mod(D(BunT ))→ D(BunT ),

left adjoint to trivn−X .

(3) The functor

invn−X : n−X -mod(D(BunT ))ren → D(BunT )


invn−Xenh : n−X -mod(D(BunT ))ren → Ω(n−X)-mod(D(BunT )),


(4) The functor invn−Xenh is an equivalence. Its inverse is the functor

coinvΩ(n−X)

enh : Ω(n−X)-mod(D(BunT ))→ n−X -mod(D(BunT ))ren,


coinvΩ(n−X) : Ω(n−X)-mod(D(BunT ))→ D(BunT ),

left adjoint to trivΩ(n−X).

(5) The above equivalences make the following diagrams commutative:

(4.2)

Ω(n−X)-mod(D(BunT )) −−−−→ n−X -mod(D(BunT ))ren

ΞΩ(n−X

)

x Ψn−X

xΩ(n−X)-mod(D(BunT ))ren −−−−→ n−X -mod(D(BunT ))

16 D. GAITSGORY

and

(4.3)

Ω(n−X)-mod(D(BunT )) −−−−→ n−X -mod(D(BunT ))ren

ΨΩ(n−X

)

y Ξn−X

yΩ(n−X)-mod(D(BunT ))ren −−−−→ n−X -mod(D(BunT ))

4.3. Implications for Eisenstein series.

4.3.1. Consider the functor

n−X -mod(D(BunT ))→ Ω(n−X)-mod(D(BunT ))

given by (either of the) two circuits of the diagram (4.2).

From Corollary 3.5.4, we obtain:

Corollary 4.3.2. The following diagram of functors commutes:

n−X -mod(D(BunT )) −−−−→ Ω(n−X)-mod(D(BunT ))

Eisint!∗

y Eisint!

yD(BunG)

Id−−−−→ D(BunG)

4.3.3. On the other hand, we should point out that the diagram involving the functors Eisint! andEisint!∗ , and the diagram (4.3) will not commute. However, by Corollary 4.3.2, the calculationof the resulting functor

Ω(n−X)-mod(D(BunT )) ⇒ n−X -mod(D(BunT ))Eisint!∗−→ D(BunG)

boils down to the calculation of the composite functor

(4.4) ΞΩ(n−X) ΨΩ(n−X) : Ω(n−X)-mod(D(BunT ))→ Ω(n−X)-mod(D(BunT )).

4.4. ε-factors.

4.4.1. In this subsection we’ll study the functor (4.4), introduced above, and the correspondingfunctor

(4.5) Ξn−X Ψn−X : n−X -mod(D(BunT ))→ n−X -mod(D(BunT )).

Note that by Corollary 4.2.4, the study of the above functors is equivalent to that of thecomposed functors

ΨΩ(n−X) ΞΩ(n−X) : Ω(n−X)-mod(D(BunT ))ren → Ω(n−X)-mod(D(BunT ))ren

and

Ψn−X Ξn−X : n−X -mod(D(BunT ))ren → n−X -mod(D(BunT ))ren.

We’ll calculate these compositions on certain subcategories in both cases.


4.4.2. For every coroot α consider the functor D(BunT )→ D(BunT ) given by

D(BunT )mult!

−→ D(BunT ×BunT )(id×iα)!

−→

→ D(BunT ×Pic(X))→ D(BunT ×Pic′(X))pr∗−→ D(BunT ),

where

iα : Pic(X)→ BunT

is the map induced by α; Pic′(X)→ Pic(X) is the coarse moduli space (i.e., the Picard scheme).

Let Dreg(BunT ) ⊂ D(BunT ) denote the full subcategory spanned by objects annihilated bythe above functors for all coroots α. I.e., this is the right orthogonal of the category generatedby the images of the pull-push functors corresponding to the diagram

D(Bun′Tα)← D(Bun′T )→ D(BunT ),

where Tα is the quotient torus of T by the corresponding copy of Gm, and for a torus T ′, BunT ′

denotes the corresponding coarse moduli space.

Thus, the embedding Dreg(BunT ) → D(BunT ) admits a right adjoint, so we can think ofDreg(BunT ) as a localization of Dreg(BunT ).

4.4.3. Let

Ω(n−X)-mod(Dreg(BunT )), Ω(n−X)-mod(Dreg(BunT ))ren,

n−X -mod(Dreg(BunT )) and n−X -mod(Dreg(BunT ))ren

be the preimages of in the corresponding categories of Dreg(BunT ) ⊂ D(BunT ) under theforgetful functors.

We’ll prove:

Proposition 4.4.4.

(1) The vertical functors in the diagram

Ω(n−X)-mod(Dreg(BunT )) −−−−→ n−X -mod(Dreg(BunT ))ren

ΨΩ(n−X

)

y Ξn−X

yΩ(n−X)-mod(Dreg(BunT ))ren −−−−→ n−X -mod(Dreg(BunT ))

are localizations, and the vertical functors in the diagram

Ω(n−X)-mod(Dreg(BunT )) −−−−→ n−X -mod(Dreg(BunT ))ren

ΞΩ(n−X

)

x Ψn−X

xΩ(n−X)-mod(Dreg(BunT ))ren −−−−→ n−X -mod(Dreg(BunT ))

are fully faithful.

(2) The composition (4.5) restricted to n−X -mod(Dreg(BunT )), is isomorphic to the shift functor

F 7→ (−2ρ(ωX)) ? F[(2g − 2) dim(n−)],

where 2ρ(ωX) is the point of BunT induced from ωX ∈ Pic(X), using the cocharacter 2ρ.

Te rest of this section is devoted to the proof of this proposition.

18 D. GAITSGORY

4.4.5. Let us apply the Fourier-Mukai equivalence

ΨT : D(BunT ) ' QCoh(LocSysT ).

Under this equivalence, n−X -mod(D(BunT )) corresponds to quasi-coherent sheaves, endowed

with an action of the sheaf of DG Lie-algebras n−univ:

E 7→ Γ(X, n−E).

The category n−X -mod(D(BunT ))ren is the ind-completion of the full subcategory of

n−X -mod(D(BunT )), conisiting of objects that are compact as objects of QCoh(LocSysT ).

Note also that under Fourier-Mukai, Dreg(BunT ) corresponds to localization on the opensubstack

QCohreg(LocSysT ) ⊂ QCoh(LocSysT )

consisting of T -local systems E, such that the 1-dimensional local system α(E) is non-trivialfor every coroot α of G (=root α of G).

Over this substack, the sheaf n−univ is concentrated in cohomological degree 1. This impliesthat

Ξn−X : n−X -mod(Dreg(BunT ))ren ' n−univ-mod(QCohreg(LocSysT ))ren →→ n−univ-mod(QCohreg(LocSysT ))ren ' n−X -mod(Dreg(BunT ))

is a localization.

The composition Ξn−X Ψn−X is given by the n−univ-module equal to U(n−univ)∨, where F 7→ F∨

denotes the natural duality on QCoh(LocSysT ). Since n−univ[1] is a locally free coherent sheaf,we have a canonical isomorphism of n−X -modules:

U(n−univ)∨ ' U(n−univ)⊗ det(n−univ[1]∨)[rk(n−univ)].

By Riemann-Roch, rk(n−univ) = (2g−2) dim(n−). Finally, we need to identify the line bundledet(n−univ[1]) with the line bundle

E 7→ 〈E,−2ρ(ωX)〉 ' Πα〈−α(E), ωX〉,

where 〈−,−〉 is the Weil pairing.

The required identification follows from the next general observation:

Lemma 4.4.6. Let E′ be a non-trivial 1-dimensional local system on X. Then we have acanonical isomorphism

det(H1(X,E′)) ' 〈E′, ωX〉.

5. Verdier dualily and the functional equation

5.1. The dual categories.

5.1.1. Let us return to the general setting of Sect. 4.1. For an associative algebra A+ ∈D(Ran(X,Λpos)), let’s denote by A− ∈ D(Ran(X,Λneg)) the algebra obtained from the tauto-logical map

(5.1) λ 7→ −λ : Λneg → Λpos.


5.1.2. Note that Verdier duality on D(BunT ) gives rise to a canonical identification(A+-mod(D(BunT ))

)∨ ' A−-mod(D(BunT )),

in such a way that the following diagram is commutative

(D(BunT ))c D−−−−→ (D(BunT ))

c

indA+

y indA−y

(A+-mod(D(BunT )))c D−−−−→ (A−-mod(D(BunT )))

c,

where we use the notation D to denote the canonical anti-equivalence

Cc → (C∨)c

for a compactly generated category C and its subcategory Cc of compact objects.

In other words, the dual of the forgetful functor A+-mod(D(BunT )) → D(BunT ) is the

induction functor indA− , and vice versa.

5.1.3. In addition, we have a canonical identification(A+-mod(D(BunT ))ren

)∨ ' A−-mod(D(BunT ))ren,

in such a way that the diagram

(D(BunT ))c D−−−−→ (D(BunT ))

cx x(A+-mod(D(BunT ))ren)

c D−−−−→ (A−-mod(D(BunT ))ren)c,

where the vertical arrows are the forgetful functors. In other words, the functor dual to the

forgetful functor A+-mod(D(BunT ))ren → D(BunT ) is the co-induction functor coindA− , andvice versa.

The next assertion follows from the definitions:

Lemma 5.1.4. The dual of the functor

ΞA+

: A+-mod(D(BunT ))ren → A+-mod(D(BunT ))

is

ΨA− : A−-mod(D(BunT ))→ A−-mod(D(BunT ))ren,

and vice versa.

5.1.5. Let B+ be the Koszul dual algebra to A+. Then B− is the Koszul dual of A−.

Let us observe that the equivalences of Proposition 4.2.2 are compatible with those above,in the sense that the duals of the functors

coinvA+

enh : A+-mod(D(BunT )) B+-mod(D(BunT ))ren : invB+

enh

are

invB−

enh : B−-mod(D(BunT ))ren A−-mod(D(BunT )) : coinvB+

enh.

5.2. Verdier duality on BunB.

20 D. GAITSGORY

5.2.1. Theorems 3.3.3 and 3.5.2 have the following Verdier dual cousins. Consider theΛpos-graded Lie-* algebra n+

X , and the corresponding co-commutative co-algebra Υ(n+X) in

D(Ran(X,Λpos)) with respect to ?.

Theorem 5.2.2. The object ∗(ICBunB ) ∈ D(BunB) is naturally a co-module with respect toΥ(n+

X). Th corresponding map

ιλ∗ ∗(ICBunB )→ Υλ(n+X) ICBunB

identifies the latter with H0 of the former.

5.2.3. Consider now the object

(5.2) coBar(Υ(n+

X), ∗(ICBunB ))∈ D(BunB).

NB: The formation of the co-Bar complex involves a limit, so such is the case in formingthe object (5.2). However, the corresponding inverse system is easily seen to stabilize whenrestricted to every open substack of BunB of finite type.

Theorem 5.2.4. There exists a canonical isomorphism in D(BunB):

coBar(Υ(n+

X), ∗(ICBunB ))' ICBunB

.

Corollary 5.2.5. The object ICBunB∈ D(BunB) is naturally a U(n+

X)-module, or, equivalently,

a module for the Lie algebra n+X ∈ D(Ran(X,Λneg)).

5.2.6. We observe the following phenomenon: the object ICBunBhas a structure of module with

respect to n+X and co-module with respect to (n−X)∨.

It is natural to ask:

Question 5.2.7. Can one formulate in what sense ICBunBcarries an action of the entire g?

This is closely related to our goal 3 stated in the introduction. In fact, we’ll consider onobject (an algebra in D(Ran(X,Λpos))), which is richer than Ω(n−X), which acts on !(ICBunB ),and such as this action encodes the required structure.

5.3. Verdier dual picture for Eisenstein series.

5.3.1. In addition to D(BunG), we can consider its dual category, D(BunG)∨. We will distin-guish them notationally, by denoting the former by D(BunG)! and the latter by D(BunG)∗.

The functor Eis(∗(ICBunB ),−) is naturally a functor

Eis∗ : D(BunT )→ D(BunG)∗.

As in Corollary 3.4.2, from Theorem 5.2.2, we obtain:

Corollary 5.3.2. The functor Eis∗ canonically extends to a functor

Eisint∗ : Ω(n+X)-mod(D(BunT ))→ D(BunG),

so that Eis∗ ' Eisint∗ indΩ(n+X).


5.3.3. The following has been established in [Eis]:

Proposition 5.3.4. The functor Eis! : D(BunT ) → D(BunG)! sends compact objects to com-pacts. For a compact F ∈ D(BunT ) we have:

D Eis!(F) ' Eis∗(D(F)).

Corollary 5.3.5. The functors

Eisint! : Ω(n−X)-mod(D(BunT ))→ D(BunG)! and Eisint∗ : Ω(n+X)-mod(D(BunT ))→ D(BunG)∗

both send compact objects to compacts, and for a compact F ∈ Ω(n−X)-mod(D(BunT )) we have:

D Eisint! (F) ' Eisint∗ (D(F)).

5.3.6. In addition to Eisint∗ , one would wish to use Theorem 5.2.4 to define a functor

Eisint∗! : U(n+X)-mod(D(BunT ))→ D(BunG)∗,

so that

Eisint∗! indn+X ' Eis∗!,

where Eis∗! is a functor

D(BunT )→ D(BunG)∗,

defined using the kernel ICBunB. According to Theorem 5.2.4, this functor should also make

the following diagram commutative:

(5.3)

Ω(n+X)-mod(D(BunT ))

Eisint∗−−−−→ D(BunG)∗

ΨΩ(n+X

)

y Eisint∗!

xΩ(n+

X)-mod(D(BunT ))ren∼−−−−→ U(n+

X)-mod(D(BunT )).

Unfortunately, we do not know how to define Eis∗! (making sure that it commutes with directsums). So, we do not know how to define Eisint∗! either. However, we can define the functorEisint∗! (and, hence, Eis∗!) on the subcategories

U(n+X)-mod(Dreg(BunT )) ⊂ U(n+

X)-mod(D(BunT )) and Dreg(BunT ) ⊂ D(BunT ),

respectively.

Namely, we have the following assertion:

Lemma 5.3.7. The functor Eisint∗ : Ω(n+X)-mod(D(BunT )) → D(BunG)∗, when restricted to

U(n+X)-mod(Dreg(BunT )) factors through the localization

U(n+X)-mod(Dreg(BunT ))→ U(n+

X)-mod(Dreg(BunT ))ren.

5.4. Functional equation.

5.4.1. Recall that we have the functor

F : D(BunG)∗ → D(BunG)!

with the following property:

(5.4) F Eis∗ ' Eis! w0,

where w0 : D(BunT )→ D(BunT ) denotes the functor corresponding to the action of w0 on T .

22 D. GAITSGORY

5.4.2. Note that the action of w0 on D(BunT ) extends to an equivalence

w0 : Ω(n+X)-mod(D(BunT ))→ Ω(n−X)-mod(D(BunT )).

Thus, from (5.4), we obtain the following form of the functional equation:

Corollary 5.4.3. There exists a canonical isomorphism

F Eisint∗ ' Eisint! w0.

5.4.4. We shall now establish another form of the functional equation, this time for the functorEis!∗. Namely, we claim:

Proposition 5.4.5. For F ∈ Dreg(BunT ), we have a canonical isomorphism

F Eis∗!(F′) ' Eis!∗(w0(F)),

where F′ is the shift of F equal to (−2ρ(ωX)) ? F[(2g − 2) dim(n−)].

Proof. By Corollary 5.4.3, the RHS is isomorphic to

F Eisint∗

(trivΩ(n+

X)(F)).

Hence, the assertion follows from Proposition 4.4.4.

6. Eisenstein series, take II

6.1. New algebras.

6.1.1. We’ll change the framework of Sect. 2 slightly. Instead of D(Ran(X,Λpos)), we’ll considerthe category D(Ran(X,Λpos)× BunT ). The only difference is that the operation

Fλ ∈ D(Xλ), Fµ ∈ D(Xµ) 7→ Fλ Fµ ∈ D(Xλ ×Xµ)

gets replaced by a twisted one, namely,

Fλ ∈ D(Xλ × BunT ), Fµ ∈ D(Xµ × BunT ) 7→

FλFµ := (idXλ ×multµ)!(Fλ)!⊗ (πλ × idBunT )!(Fµ) ∈ D(Xλ ×Xµ × BunT ).

This makes D(Ran(X,Λpos)× BunT ) into a monoidal category by means of

Fλ,Fµ 7→ Fλ ? Fµ := (addλ,µ× idBunT )∗(FλFµ).

As such it acts on D(BunT ) via

Fλ, S 7→ Fλ ? S := multλ ∗(FλS),

where

FλS := Fλ!⊗ (πλ × idBunT )!(S),

and similarly for D(BunB).

The discussion of Koszul dualities (in the associative/co-associative setting) goes throughwithout change.

We can also talk about factorization algebras in D(Ran(X,Λpos)×BunT ). By this we meanan object A := Aλ ∈ D(Xλ × BunT ) endowed with an isomorphism

(6.1) AλAµ|(Xλ×Xµ)disj×BunT ' add!λ,µ(Aλ+µ)|(Xλ×Xµ)disj×BunT ,

satisfying a natural associativity condition.


We’ll say that a factorization algebra structure on A is compatible with an associative algebra(resp., co-associative co-algebra) structure if the morphisms

AλAµ → add!λ,µ(Aλ+µ) and add∗λ,µ(Aλ+µ)→ AλAµ

corresponding to the algebra (resp., co-algebra) structure restrict to the map (6.1) on the openpart (Xλ ×Xµ)disj × BunT .

6.1.2. We introduce an associative algebra Ω(n−X) in D(Ran(X,Λpos)×BunT ) with a compatiblefactorization structure as follows:

Proposition-Construction 6.1.3. There exists a canonically defined factorization algebraΩ(n−X) equipped with a structure of associative algebra in D(Ran(X,Λpos)× BunT ) such that

ιλ!(!(ICBunB )) ' Ω(n−X)λ ICBunB := (idXλ ×q)!(Ω(n−X)λ)!⊗ (πλ × idBunB )!(ICBunB ),

and the resulting mapΩ(n−X) ? !(ICBunB )→ !(ICBunB )

is an algebra action.

6.1.4. Let Ω(n−X)-mod(D(BunT )) denote the category of Ω(n−X)-modules in D(BunT ).

We obtain that the functor Eis! can be canonically extended to a functor

Eisult! : Ω(n−X)-mod(D(BunT ))→ D(BunG)!,

such thatEisult! indΩ(n−X) ' Eis! .

6.1.5. Let Υ(n−X) denote the Verdier dual of Ω(n−X); this is a factorization algebra in

D(Ran(X,Λneg)) with a compatible structure of co-associative co-algebra. Let Υ(n+X) (resp.,

Ω(n+X)) be the corresponding objects obtained via (5.1).

Applying Verdier duality to Proposition 6.1.3, we obtain:

Corollary 6.1.6. The object ∗(ICBunB ) ∈ D(BunB) is naturally a Υ(n+X)-comodule.

6.1.7. Consider the corresponding category

Υ(n+X)-comod(D(BunT )) ' Ω(n+

X)-mod(D(BunT )).

As in Sect. 5.1, we have a canonical equivalence(Ω(n−X)-mod(D(BunT ))

)∨' Ω(n+

X)-mod(D(BunT )).

As in Sect. 5.3, we can canonically extend the functor Eis∗ : D(BunT ) → D(BunG)∗ to afunctor

Eisult∗ : Ω(n+X)-mod(D(BunT ))→ D(BunG)∗,

such thatEisult∗ indΩ(n−X) ' Eis∗ .

The functors Eisult! and Eisult∗ are Verdier conjugate of each other in the same sense as inCorollary 5.3.5.

Finally, the action of w0 ∈W defines an equivalence

Ω(n+X)-mod(D(BunT ))→ Ω(n−X)-mod(D(BunT )),

and as Corollary 5.4.3 we have the functional equation:

(6.2) F Eisult∗ ' Eisult! w0.

24 D. GAITSGORY

6.2. Koszul duality for Ω.

6.2.1. A crucial observation concerning the quadruple of (co)-algebras

Ω(n+X), Ω(n−X), Υ(n+

X), Υ(n−X)

is that it possesses an extra symmetry with respect to Koszul duality:

Proposition 6.2.2. We have a canonical isomorphism of co-associative co-algebras:

(6.3) Bar(Ω(n−X)) ' Υ(n+X),

and of resulting co-modules in D(BunB)

(6.4) Bar(

Ω(n−X)), !(ICBunB ))' ∗(ICBunB ).

Proof. Consider the object

Bar(

Ω(n−X)), !(ICBunB ))∈ D(BunB).

We claim that it is isomorphic to ∗(ICBunB ). Indeed, it suffices to show that when we applyιλ! to it for λ 6= 0, we obtain 0. However,

ιλ!(

Bar(

Ω(n−X)), !(ICBunB )))' Bar

(Ω(n−X)), Ω(n−X)λ

) ICBunB = 0.

Now, applying ιλ∗ to both sides of (6.4), we obtain:

Bar(Ω(n−X))λ) ICBunB ' ιλ∗ (∗(ICBunB )) ,

Applying Verdier duality to Proposition 6.1.3, we obtain that

ιλ∗ (∗(ICBunB )) ' Υ(n+X) ICBunB ,

whence the identification (6.3).

6.2.3. Let

ΞΩ(n−X) : Ω(n−X)-mod(D(BunT ))ren Ω(n−X)-mod(D(BunT )) : ΨΩ(n−X)

andΞΩ(n+

X) : Ω(n+X)-mod(D(BunT ))ren Ω(n+

X)-mod(D(BunT )) : ΨΩ(n+X)

be the corresponding categories and functors as in Sect. 4.1.We have the corresponding commutative diagram of functors with the rows being mutually

inverse equivalences:

Ω(n−X)-mod(D(BunT ))coinv

Ω(n−X

)

enh−−−−−−−−→ Ω(n+X)-mod(D(BunT ))ren

ΨΩ(n−X

)

y ΞΩ(n+X

)

yΩ(n−X)-mod(D(BunT ))ren

invΩ(n−X

)

enh−−−−−−→ Ω(n+X)-mod(D(BunT ))

and

Ω(n−X)-mod(D(BunT ))inv

Ω(n+X

)

enh←−−−−−− Ω(n+X)-mod(D(BunT ))ren

ΞΩ(n+X

)

x ΨΩ(n+X

)

xΩ(n−X)-mod(D(BunT ))ren

coinvΩ(n

+X

)

enh←−−−−−−−− Ω(n+X)-mod(D(BunT )).


6.2.4. Recall now that in addition to the functor F , there exists the tautological functor

T : D(BunG)∗ → D(BunG)!.

We have:

Proposition 6.2.5. The following diagram of functors is commutative:

Ω(n+X)-mod(D(BunT ))

ΞΩ(n−X

)coinvΩ(n

+X

)

enh−−−−−−−−−−−−−→ Ω(n−X)-mod(D(BunT ))

Eisult∗

y Eisult!

yD(BunG)∗

T−−−−→ D(BunG)!.

Proof. The statement of the proposition is equivalent to the following one:

∗(ICBunB ) ' Bar(

Ω(n−X), !(ICBunB ))

in a way compatible with the co-action of Υ(n+X). However, this follows from Proposition 6.2.2.

6.3. Summary of the functional equation. We can now summarize what we have obtainedso far regarding the functional equation (our goal 4):

For non-compactified Eisenstein series we have the isomorphisms:

F Eis∗ ' Eis! w0, F Eisint∗ ' Eisint! w0, and Eisult∗ ' Eisuld! w0,

DBunGEis∗ ' Eis! DBunT , DBunGEisint∗ ' Eisint! DBunT , and DBunGEisult∗ ' Eisult! DBunT ,

and

T Eisult∗ ' Eisult! (

ΞΩ(n−X) coinvΩ(n+

X)

enh

).

For compactified Eisenstein series for F ∈ Dreg(BunT ) we have:

F Eis∗!(F) ' Eis!∗ w0 shift(F),

where shift is the shift functor on BunT by −2ρ(ωX).

Combined from the ”real” functional equation of [BG1]:

Eis!∗(F) ' Eis!∗ w0 shift(F)

for F ∈ Dreg(BunT ), we obtain yet one more isomorphism:

F Eis∗!(F) ' Eis!∗(F).

7. The functor of constant term

7.1. Two versions of the constant term functor.

26 D. GAITSGORY

7.1.1. Recall that the functor

Eis! : D(BunT )→ D(BunG),

has a right adjoint, known as the constant term functor CT∗ : D(BunG)→ D(BunT ), given by

F′ ∈ D(BunG) 7→ q∗ p!(F′),

up to a cohomological shift by 2 dim(BunT ) − dim(BunB), the latter being different for each

connected component BunλB .

Therefore, we have a monad Γ := CT∗ Eis! acting on D(BunT ) such that there exists acanonical isomorphism

HomD(BunG)(Eis!(F1),Eis!(F2)) ' HomD(BunT ) (F1,Γ(F2)) .

7.1.2. Recall, however, that according to [Eis], the functor

CT! : D(BunG)! → D(BunT )

given by q! p∗ is also well-defined, and there is a canonical isomorphism:

(7.1) w0 CT! ' CT∗ .

The latter observation allows to calculate the composition Γ explicitly.

7.1.3. Consider the Cartesian product

BunB ×BunG

BunB− .

Let ′p− and ′p denote its projections to BunB and BunB− , respectively:

BunT BunG BunT

BunB BunB−

BunB ×BunG

BunB−

q

p

p−

q−

′p−

p−

Let’s denote the composed arrows

r := q ′p− and r− := q− ′p.

By base change and taking into account the fact that q is smooth, we obtain:

Γ ' w0 CT! Eis! ' (q− ′p)! (′p− q)∗,

up to the cohomological shift by dim(BunB)− dim(BunB−), which again depends on the con-nected component of BunB and BunB− .


7.1.4. Bruhat decomposition defines a decomposition of this stack into locally closed substacksaccording to the relative position of the two reductions at the generic point of the curve. Forw ∈W let (

BunB ×BunG

BunB−

)w

denote the corresponding locally closed substack; our conventions are such that w = 1 is theopen stratum, which we also denote by Z, the Zastava space. Note also that for w = w0, wehave (

BunB ×BunG

BunB−

)w0

' BunB .

Let ′p−w and ′pw denote the projections from

(BunB ×

BunGBunB−

)w

to BunB and BunB− ,

respectively. Letrw := q ′p−w and r−w := q− ′pw.

We obtain that the functor Γ admits a canonical filtration indexed by w, with the w-th sub-quotient denoted Γw given by

(7.2) Γw := r−w ! r∗w,again up to the cohomological shift by dim(BunB)− dim(BunB−).

7.2. Relation to Ω. Our current goal is to establish the following:

Proposition 7.2.1. The functor Γ1 is canonically isomorphic to

F 7→ Ω(n−X) ? F.

Remark. With a little extra work one can show that the resulting map

Ω(n−X) ? F → Γ(F)

is a map of monads.

The rest of this section is devoted to the proof of this proposition.

7.2.2. For λ ∈ Λpos let Zλ be the union of connected components of Z equal to⊔λ1−λ2=λ

(Bunλ1

B ×BunG

Bunλ2

B−

)1

.

There exists a natural projection (defect of transversality):

pλ : Zλ → Xλ,

such that the composition

Zλpλ×r1−→ Xλ × BunT

multλ−→ BunT

identifies with r−1 .

Hence, by the projection formula, Γ1 is given by

F 7→ multλ !

((pλ × r1)!(CZλ)⊗ (πλ × idBunT )∗(F)

)[2|λ|],

where CZλ denotes the D-module corresponding to the constant sheaf on Zλ, and 2|λ| appears

as the difference dim(Bunλ2

B−)− dim(Bunλ1

B ).

On the other hand, the functor F 7→ Ω(n−X)λ ? F can be written as

F 7→ multλ !

(Ω(n−X)λ ⊗ (πλ × idXλ)∗(F)

)[−2 dim(BunT )].

28 D. GAITSGORY

This is due to the fact that Ω(n−X) is ULA with respect to the projection

πλ × idBunT : Xλ × BunT → BunT .

Thus, the required isomorphism would follow from the following one:

(7.3) (pλ × r1)!(CZλ)[2|λ|] ' Ω(n−X)λ[−2 dim(BunT )].

7.2.3. The proof of (7.3) follows from the usual contraction picture for Zastava spaces:

Let

(7.4) ZλZ← Zλ

be the partial compactification of Zλ, i.e., the open substack(BunB ×

BunGBunB−

)1

⊂ BunB ×BunG

BunB− .

The corresponding morphism

pλ : Zλ → Xλ × BunT

admits a section, denoted sλ. Moreover, there is a Gm-action on Zλ that contracts it on theimage of sλ. Hence,

(pλ × r1)!(CZλ) ' (pλ × r1)! Z!(CZλ) ' sλ! Z!(CZλ).

The assertion follows now from the fact that the pair in (7.4) is smoothly equivalent to thepair

BunB← BunB ,

and dim(Zλ) = 2|λ|.

8. The space of rational reductions to B

The rest of the paper is devoted to addressing our goal 5.

8.1. The category.

8.1.1. What follows is an attempt to realize Drinfeld’s idea of the space of G-bundles with arational reduction to B. Unfortunately, we won’t be able to construct the space itself, butrather the category of D-modules on it. Our approach will be very naive: we’ll start with BunBand we’ll ”contract” the strata

ιλ : Xλ × BunB → BunB

under πλ : Xλ × BunB → BunB .


8.1.2. Let H be the closed substack of BunB ×BunG

BunB corresponding to G-bundles with a

pair of generalized B-reductions which agree at the generic point of X. This is a groupoid overBunB , and let

p1, p2 : H ⇒ BunB

denote its two projections to BunB . The maps pi are proper, since BunB is proper over BunG.

Let D(BunratB ) denote the category of H-equivariant objects on D(BunB). I.e., it consists ofF ∈ D(BunB), endowed with an isomorphism:

p!1(F) ' p!

2(F),

which satisfies a natural associativity condition.

We have a natural forgetful functor D(BunratB ) → D(BunB), which admits a left adjoint

denoted indBunratB

BunBand given by

F 7→ p1∗ p!2(F).

Our current goal is to describe the category D(BunratB ) more explicitly. Note that from thedefinitions we obtain:

Lemma 8.1.3. The functor p! factors as

D(BunG)prat!−→ D(BunratB )→ D(BunB).

By adjunction, the functor p∗ : D(BunB)→ D(BunG) naturally actors as

D(BunB)ind

BunratBBunB−→ D(BunratB )

prat∗−→ D(BunG).

8.2. A monoid(al) approach.

8.2.1. We define the category ′D(BunratB ) to consist of the data of F ∈ D(BunB), endowed withisomorphisms

αλ : πλ!(F)→ ιλ!(F),

for λ ∈ Λpos, that are associative in the sense that for λ = λ1 + λ2 the two isomorpisms takingplace on Xλ1 ×Xλ2 × BunB :

πλ1,λ2 !(F) ' (idXλ1 ×πλ2)! πλ1 !(F)αλ1

' (idXλ1 ×πλ2)! ιλ1 !(F) '

' (idXλ2 ×ιλ1)! πλ2 !(F)αλ2

' (idXλ2 ×ιλ1)! ιλ2 !(F) ' ιλ1,λ2 !(F),

and

πλ1,λ2 !(F) ' (addλ1,λ2 × idBunB)! πλ1+λ2 !(F)

αα1+α2

'

' (addλ1,λ2 × idBunB)! ιλ1+λ2 !(F) ' ιλ1,λ2 !(F)

coincide, where

πλ1,λ2 : Xλ1 ×Xλ2 × BunB → BunB and ιλ1,λ2 : Xλ1 ×Xλ2 × BunB → BunB

are the natural maps.

30 D. GAITSGORY

8.2.2. One can view the definition of ′D(BunratB ) in the following framework.

Consider D(Ran(X,Λpos)) as a monoidal category acting on D(BunB) as in Sect. 3.2, andalso as acting on Vect by

Fλ 7→ ⊕λH(Xλ,Fλ).

Then

′D(BunratB ) ' Vect ⊗D(Ran(X,Λpos))

D(BunB),

where the forgetful functor ′D(BunratB ) → D(BunB) is the right adjoint to the tautologicalfunctor

D(BunB) ' Vect⊗D(BunB)→ Vect ⊗D(Ran(X,Λpos))

D(BunB).

We’ll denote the above left adjoint D(BunB)→ ′D(BunratB ) by ′indBunratB

BunB.

8.2.3. We shall now discuss two ways how to compute Hom’s in the category ′D(BunratB ).

Let ′D(BunratB )lax,1 denote the category consisting of pairs (F, αλ) as in the definition of′D(BunratB ), but without the requirement that αλ’s be isomorphisms. Let ′D(BunratB )lax,2 be asimilarly defined category of pairs (F, βλ), where

βλ : ιλ!(F)→ πλ!(F).

By definition, we have:

Lemma 8.2.4. The functors

′D(BunratB )→′ D(BunratB )lax,1 and ′D(BunratB )→′ D(BunratB )lax,2

are fully faithful.

Next, we claim that Hom’s in the categories ′D(BunratB )lax,1 and ′D(BunratB )lax,2 can be

computed ”algorithmically” in terms of Hom’s in D(BunB).

In fact, we claim that ′D(BunratB )lax,1 (resp., ′D(BunratB )lax,2) is the category of modules for

a certain monad M1 (resp., M2) acting on D(BunB). Indeed, the monads in question are

M1(F) = ⊕λ∈Λpos

ιλ! πλ!(F) and M2(F) = ⊕λ∈Λpos

πλ∗ ιλ!(F),

respectively.

So, Hom’s in ′D(BunratB )lax,i ' Mi-mod(D(BunB)) can be expressed in terms of Hom’s in

D(BunB) via the corresponding bar-complex.

8.3. A comparison.


8.3.1. Note that the maps

πλ : Xλ × BunB → BunB and ιλ : Xλ × BunB → BunB

combine to a map

Xλ × BunB → H.

Hence, we obtain that the forgetful functor D(BunratB )→ D(BunB) factors through a functor

Φ1 : D(BunratB )→ ′D(BunratB ).

From the same geometric picture we obtain that the functor

indBunratB

BunB: D(BunB)→ D(BunratB )

canonically factors as

D(BunB)

′indBunratBBunB−→ ′D(BunratB )→ D(BunratB ),

by the definition of tensor product. We’ll denote the resulting functor ′D(BunratB )→ D(BunratB )by Φ2. By construction, Φ2 is the left adjoint to Φ1.

Proposition 8.3.2. The functors

Φ2 : ′D(BunratB ) D(BunratB ) : Φ1

are mutually inverse equivalences of categories.

The rest of this subsection is devoted to the proof of Proposition 8.3.2.

8.3.3. Let us denote the monads

D(BunB)ind

BunratBBunB−→ D(BunratB )→ D(BunB)

and

D(BunB)

′indBunratBBunB−→ ′D(BunratB )→ D(BunB)

by Av and ′Av, respectively.

By construction, we have a natural transformation

(8.1) ′Av(F)→ Av(F).

It is easy to see that the statement of Proposition 8.3.2 is equivalent to the assertion thatthe map (8.1) is an isomorphism. To prove the latter, it suffices to show that the map

ιλ! ′Av(F)→ ιλ! Av(F)

is an isomorphism for any λ ∈ Λpos. However, since both functors appearing in (8.1) factorthrough ′D(BunratB ), it suffices to show that the map

(8.2) ! ′Av(F)→ ! Av(F)

is an isomorphism.

32 D. GAITSGORY

8.3.4. Let’s first calculate the RHS of (8.2). We claim that

(8.3) ! Av(F) ' ⊕λπλ∗ ιλ!(F).

The above claim follows from the next geometric assertion: there exists a natural isomor-phism ⋃

λ∈Λpos

Xλ × BunB → BunB ×BunB

H.

Note that (8.3) allows to give the following fairly explicit description of the categoryD(BunratB ):

Let Fi, i = 1, 2, 3 be compact objects of D(BunB), such that the objects !(Fi) ∈ D(BunB)are well-defined. We obtain:

HomD(BunratB )

(ind

BunratB

BunB(!(F1)), ind

BunratB

BunB(!(F2))

)'

' ⊕λ

HomD(BunB)

(F1, π

λ∗ ιλ! !(F2)

)'

' ⊕λ

HomD(BunB)

(ιλ∗(CXλ !(F1)), !(F2)

).

The composition

HomD(BunratB )

(ind

BunratB

BunB(!(F1)), ind

BunratB

BunB(!(F2))

)×

×HomD(BunratB )

(ind

BunratB

BunB(!(F2)), ind

BunratB

BunB(!(F3))

)→ HomD(BunratB )

(ind

BunratB

BunB(!(F1)), ind

BunratB

BunB(!(F3))

)can be described as follows. For

ιλ∗(CXλ !(F1))→ !(F2) and ιµ∗ (CXµ !(F2))→ !(F3),

the resulting morphism

ιλ+µ∗ (CXλ+µ !(F1))→ !(F3)

equals the composition

ιλ+µ∗ (CXλ+µ !(F1))→ ιλ+µ

∗ (addλ,µ ∗(CXλ CXµ) !(F1)) '

' ιλ,µ∗ (CXλ CXµ !(F1)) ' ιµ∗ (idXµ ×ιλ)∗ (CXµ (CXλ !(F1)))→→ ιµ∗ (CXµ !(F2))→ !(F3).

8.3.5. Next, we’ll show that the functor F 7→ ! ′Av(F) is also isomorphic to

(8.4) F 7→ ⊕λπλ∗ ιλ!(F).

(It will be clear from the construction that the map in (8.2) corresponds to the identity mapon (8.4)).

For a coweight µ, consider the following category equipped with a natural action ofD(Ran(X,Λpos)):

indRan(D(BunµB)) := D

( ⋃λ∈Λpos

Xλ × BunµB

)' D(Ran(X,Λpos))⊗D(BunB).


Note that

Vect ⊗D(Ran(X,Λpos))

indRan(D(BunµB)) ' D(BunµB),

such that the functors

indRan(D(BunµB)) D(BunµB)

identify with

Fλ 7→ ⊕λ∈Λpos

πλ∗ (Fλ)

and

F 7→ πλ!(F),respectively. Let Avind denote the composition

indRan(D(BunµB))→ D(BunµB)→ indRan(D(BunµB)).

Consider the functor

φ : D(BunB)→ indRan(D(BunµB))

given by

F 7→ ιλ!(F).We claim that φ commutes with the action of D(Ran(X,Λpos)). Indeed, this follows from

the fact that for λ, µ ∈ Λpos, the diagram

Xλ+µ × BunBιλ+µ

−−−−→ BunB

addλ+µ

x ιλ

xXλ ×Xµ × BunB

idXλ×ιµ

−−−−−−→ Xλ × BunB

is Cartesian.

The required result regarding the functor ! ′Av follows from the next assertion:

Proposition 8.3.6. There exists a natural isomorphism of functors

Avind φ ' φ ′Av .

8.3.7. Recall the following general pattern. Let A be a monoidal category acting on the left onC2 and on the right on C1. We assume that the multiplication and action functors

A⊗A→ A and A⊗C2 → C2, C1 ⊗A→ C1

have right adjoints that commute with direct sums. In this case the natural functor

C1 ⊗C2 → C1 ⊗A

C2

admits a right adjoint that commutes with direct sums. Let now C′1 and C′2 be another suchpair, and φ1 : C1 → C′1 and φ2 : C2 → C′2 be functors that commute with the A-actions. Inthis case, we have a commutative diagram of functors:

(8.5)

C1 ⊗C2 −−−−→ C′1 ⊗C′2y yC1 ⊗

AC2 −−−−→ C′1 ⊗

AC′2.

Consider the right adjoint functors

C1 ⊗A

C2 → C1 ⊗C2 and C′1 ⊗A

C′2 → C′1 ⊗C′2.

34 D. GAITSGORY

Assume now that the functors φ1 and φ2 have the property that they admit left adjoints thatalso commute with the actions of A. In this case the following diagram

(8.6)

C1 ⊗C2 −−−−→ C′1 ⊗C′2x xC1 ⊗

AC2 −−−−→ C′1 ⊗

AC′2.

commutes as well. Hence, if we denote by N and N ′ the monads

C1 ⊗C2 → C1 ⊗A

C2 → C1 ⊗C2 and C′1 ⊗C′2 → C′1 ⊗A

C′2 → C′1 ⊗C′2,

respectively, we obtain a natural isomorphism

(8.7) (φ1 ⊗ φ2) N ' N ′ (φ1 ⊗ φ2),

as functors

C1 ⊗C2 → C′1 ⊗C′2.

In addition, we note the diagram (8.6) commutes and isomorphism (8.3.6) holds also if φ1

and φ2 admit fully faithful right adjoints (i.e., C′i is a localization of Ci for i = 1, 2).

8.3.8. We apply the above general discussion in the following circumstances. Throughout, we’lltake A = D(Ran(X,Λpos)) and C1 = C′1 = Vect, and φ1 to be the identity map.

Consider now the following category:

D(Bunµ,≤B ) := D

( ⋃λ∈Λpos

Bunµ−λ,≤λB

),

where

Bunµ−λ,≤λB

λ

→ Bunµ−λB

is the open substack corresponding to the condition that the total degeneration is ≤ λ. Forevery λ, the map ιλ defines a closed embedding:

Xλ × BunµB → Bunµ−λ,≤λB .

We have a natural action of D(Ran(X,Λpos)) on D(Bunµ,≤B ). Let Av≤ denote the corre-

sponding functor

D(Bunµ,≤B )→ Vect ⊗

D(Ran(X,Λpos))D(Bun

µ,≤B )→ D(Bun

µ,≤B ).

In the setting of Sect. 8.3.7, let’s first take C2 := D(Bunµ,≤B ), C′2 := indRan(D(BunµB)) and

φ2 to be the functor φ≤ given by

Fµ−λ 7→ ιλ!(F).As in the case of the functor φ, it is easy to see that the functor φ≤ commutes with the

action of D(Ran(X,Λpos)). However, unlike φ, the functor φ≤ admits a left adjoint, given by

Fλ 7→ ιλ! (Fλ),which is easily seen to commute with the action of D(Ran(X,Λpos)). Hence, we obtain thatthe corresponding isomorphism (8.7) holds, i.e.,

φ≤ Av≤ ' Avind φ≤.


We shall now apply the formalism of Sect. 8.3.7 again, this time to

C2 := D(BunB) and C′2 := D(Bunµ,≤B ),

with the functor φ2 being ≤! given by definition by

F 7→ λ!(F).

In this case, the isomorphism (8.7) holds again, sinceD(Bunµ,≤B ) is a localization ofD(BunB).

So,! ′Av ' Av≤ ≤!.

Together, we obtain thatφ≤ j≤! ′Av ' Avind φ≤ ≤!.

Finally, it remains to observe that φ ' φ≤ !.(Proposition 8.3.6 and Theorem 8.3.2)

8.4. Verdier duality on BunratB .

8.4.1. Let’s fix an element λ ∈ Λ, and consider a closed substack of BunB , denoted by ≥λBunB ,whose intersection with every Bun

µ

B equals⋃λ′≥λ,µ

Im(ιλ′−µ(Bun

λ′

B )).

In other words, the intersection≥λBunB ∩ Bun

µ

B

is empty unless µ − λ belongs to the coroot lattice, and in the latter case is the union of thestrata

Xν × Bunλ′

B , λ′ ≥ λ, ν ∈ Λpos.

It is clear that the substack ≥λBunB is stable with respect to the correspondence H. There-fore, the category D(≥λBunratB ) makes sense. This is a full subcategory of D(BunratB ), and itsembedding admits a right adjoint. We have:

D(≥λ′BunratB ) ⊂ D(≥λBunratB )

whenever λ′ ≥ λ. It is clear that the entire D(BunratB ) is the direct limit of D(≥λBunratB )’s asλ ranges over Λ.

8.4.2. For two elements λ1 ≤ λ2, consider the open substack≤λ2,≥λ1BunB ⊂ ≥λ1BunB

equal to≥λ1BunB −

⋃λ/∈λ2−Λpos

≥λBunB .

In other words, for a coweight µ, the intersection≤λ2,≥λ1BunB ∩ Bun

µ

B

is the union of the strata

Xν × Bunλ′

B , λ2 ≥ λ′ ≥ λ1, ν ∈ Λpos.

The stack ≤λ2,≥λ1BunB is stable with respect to H, and we can consider the correspond-ing category D(≤λ2,≥λ1 BunratB ). In fact, D(≤λ2,≥λ1 BunratB ) identifies with a localization ofD(≥λ1BunratB ) with respect to the subcategory generated by

D(≥λBunratB ) with λ ≥ λ2, but λ /∈ λ2 − Λpos.

36 D. GAITSGORY

8.4.3. The stack ≤λ2,≥λ1BunB is a disjoint union of stacks of finite type. Therefore, Verdierduality identifies D(≤λ2,≥λ1BunB) with its own dual. Let

evBunB: D(≤λ2,≥λ1BunB)⊗D(≤λ2,≥λ1BunB)→ Vect

denote the corresponding pairing.

Proposition-Construction 8.4.4.

(1) The functor

D(≤λ2,≥λ1BunB)⊗D(≤λ2,≥λ1 BunratB )→ D(≤λ2,≥λ1BunB)⊗D(≤λ2,≥λ1BunB)evBunB−→ Vect


evBunratB: D(≤λ2,≥λ1 BunratB )⊗D(≤λ2,≥λ1 BunratB )→ Vect

via

indBunratB

BunB: D(≤λ2,≥λ1BunB)→ D(≤λ2,≥λ1 BunratB ).

(2) The resulting pairing evBunratBidentifies D(≤λ2,≥λ1 BunratB ) with its own dual in such a way

that the diagram

D(≤λ2,≥λ1BunB) D(≤λ2,≥λ1 BunratB )

is self-dual.

Proof. Consider the following framework. Let A be an augmented monoidal category actingon C. We assume that the functors

A⊗A→ A and A⊗C→ C

admit right adjoints. Assume now have an identification A∨ ' A, such that the dual of themap A⊗A→ A identifies with its right adjoint. Let C∨ be the dual category of C. We definethe action A ⊗ C∨ → C∨ to be dual map to the right adjoint of A ⊗ C → C. Under suchcircumstances, it is easy to see that the dual of CA identifies with C∨A.

We apply the above discussion to A := D(Ran(X,Λpos)) and C := D(≤λ2,≥λ1 BunratB ).Verdier duality defines identifications A ' A∨ and C ' C∨, such that the resulting A-actionon C∨ coincides with the original action on C.

Hence, the assertion of the proposition follows from the above general principle.

9. Eisenstein series, take III

9.1. The ”stack” BunT . The projection q : BunB → BunT has contractible fibers. Hence thefunctor q∗ is a fully faithful embedding. In this subsection we’ll define the category D(BunT )which will play a similar role vis-a-vis D(BunB).

9.1.1. We let D(BunT ) be the full subcategory of D(BunB) consisting of objects F satisfyingthe following: for each λ ∈ Λpos,

(ιλ)!(F) ∈ D(Xλ × BunB)

belongs to the essential image of

(id×p)! : D(Xλ × BunT )→ D(Xλ × BunB).

We claim:


Proposition 9.1.2.

(1) For F ∈ D(Xλ × BunT ) the object

(ιλ)!

((id×q)!(F)

)∈ D(BunB)

is defined, and belongs to D(BunT ).

(2) Objects as above generate D(BunT ).

Proof. First, we need to show that the object as in (1) is indeed defined. Since ιλ = ιλ (id×),it suffices to show that

(id×)! (id×q)! : D(Xλ × BunT )→ D(Xλ × BunB)

is well-defined. However, since the latter categories are obtained by tensoring up with D(Xλ),it is sufficient to show that

! q! : D(BunT )→ D(BunB)

is well-defined. However, this follows from the ULA property of !(KBunB ) with respect to q.Indeed, we have:

! q!(F) ' !(KBunB )!⊗ q!(F).

To prove that (ιλ)!

((id×q)!(F)

)belongs to D(BunT ) we need to calculate the composition

(9.1) (ιµ)! (ιλ)!

((id×q)!(F)

)' (ιµ)! (ιλ)∗

((id×q)!(F)

!⊗ (id×)!(KXλ×BunB )

).

Note that we have a Cartesian square:

Xµ × BunBιµ−−−−→ BunB

addλ,λ−µ× id

x ιλ

xXλ ×Xµ−λ × BunB

id×ιµ−λ−−−−−−→ Xλ × BunB

The expression in (9.1) can thus be rewritten as:

(addλ,λ−µ× id)∗ (id×ιµ−λ)!

((id×q)!(F)


).

It would be sufficient to show that

(id×ιµ−λ)!

((id×q)!(F)


)belongs to the essential image of D(Xλ ×Xµ−λ × BunT ) under. (idXλ×Xµ−λ ×q)!. Again, thefactor Xλ comes in along for the ride, so we have to show that for ν ∈ Λpos,

(ιν)!

(q!(F)

!⊗ !(KBunB )

)belongs to the essential image of (idXν ×q)!. However, the latter expression is isomorphic to

(idXν ×q)!

(Ω(n−X)ν

!⊗mult!

ν(F)

).

The second point of the proposition is tautological from the first one.

9.2. The ”stack” BunratT .

38 D. GAITSGORY

9.2.1. Let D(BunratT ) denote the full subcategory of D(BunratB ) equal to the preimage ofD(Bun) ⊂ D(BunB) under the forgetful functor.

Lemma 9.2.2. The functor

indBunratB

BunB: D(BunB)→ D(BunratB )

sends D(BunT ) to D(BunratT ).

Proof. By definition, we need to show that the monad Av sends D(BunT ) to itself, i.e., thatfor an object F ∈ D(BunT ), the object (ιλ)! Av(F) belongs to the essential image of (id×q)!.However,

(ιλ)! Av(F) ' (πλ × id)! ! Av(F),

so it is enough to show that ! Av(F) belongs to the essential image of q!. Now, the resultfollows from the explicit description of ! Av given in Sect. 8.3.4.

We’ll denote by indBunratT

BunTthe resulting functor

D(BunT )→ D(BunratT ).

9.2.3. One can view the category D(BunratT ) as the universal source of the Eisenstein seriesfunctor by means of

D(BunratT ) → D(BunratB )prat∗−→ D(BunG).

We shall now investigate its relationship with the functor Eisult! . Namely, we’ll establish:

Proposition 9.2.4. The functor

F 7→ indBunratT

BunT(! q!(F)) : D(BunT )→ D(BunratT )

canonically extends to a functor

(qrat)! : Ω(n−X)-mod(D(BunT ))→ D(BunratT ),

and the latter is an equivalence of categories. Moreover:

(1) The forgetful functor

Ω(n−X)-mod(D(BunT ))→ D(BunT )

identifies with

D(BunratT )→ D(BunT )!−→ D(BunT ).

(2) The functor prat∗ identifies with Eisult! .

9.2.5. Proof of Proposition 9.2.4. All we need to show is that the monad on D(BunT ) given by

F 7→ ! Av ! q!(F)

identifies with

F 7→ q! (Ω(n−X) F).

However, this again follows immediately from the description of ! Av in Sect. 8.3.4.

9.3. An equivalence with BunG.


9.3.1. Let λ1 ≤ λ2 be a pair of weights. Consider the corresponding categories

D(≤λ2,≥λ1BunB) and D(≤λ2,≥λ1 BunratB ).

By the same taken as in Sect. 9.1 and Sect. 9.2, we define the corresponding full subcategories

D(≤λ2,≥λ1BunT ) and D(≤λ2,≥λ1 BunratT ).

It is clear that the Verdier duality functor (defined on the subcategory of compact objectsof D(≤λ2,≥λ1BunB)) sends D(≤λ2,≥λ1BunT ) to itself. Indeed, it’s enough to check this fact oncompact generators provided by Proposition 9.1.2. Hence, the same is true for the subcategory(

D(≤λ2,≥λ1 BunratT ))c ⊂ (D(≤λ2,≥λ1 BunratB )

)c.

9.3.2. Recall that for every regular dominant weight λ, there exists a canonically defined locallyclosed substack

BunλG ⊂ BunG .

In fact, the projection pλ maps BunλB isomorphically onto BunλG.

Let now λ1 ≤ λ2 be as above, and assume that they are deep enough in the dominantchamber so that every weight λ satisfying

λ1 ≤ λ ≤ λ2

is dominant and regular. In this case, the union of strata

≤λ2,≥λ1 BunG := ∪λ1≤λ≤λ2

BunλG

is also a locally closed substack of BunG.

The preimage of ≤λ2,≥λ1 BunG under p equals the substack

≤λ2,≥λ1BunB ⊂ BunB .

Hence, the functor p! gives rise to a well-defined functor

D(≤λ2,≥λ1BunB)→ D(≤λ2,≥λ1 BunG),

which canonically factors through a functor

prat∗ : D(≤λ2,≥λ1 BunratB )→ D(≤λ2,≥λ1 BunG).

9.3.3. We are going to prove:

Theorem 9.3.4. Assume that λ1, λ2 are deep enough the dominant chamber so that everyweight λ satisfying

λ1 ≤ λ ≤ λ2

satisfies also

〈λ, αi〉 > 2g − 2,

for every simple root αi. In this case the composed functor

D(≤λ2,≥λ1 BunratT ) → D(≤λ2,≥λ1 BunratB )prat∗→ D(≤λ2,≥λ1 BunG)

is an equivalence.

The rest of this section is devoted to the proof of this theorem.

40 D. GAITSGORY

9.3.5. First, we note that under the assumptions on λ1, λ2 for every λ between them the map

q! : D(BunλT )→ D(BunλB)

is an equivalence. Hence, the functors

D(≤λ2,≥λ1BunT ) → D(≤λ2,≥λ1BunB)

and

D(≤λ2,≥λ1 BunratT ) → D(≤λ2,≥λ1 BunratB )

are equivalences as well.

This implies that the essential image of the functor in the theorem generates the targetcategory. Therefore, it remains to establish fully faithfulness. The latter is, in turn, equivalentto the fact that the monad

(9.2) p! p! : D(≤λ2,≥λ1BunB)→ D(≤λ2,≥λ1BunB)

is canonically isomorphic to the monad Av.

The monad in the LHS of (9.2) is given by pull-push along the diagram

≤λ2,≥λ1BunB ←≤λ2,≥λ1 BunB ×≤λ2,≥λ1 BunG

≤λ2,≥λ1BunB → ≤λ2,≥λ1BunB .

The monad in the RHS of (9.2) is given by pull-push along the diagram

≤λ2,≥λ1BunB ← ≤λ2,≥λ1BunB ×BunB

H ×BunB

≤λ2,≥λ1BunB → ≤λ2,≥λ1BunB .

We have a canonical closed embedding from the latter diagram to the former, and it remainsto show that this embedding is in fact an isomorphism. This is equivalent to the followingwell-known assertion:

Lemma 9.3.6. Let PG be an unstable G-bundle, equipped with two B-reductions both of whichhave dominant Chern classes. Then these two reductions coincide.

10. The local nature of Ω(n−X)

10.1. The Hecke stack.

10.1.1. For a reductive group G let SatG be the Satake category, i.e., the (slightly renormalizedversion of) category D(HeckeG,x,loc),

3 where HeckeG,x,loc is the local Hecke stack at x, i.e.,

HeckeG,x,loc := BunG(Dx) ×BunG(

Dx)

BunG(Dx),

where Dx andDx are the formal and formal punctured discs around x, respectively.

3It’s crucial for what follows that we consider the ”full” i.e., derived category and not the heart of itst-structure.


We can also consider the global Hecke stack

BunG(Dx) BunG(Dx)

HeckeG,x,loc

BunG BunG

HeckeG,x,glob

←h loc

→h loc

←h glob

→h glob

There is a natural functor

SatG,x → D(HeckeG,x,glob).

10.1.2. We regard the assignment x 7→ SatG,x as a chiral monoidal category, which we denoteby SatG. In particular, the category SatG(Ran(X)) makes sense, and there is a functor

(10.1) SatG(Ran(X))→ D(HeckeG,glob,Ran(X)),

where HeckeG,glob,Ran(X) is the Ran version of the global Hecke stack.

10.1.3. We apply this to the group being T . For λ ∈ Λpos we can regard the product Xλ×BunTwith its two projections

BunTmultλ←− Xλ × BunT

πλ×idBunT−→ BunT

as a closed substack of the λ-graded component of HeckeT,glob,Ran(X).

In what follows, we’ll describe a chiral algebra in SatT , such that the corresponding objectof SatG(Ran(X)) conjecturally gives rise to Ω(n−X) via (10.1).

10.2. Geometric Satake.

10.2.1. For a group G1, let us denote by HeckeG1,x,loc,spec the local spectral Hecke (DG)-stack,i.e.,

HeckeG1,x,loc,spec := LocSysG1(Dx) ×

LocSysG1(Dx)

LocSysG1(Dx),

and let

SatG1,x,spec := IndCoh(HeckeG1,x,loc,spec).

The assignment x 7→ SatG1,x,spec forms a monoidal chiral category that we denote SatG1,spec.

Recall that for a reductive group G we have a canonical equivalence of monoidal chiralcategories.

(10.2) SatG ' SatG,spec .

42 D. GAITSGORY

10.2.2. The following general construction will be used in the sequel. Let G1 → G2 be a grouphomomorphism. Consider the following relative version of the local spectral Hecke stack:

HeckeG1,G2,x,loc,spec := LocSysG1(Dx) ×LocSysG2

(Dx) ×LocSysG2

(Dx)

LocSysG1(Dx)

LocSysG1

(Dx).

endowed with the corresponding maps

LocSysG1(Dx) LocSysG1

(Dx).

HeckeG1,G2,x,loc,spec

←h rel,loc

zz

→h rel,loc

$$

The assignment

x 7→ IndCoh(HeckeG1,G2,x,loc,spec)

also forms a monoidal chiral category, which we denote SatG1,G2,spec.

We have a natural forgetful morphism

HeckeG1,G2,x,loc,spec → HeckeG1,x,loc,spec,

which defines a monoidal chiral functor

SatG1,G2,spec → SatG1,spec .

We let

AG1,G2∈ SatG1,spec

be the chiral algebra equal to image under the above functor of the chiral algebra in

SatG1,G2,spec, whose value on each HeckeG1,G2,x,loc,spec is the (relative with respect to←hrel,loc)

of the dualizing sheaf on this (DG)-stack. By construction, this chiral algebra carries anassociative algebra structure with respect to the monoidal structure on SatG1,spec. Todisambiguate the notation, we’ll denote by AG1,G2

(Ran(X)) the corresponding factorizableobject of SatG1,spec(Ran(X)).

10.3. The conjecture.

10.3.1. We apply the discussion of Sect. 10.2.2 to G1 = B and G2 = G. We obtain the chiralalgebra

AB,G ∈ SatB,spec .

Note that the (DG)-stack

LocSysG(Dx) ×LocSysG(

Dx)

LocSysB(Dx)

identifies with the derived Springer fiber over 0 ∈ g, denoted Spr0.


10.3.2. Consider now the canonical map

qHecke,x,spec : HeckeB,x,loc,spec → HeckeT ,x,loc,spec,

induced by the projection B → T . Direct image defines a chiral functor

(10.3) (qHecke,spec)∗ : SatB,spec → SatT ,spec .

The above functor (10.3) has an additional structure of lax monoidal functor.

We let

AT ,B,G ∈ SatT ,spec

be the image of AB,G under the above functor (qHecke,spec)∗.

By construction, this is a chiral algebra in SatT ,spec with an additional structure of associativealgebra with respect to the monoidal structure on SatT ,spec.

10.3.3. We propose:

Conjecture 10.3.4. The pull-back by means of

(idXλ − ρ(ωX)) : Xλ × BunT → Xλ × BunT

of Ω(n−X), when viewed as an associative algebra in D(HeckeT,glob,Ran(X)) arises via (10.1) fromAT ,B,G(Ran(X)) via (10.2).

This conjecture doesn’t seem very far-fetched. We expect that it results from the descriptionof the local category

D(G(Ox)\G(Kx)/Nρ(ωDx )(Kx))

as QCoh(Spr0) a la Bezrukavnikov. Here Nρ(ωDx ) is the twist of N , veiwed as a groop-schemeover Dx by the T -torsor ρ(ωX).

In the next two sections we’ll explain the role that Conjecture 10.3.4 plays in viewing geo-metric Eisenstein series within geometric Langlands correspondence.

11. Eisenstein series and Langlands correspondence

This section is devoted to our goal 1, namely the analysis of Eisenstein series in the frameworkof Langlands correspondence.

11.1. Spectral Eisenstein series.

11.1.1. Let QCohN(LocSysG) ⊂ IndCoh(LocSysG) be the category introduced in [Sum]. Con-sider the diagram

LocSysG LocSysB

LocSysG

pspec

qspec

As was explained in loc. cit., the functor

Eisspec := pspec∗ q∗specis well-defined as a functor

Perf(LocSysT )→ CohN(LocSysG),

44 D. GAITSGORY

and hence gives rise to (the same named functor)

QCoh(LocSysT )→ QCohN(LocSysG).

The functor Eisspec admits a right adjoint CTspec given by

CTspec := qspec∗ p!spec.

Let Γspec denote the monad on QCoh(LocSysT ) given by CTspec Eisspec.

11.1.2. The morphism

pspec : LocSysB → LocSysG

is easily seen to be proper. Let

(11.1) QCohN(LocSysG,Eis) ⊂ QCohN(LocSysG)

be the full subcategory consisting of objects that vanish when localized on the complement tothe image of pspec. I.e., the functor Eisspec factors through a functor

QCoh(LocSysT )→ QCohN(LocSysG,Eis),

which we’ll denote by the same symbol by a slight abuse of notation. The adjoint functorCTspec factors through the right adjoint

QCohN(LocSysG,Eis)← QCohN(LocSysG).

The following has been recently established by D. Arinkin:

Theorem 11.1.3. The image of the functor (pspec)∗ (or, equivalently, Eisspec) generatesQCohN(LocSysG,Eis).

11.1.4. Recall that our goal 5 stated in the introduction was to construct an equivalence

ΨG,Eis : QCohN(LocSysG,Eis)→ D(BunG)Eis,

which would make the following diagram

QCohN(LocSysG,Eis)ΨG,Eis−−−−→ D(BunG)Eis

Eisspec

x Eis! −ρ(ωX)

xQCoh(LocSysT )

ΨT−−−−→ D(BunT )

commute, where −ρ(ωX) is the functor of shift by −ρ(ωX) ∈ BunT . Equivalently, we’d like thefollowing diagram to commute:

QCohN(LocSysG,Eis)ΨG,Eis−−−−→ D(BunG)Eis

(pspec)∗

x Eisint! −ρ(ωX)

xQCoh(LocSysB)

ΨB−−−−→ Ω(n−X)-mod(D(BunT )),

where ΨB is the enhancement of the Fourier-Mukai transform ΨT obtained by consideringmodules over the corresponding monads.

Taking into account Theorem 11.1.3, the construction of such a functor amounts to estab-lishing the following plausible conjecture:


Conjecture 11.1.5. The Fourier-Mukai equivalence ΨT intertwines the two monads:

(11.2) ρ(ωX) Γ −ρ(ωX) : D(BunT )→ D(BunT )

and

(11.3) Γspec : QCoh(LocSysT )→ QCoh(LocSysT ).

Unfortunately, we can’t prove this conjecture at the moment. However, we’ll able able tomatch certain parts of both sides.

11.2. Bruhat decomposition of Γspec.

11.2.1. Consider the diagram

LocSysT LocSysG LocSysT

LocSysB LocSysB

LocSysB ×LocSysG

LocSysB

qspec

pspec

pspec

qspec

plspec

prspec

The functor Γspec is isomorphic to the composition

(qspec)∗ (prspec)∗ (plspec)! (pspec)

∗.

11.2.2. For each w ∈W let(LocSysB ×

LocSysG

LocSysB

)w

⊂(

LocSysB ×LocSysG

LocSysB

)∼w

be the corresponding locally closed substack and its completion. (Our normalization is such

that for w = 1, the substack

(LocSysB ×

LocSysG

LocSysB

)1

is closed. Let us denote its two

projections as in the diagram below:

LocSysB LocSysB .

(LocSysB ×

LocSysG

LocSysB

)∼w

plw,spec

prw,spec

We define the corresponding functor

Γw,spec : QCoh(LocSysT )→ QCoh(LocSysT )

by

(11.4) (qspec)∗ (prw,spec)∗ (plw,spec)! (qspec)

∗.

46 D. GAITSGORY

11.2.3. The following seems to be a local (and hence, more tractable) case of Conjecture 11.1.5:

Conjecture 11.2.4. Fourier-Mukai equivalence ΨT intertwines the functors

ρ(ωX) Γw,spec −ρ(ωX) and Γw.

Let us note that one case of this conjecture is easy, namely w = w0. Indeed, in this caseboth functors are given by the action of w0 on BunT .

In what follows we’ll concentrate on ”the most interesting” case of Conjecture 11.2.4, namelyfor w = 1, i.e.:

Conjecture 11.2.5. Fourier-Mukai equivalence ΨT intertwines the monads

ρ(ωX) Γ1,spec −ρ(ωX) and Γ1.

We’ll show that Conjecture 11.2.5 follows from Conjecture 10.3.4. It’s plausible that a similarargument will show that Conjecture 11.2.4 also follows from Conjecture 10.3.4.

11.3. ”Proof” of Conjecture 11.2.5.

11.3.1. Let’s note that the monad

pr1,spec)∗ (pl1,spec)! : QCoh(LocSysB)→ QCoh(LocSysB)

falls into the following general paradigm.

Let f : Z1 → Z2 be a map between Artin (DG)-stacks of finite type. Consider the completionof the diagonal:

Z1 Z1

(Z1 ×

Z2

Z1

)∼

Z2.

′fl

′fr

f

f

Then we can consider the monad

(11.5) IndCoh(Z1)(′fl)!

−→ IndCoh

((Z1 ×

Z2

Z1

)∼)(′fr)∗−→ IndCoh(Z1).

We’ll denote it by U(ΘZ1/Z2) for the following reason: this monad is given by tensoring with

the unversal enveloping DZ1-algebra of the relative tangent Lie algebroid ΘZ1/Z2

.

By construction, the functor f∗ : IndCoh(Z1) → IndCoh(Z2) factors via the categoryU(ΘZ1/Z2

)-mod(IndCoh(Z1)).

Note that when f is a closed embedding (or, more generally, radicial) the category of modulesover this monad is, by the Barr-Beck theorem, equivalent to the full subcategory of IndCoh(Z2)consisting of modules supported set-theoretically on Z1.

In another extreme case, when Z2 = pt and Z1 is l.c.i., the category of modules over thismonad is equivalent to that of right D-modules on Z1.


Note that the composition

IndCoh(Z1)U(ΘZ1/Z2

)−→ IndCoh(Z1)→ QCoh(Z1)

factors through the localization IndCoh(Z1) → QCoh(Z1). We’ll abuse the notation slightlyand use the same symbol U(ΘZ1/Z2

) to denote the resulting monad QCoh(Z1) → QCoh(Z1),i.e., we have a commutative diagram:

IndCoh(Z1)U(ΘZ1/Z2

)−−−−−−−→ IndCoh(Z1)y y

QCoh(Z1)U(ΘZ1/Z2

)−−−−−−−→ QCoh(Z1).

11.3.2. Let Y be a crystal of formally smooth schemes or Artin stacks over X. Along with Y, wecan consider the crystal of formal schemes/Artin stacks, denoted Ymer,form, whose fiber over

x ∈ X is the formal neighborhood of Yx = Y(Dx) inside the functor Y(Dx).

Consider the ind-stack

HeckeY,x,loc := Y(Dx) ×Y(Dx)

Y(Dx),

and category

SatY,x := IndCoh (HeckeY,x,loc) .

The assignment x 7→ SatY,x forms a chiral category; moreover convolution defines on it astructure of monoidal chiral category, which we denote SatY.

Note that the projections

Y(Dx)←h loc←− HeckeY,x,loc

→h loc−→ Y(Dx)

and ind-finite. Let AY denote an associative factorization algebra in SatY equal to the dualizing

sheaf of HeckeY,x,loc with respect to←h loc.

Let Y(X) be the scheme/stack of global sections of Y. For every x we have a diagram

Y(Dx) Y(Dx)

HeckeY,x,loc

Y(X) Y(X

HeckeY,x,glob

←h loc

→h loc

←h glob

→h glob

with both diamonds being Cartesian.

In particular, objects of SatY(Ran) define functors

IndCoh(Y(X))→ IndCoh(Y(X)),

48 D. GAITSGORY

which descent do well-defined functors on the localization

QCoh(Y(X))→ QCoh(Y(X)).

Associative algebras in SatY(Ran) give rise to monads.

The following is due to N. Rozenblyum:

Theorem 11.3.3. The monad IndCoh(Y(X)) → IndCoh(Y(X)) (resp., QCoh(Y(X)) →QCoh(Y(X))) given by AY(Ran) is canonically isomorphic to U(ΘY(X)/ pt).

In what follows, we’ll need a relative version of the above situation.

11.3.4. Let Y1 and Y2 be two crystals of schemes/stacks as above, and let floc : Y1 → Y2 be amorphism.

For a point x ∈ X consider the ind-stack

HeckeY1,Y2,x,loc := Y1(Dx) ×Y1(

Dx) ×

Y2(Dx)

Y2(Dx)

Y1(Dx),

equipped with the two projections

Y1(Dx)

←h rel,loc←− HeckeY1,Y2,x,loc

→h rel,loc−→ Y1(Dx).

These projections are ind-finite.

In addition, we have a natural map

HeckeY1,Y2,x,loc → HeckeY1,x,loc .

We let AY1,Y2,x be the associative algebra in SatY1,x equal to the direct image under the

above map of the relative (w.r. to←hrel,loc) of the dualizing sheaf of HeckeY1,Y2,x,loc.

The assignment x 7→ AY1,Y2,x defines an associative factorization algebra in SatY1 that wedenote by AY1,Y2

.

Theorem 11.3.5. The monad IndCoh(Y(X)) → IndCoh(Y(X)) (resp., QCoh(Y(X)) →QCoh(Y(X))) given by AY1,Y2

(Ran) is canonically isomorphic to U(ΘY1(X)/Y2(X)).

11.3.6. We apply the above discussion to the case Y1 = pt /G1, and Y2 = pt /G2. Note theslight discrepancy of notation

Heckept /G1,x,loc = HeckeG1,x,loc,spec

Apt /G1,pt /G2= AG1,G2,spec.

From Theorem 11.3.5, we obtain a description of the monad

(11.6) U(ΘLocSysG1/LocSysG2

) : QCoh(LocSysG1)→ QCoh(LocSysG1

).

Namely:

Lemma 11.3.7. The monad (11.6) is given by the action of the chiral algebra AG1,G2,spec(Ran).


11.3.8. We apply the discussion of Sect. 11.3.6 to G1 = B, G2 = G.

Lemma 11.3.7 provides an answer for the monad

(pr1,spec)∗ (pl1,spec)! = U(ΘLocSysB /LocSysG

).

We now wish to describe the monad

(qspec)∗ U(ΘLocSysB /LocSysG) (pspec)

∗ : QCoh(BunT )→ QCoh(BunT )

of (11.4).

Lemma 11.3.9. For a functor S : QCoh(BunB) → QCoh(BunB) given by the action ofA(Ran(X)) for a unital chiral algebra A ∈ SatB,spec, the composed functor

QCoh(BunT )(qspec)

∗

−→ QCoh(BunB)S→ QCoh(BunB)

(qspec)∗−→ QCoh(BunT )

is given by A′(Ran(X)), where A′ ∈ SatT,spec is the chiral algebra (qHecke,spec)∗(A).

11.3.10. Lemmas 11.3.7, 11.3.9 monad show that the statement of Conjecture 11.2.5 followsfrom that of Conjecture 10.3.4, using the fact that the Fourier-Mukai transform ΨT is compatiblewith the geometric Satake isomorphism (10.2) via the action of Hecke functors on both sides.

12. What acts compactified Eisenstein series

In this subsection we’ll assume Conjecture 10.3.4 and pursue our goal 3.

12.1. The local monad.

12.1.1. In Corollary 3.5.11 we’ve seen that the functor Eis!∗ : D(BunT )→ D(BunG) canonicallyextends to a functor

Eisint!∗ : U(n−X)-mod(D(BunT ))→ D(BunG),

where U(n−X) ?− was a certain explicit monad acting on D(BunT ).

The goal of this section is to extend this constructing further. We’ll construct a monad thatwe’ll denote U(g/tX) acting on D(BunT ), which receives a map from U(n−X), and a functor

Eisult!∗ : U(g/tX)-mod(D(BunT ))→ D(BunG),

such thatEisint!∗ ' Eisult!∗ ind

U(g/tX)

U(n−X).

12.1.2. Namely, consider the chiral algebra in SatT ,spec equal to AT ,G. Let U(g/tX) denote

the corresponding object of SatT (Ran) obtained via (10.2). By construction, there exists acanonical algebra homomorphism U(n−X) → U(g/tX). By a slight abuse of notation, we’lldenote by the same symbol U(g/tX) the resulting monad acting on D(BunT ).

In what follows, we’ll show that Conjecture 10.3.4 implies the following:

Conjecture 12.1.3. The functor

Eisint!∗ : U(n−X)-mod(D(BunT ))→ D(BunG)

canonically extends to a functor

Eisult!∗ : U(g/tX)-mod(D(BunT ))→ D(BunG),

such that the functor

Eisint!∗ ' Eisult!∗ indU(g/tX)

U(n−X).

50 D. GAITSGORY

Remark. Recall Ω(n−X), viewed as an object of D(Ran(X,Λpos) × BunT ) was initially defined

through the geometry of the stack BunB (as a certain local cohomology). In Sect. ??, we’ll seea similar interpretation for U(g/tX).

Note that from Theorem 11.3.5 we obtain:

Corollary 12.1.4. The monad acting on QCoh(LocSysT ) that corresponds to U(g/tX) via theFourier-Mukai equivalence ΨT identifies canonically with U(ΘLocSysT /LocSysG

).

12.2. ”Proof” of Conjecture 12.1.3.

12.2.1. We’ll work inside the monoidal chiral category SatT ; let 1SatT ∈ SatT (Ran) be the unit.The Koszul resolution

1SatT ' Ω(n−X) ?⊗U∨(n−X)

makes it a bi-module with respect to Ω(n−X) and U∨(n−X). Convolution with this bi-moduledefines a functor

U∨(n−X)-mod→ Ω(n−X)-mod

within any category acted on monoidally by SatT (Ran).

We obtain that Conjecture 12.1.3 would follow from the following isomorphism:

(12.1) Ω(n−X) ⊗Ω(n−X)

1SatT ' 1SatT ⊗U(n−X)

U(g/tX),

as bi-modules with respect to Ω(n−X) and U∨(n−X), in a way compatible with multiplication.

We’ll establish (12.1) by transferring it to a (tautological) isomorphism that takes place inSatT ,spec, and then applying Conjecture 10.3.4.

12.2.2. First, we recall that Ω(n−X) identifies with direct image of the structure sheaf under the

morphism of factorization stacks pt /B → pt /T . Here we view QCoh(pt /T ) as mapping toSatT ,spec via the diagonal map

pt /T ' pt /T (Dx)→ pt /T (Dx) ×pt /T (

Dx)

pt /T (Dx).

According to Conjecture 10.3.4, Ω(n−X) corresponds to the associative algebra AT ,B,G,spec.

Finally, we observe that U(n−X) corresponds to the associative chital algebra AT ,B,spec.

Hence, we need to establish an isomorphism

(12.2) AT ,B,G,spec ⊗Opt /B

1SatT ,spec ' 1SatT ,spec ⊗AT ,B,spec

AT ,G.

This isomorphism needs to take place in SatT ,spec(Ran), i.e., pointwise in

IndCoh(HeckeT ,spec). We will establish that it takes place in a localization of this

category, namely in QCoh(HeckeT ,spec). However, since both sides are co-connective, theisomorphism in the localization implies the original one.

12.2.3. Let us return to the context of Sect. 11.3.2. We’ll use the following result of J. Lurie:

Theorem 12.2.4. The action of QCoh(HeckeY,x,loc) on QCoh(Yx) identifies the former withthe category of endomorphisms of the latter as a chiral module category over itself.

Thus, in order to establish (12.2) we need to identify both sides as endomorphisms of

QCoh(pt B(Dx)), viewed as a category over pt B(Dx).


12.2.5. Recall the general context of Sect. 11.3.4, and suppose now that we have three stacksY1 → Y2 → Y3. Consider the forgetful functor

AY1,Y2,x-mod(QCoh(Y1,x))→ QCoh(Y2,x).

Tautologically, we have:

Lemma 12.2.6. The above functor intertwines the monad

AY1,Y3,x ×AY1,Y2,x

−

acting on the LHS, with the monad AY2,Y3,x acting on the RHS. The isomorphism of functors

in question holds as an isomorphism over the stack Y2(Dx).

This proves the required isomorphism (12.2), when we view pt /T → pt /B as a map of stacksover pt /T via the projection B → T .

References

[BG1] A. Braverman, D Gaitsgory, ”Geometric Eisenstein Series”, arXiv:math/9912097.

[BG2] A. Braverman, D Gaitsgory, ”Deformations of local systems and Eisenstein series”, arXiv:math/0605139.[BFGM] A. Braverman, M. Finkelberg, D. Gaitsgory and I. Mirkovic,

”Intersection cohomology of Drinfeld’s compactifications”, arXiv:math/0012129.

[Eis] Weird adjunction for Eisenstein series[CHA] ”Chiraral algebras”.

[FFKM] B. Feigin, M. Finkelberg, A. Kuznetsov and I. Mirkovic,

”Semi-infinite flags-II”, arXiv:alg-geom/9711009.[JNKF] John Francis, in preparation.

[Sum] An attempt to formulate the conjecture

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Introduction - Harvard Universitypeople.math.harvard.edu/~gaitsgde/GL/WhatActs.pdf · In the classical theory, the analogue of the term main is a certain intertwining operator (acting