WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES

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1 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES D. GAITSGORY 1. Introduction 1.1. The Functor. Geometric Eisenstein series is the functor Eis! : D(Bun T D(Bun G defined by F p! q (F, (up to a cohomological shift, where Bun B p q Bun G Bun T The question that we are concerned with in this paper is the description of Hom D(BunG (Eis! (F 1, Eis! (F 2 in terms of F 1, F 2. In some sense, the answer is tautological. Let CT denote the right adjoint to Eis!. The composition Φ := CT Eis! is a monad acting on D(Bun T, and we have: Hom D(BunG (Eis! (F 1, Eis! (F 2 Hom D(BunT (F 1, Φ(F 2. So, what we are after is to have a more detailed understanding of the monad Φ The Monad Parallel to what happens in the classical theory of automorphic functions, the functor Φ admits a canonical filtration by functors numbered by the Weyl group W (viewed as a poset with respect to the Bruhat order; we denote the subquotient functor corresponding to an element w W by Φ w. The term Φ main := Φ 1 happens to be the most interesting; in fact Φ main is itself a monad, and the canonical map Φ main Φ is a homomorphism. (By contrast, the term Φ w0 is the simplest: it s given by the action of w 0 on Bun T. In the classical theory, the analogue of the term Φ main is a certain intertwining operator (acting on automorphic functions on the abelian group T, and it decomposes as a product of local intertwining operators. We can now define our goal more precisely as follows: we d like to describe Φ main : D(Bun T D(Bun T in terms that are local with respect to the curve X. The latter phrase can be given a precise meaning as follows: Date: July 4,

2 2 D. GAITSGORY Recall that for a reductive group M and x X one can attach the Satake category at x, denoted Sat M,x that acts on D(Bun M by Hecke functors. The assignment x Sat M,x forms what is called a factorization (a.k.a. chiral category over X, equipped with a compatible monoidal structure. We denote this category simply by Sat M. Now, if E Sat M is a factorization (a.k.a. chiral algebra equipped with a compatible associative algebra structure in this category, the chiral homology H ch (X, E acts as a monad on D(Bun M. So, our first goal can be stated as follows: Goal 1a: Describe explicitly the factorization algebra E Sat T, such that Φ main H ch (X, E By the definition of the functor Eis!, its construction uses the stack Bun B. However, Bun B is a little stupid as a stack: it splits into connected components numbered by elements of Λ the coroot lattice of T. However, Bun B admits a relative compactification Bun B j Bun B p q with p = p j and q = q j. Bun G Bun T The stack Bun B is stratified by locally closed substacks of the form ι λ : X λ Bun B Bun B, where X λ is the space of configuration on Λ pos -colored divisors of total degree λ. (Here Λ pos Λ is 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 Bun B The space of rational reductions to B The situation with Bun B can be pushed even further. In 2004 Drinfeld proposed that there should exist a stack Bun rat B of G-bundles equipped with a rational reduction to B. This stack is supposed to glue together the connected components Bun µ B of Bun B for µ projecting to the same element of π 1 (G. In other words, the should exist a projection Bun B Bun rat B that collapses each stratum X λ Bun µ B to just Bunµ B. However, Bun B does not exist as an algebraic stack (the strata that need to collapse have wrong self-intersections. Nonetheless we will achieve: Goal 2a: Construct a category D(Bun rat B, equipped with a direct image functor Av : D(Bun B D(Bun rat B and a functor p rat! : D(Bun rat B D(Bun G so that p! p rat! Av As we shall see, the monad Φ main will have a natural interpretation in terms of the category D(Bun rat B. Namely, we ll achieve Goal 2b: Interpret the monad Φ main in terms of Homs in the category D(Bun rat B.

3 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES Compactified Eisenstein series. The existence of the compactification Bun B has lead the authors of [BG1] and [FFKM] to consider another functor D(Bun T D(Bun G, namely that of compactified Eisenstein series: Eis! (F := p! (q (F IC BunB, where IC BunB is the intersection cohomology sheaf on Bun B. Our next goal can be stated as follows: Goal 3: From the action of the monad Φ main on Eis!, find what monad acts on the functor Eis! The Langlands dual picture Let us recall now that the functor Eis! is supposed to fit into the Geometric Langlands picture (1.1 L G QCoh N (LocSysǦ D(BunG Eis spec Eis! ρ(ω X QCoh(LocSysŤ L T D(BunT. Here QCoh N (LocSysǦ is a certain modification of the category QCoh(LocSysǦ, described in [Sum]. The horizontal arrow L G is the conjectural Langlands transform. The horizontal arrow L T is the Langlands transform for T, which since T is abelian, is the Fourier-Mukai transform. The functor Eis spec is the spectral Eisenstein series functor defined using the diagram LocSys ˇB p spec q spec by LocSysǦ Eis spec := p spec q spec. LocSysŤ Finally, ρ(ω X is the functor of shift by ρ(ω X Bun T acting on D(Bun T Our main goal can be stated as follows: Goal 4a: Give an interpretation of the monad Φ main and of the chiral algebra E on the Langlands dual side as a monad acting on QCoh(LocSysŤ Note that the map of stacks pt / ˇB pt /Ť admits a canonical section, and hence so does the map of stacks e : LocSysŤ LocSys ˇB. We ll show that if the diagram (1.2 takes place, then so does the diagram (1.2 L G QCoh N (LocSysǦ D(BunG p spec e Eis! ρ(ω X QCoh(LocSysŤ L T D(BunT.

4 4 D. GAITSGORY Our final goal is: Goal 4b: Give an interpretation of the monad from Goal 3 on the Langlands dual side as a monad acting on QCoh(LocSysŤ. We observe that our Goal 4b is a categorical upgrade of the following remarkable fact discovered in [FFKM]: namely, that the object Eis! (C BunT carries an action of the Langlands dual Lie algebra ǧ Structure of the paper. This paper is divided into three parts, according to the level of sophistication at which we employ the machinery of factorization algebras and categories Part I is largely preparatory and is meant to set the scene for the more involved but similar in spirit manipulation in the later sections. In particular, instead of the factorization category Sat T, which is needed for the description of the monad Φ main, we use its more simply-minded version that deals with factorization algebras of Λ pos -vector space. Although this restricted framework will not allow us to get to Φ main or our goals that have to 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 In Part II we ll achieve some of the goals stated above. First, we ll show what the monad Φ main has to do with the stacks Bun B. Secondly, we ll define the category D(Bun rat B with the expected properties Finally, in Part III we ll state a certain local conjecture that will relate the monad Φ main to the factorization category Sat T, and assuming this conjecture, we ll be able to make a contact with the Langlands dual picture. We note that the results proved in Part III heavily result on the yet unpublished theory of factorization categories.

5 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES 5 Part I 2. Factorization algebras in a simplified context This section introduces a language of factorization algebras graded by a semi-lattice Λ pos. 1 This is nothing but a particular case of the set-up of factorization algebras from [CHA], with the difference that the presence of Λ allows to replace the Ran space by genuine schemes The graded Ran space For λ Λ pos, let X λ denote the corresponding partially symmetrized power of X. For λ = λ 1 + λ 2 let add λ1,λ 2 : X λ1 X λ2 X λ denote the canonical map. For F i D(X λi, with λ = λ 1 + λ 2 we ll denote by F 1 F 2 the object of D(X λ equal to (add λ1,λ 2 (F 1 F We regard the disjoint union λ X λ is a Λ pos -graded version of the Ran space Ran(X and denote it by Ran(X, Λ pos. The category D(Ran(X, Λ pos := {λ F λ D(X λ } has a natural monoidal structure with respect to : for two families {F1 λ } and {F2 λ } the value of their tensor product on X λ is F λ1 1 Fλ2 2. λ=λ 1+λ 2 This monoidal structure is naturally symmetric For λ Λ pos consider the natural closed embedding λ : X X λ. The functor := { λ } makes the category D(X Λpos of Λ pos -graded objects of D(X a full subcategory of D(Ran(X, Λ pos. 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 F 1,..., F n, F D(X Λpos we have a natural map (2.1 Hom D(X Λ pos ( F, F 1!...! F n Hom D(Ran(X,Λ pos ( ( F, (F 1... (F n, and these maps are compatible with iterated tensor products. However, we have the following straightforward assertion: Lemma The maps (2.1 are isomorphisms One can speak about associative, commutative and Lie algebras and co-algebras in either D(X Λpos or D(Ran(X, Λ pos. The functor! maps such objects in D(Ran(X, Λ pos to objects of a similar nature in D(X Λpos. In addition, Lemma implies that for an object M D(X Λpos, a structure of co-algebra of any kind on it is equivalent to one on (M. 1 Unless specified otherwise, we consider Λ pos without the 0 element.

6 6 D. GAITSGORY The usual Koszul duality defines equivalences and in either context. KD A ca : Assoc. Alg co-assoc. co-alg, KD cc L : co-com. co-alg Lie alg, KD C cl : Com. Alg Lie co-alg NB: We ll use the subscripts or! to indicate which category we re in when an ambiguity is likely to occur. Explicitly, KD A ca attaches to an associative augmented algebra A the co-associative coalgebra Bar(A that computes Tor s of the augmentation module with itself over A. The inverse functor sends a co-associative co-algebra A to cobar(a, where the latter computes Exts s of the augmentation module with itself over A. The inverse to KD cc L sends a Lie algebra L to its homological Chevalley complex C (L. The inverse to KD C cl sends a Lie co-algebra L to its homological Chevalley complex C (L Factorization algebras By definition, a Λ pos -factorization algebra is the same as a chiral algebra on X, graded by Λ pos. The corresponding functor is given by A C(A, where C stands for the chiral Chevalley-Cousin complex, following the conventions from [CHA] Explicitly, we can think of a Λ pos -graded factorization algebra A as follows. To each λ Λ pos we assign A λ D(X λ and whenever λ = λ 1 + λ 2, we have an isomorphism (2.2 A λ (X λ 1 X λ 2 disj A λ1 A λ2 (X λ 1 X λ 2 disj, satisfying the natural compatibilities. NB: The restriction of the map add λ1,λ 2 to the open subscheme (X λ1 X λ2 disj X λ1 X λ2 is étale, so the notation F F (X λ 1 X λ 2 disj is unambiguous Whenever we have two factorization algebras A 1 and A 2 their -product A 1 A 2 has a natural factorization algebra structure Commutative factorization algebras We shall say that A is commutative if the above isomorphisms have been extended to maps A λ1 A λ2 add! λ 1,λ 2 (A λ, which also satisfy the natural compatibilities. By adjunction, for a commutative A, we obtain the maps A λ1 A λ2 A λ1+λ2, which make A into a commutative algebra in D(Ran(X, Λ pos with respect to. Note that commutative factorization algebras form a full subcategory among commutative algebras in D(Ran(X, Λ pos. Indeed, for a commutative algebra A := {A λ } D(Ran(X, Λ pos we have the maps A λ1 A λ2 (X λ 1 X λ 2 disj A λ (X λ 1 X λ 2 disj and A is a commutative factorization algebra if and only if these maps are isomorphisms.

7 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES As in [CHA], one can prove the following: Proposition The assignment A! (A establishes an equivalence between the category of commutative factorization algebras and that of 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 A Ran the corresponding commutative factorization algebra The following proposition is due to [JNKF]. Proposition The following diagram of functors is commutative: Lie co-alg(d(x Λpos,! KD cl C(D(X Λpos,! Com. Alg(D(X Λpos,! A A Ran Lie co-alg(d(ran(x, Λ pos, KD cl C(D(Ran(X,Λ pos, Com. Alg(D(Ran(X, Λ pos,. 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, Λ pos, and consider its cohomological Chevalley 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 to the heart of the t-structure, then so does Ω(L 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, Λ pos with respect to, and consider its universal co-enveloping co-algebra in D(Ran(X, Λ pos ; denote the resulting object of D(Ran(X, Λ pos by U (L Ran. 2 By construction, U (L Ran is a commutative algebra in D(Ran(X, Λ pos with a compatible co-associative co-algebra structure, i.e., it s a commutative Hopf algebra. In addition, U (L has the following properties: Lemma (1 As a commutative algebra in D(Ran(X, Λ pos, 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. 2 We emphasize that U (L is not the Verdier dual of the chiral universal envelope of the Verdier dual L of L, assuming it was dualizable. As we shall see, U (L is that for the loop object of L[ 1].

8 8 D. GAITSGORY Note also that the universal co-enveloping co-algebra U (L Ran of a Lie co-algebra L (in any tensor category has the following additional interpretations: For a Lie co-algebra L D(X Λpos consider its suspension L [1]. This is a co-algebra object 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, Λ pos equipped with a co-algebra structure Another (in a sense, tautologically equivalent interpretation is as follows: for a Lie coalgebra L, consider C (L as an associative algebra. Consider the co-associative co-algebra Bar (C (L. The commutative algebra structure on C (L makes Bar (C (L into a commutative Hopf algebra. We have a canonical isomorphism as commutative Hopf algebras. So, we have: U (L Bar (C (L, U (L Ran Bar(Ω(L, as commutative factorization algebras in D(Ran(X, Λ pos equipped with a co-algebra structure Co-commutative factorization algebras We say that a Λ pos -factorization algebra is co-commutative if the factorization isomorphisms (2.2 come by adjunction from maps A λ1+λ2 A λ1 A λ2, which make A into a co-commutative co-algebra in D(Ran(X, Λ pos. As in the case of commutative factorization algebras, the category of co-commutative factorization algebras is a full subcategory in the category of co-commutative co-algebras in D(Ran(X, Λ pos with respect to the tensor structure. NB: We refrain from formulating the structure of co-commutative factorization algebra as a map on X λ1 X λ2 because the latter would involve the functor add λ 1,λ 2, which is not defined for all D-modules The following is parallel to Proposition combined with Proposition 2.3.3: Proposition (1 For a Lie-* algebra L in D(X Λpos, the co-commutative co-algebra in D(Ran(X, Λ pos with respect to the tensor structure, given by C ( (L is factorizable. (2 The above assignment L 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 factorization algebra by Υ(L. It follows from the construction that it L is such that L[1] is a D-module (i.e., belongs to the heart of the t-structure, then so does Υ(L. Let U(L denote the chiral universal enveloping algebra of L; this is a Λ pos -graded chiral algebra on X. By construction, we have: C(U(L Υ(L.

9 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES The following is parallel to Sect For a Lie-* algebra L in D(X Λpos, consider (L as a Lie algebra in D(Ran(X, Λ pos with respect to the tensor structure, and let U(L Ran denote its universal enveloping algebra. This is a co-commutative factorization algebra in D(Ran(X, Λ pos 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-commutative Hopf algebra. In addition, we have a canonical isomorphism as co-commutative Hopf algebras Verdier duality. Υ(L[ 1] U(L Ran, Parallel to the above discussion, we can consider the semi-group Λ neg. For a compact object F D(X λ, we ll think of its Verdier dual D(F as an object of D(X λ. We shall say that an object F = {F λ } of D(Ran(X, Λ pos is locally compact if each of its components F λ is compact as an object of D(X λ. Thus, we obtain that Verdier duality defines a contravariant equivalence between the subcategories D(Ran(X, Λ pos and D(Ran(X, Λ neg, consisting of locally compact objects The tensor product sends (locally compact objects to (locally compact ones, and satisfies: D(F 1 F 2 D(F 1 D(F 2. In particular, Verdier duality defines anti-equivalences between the categories of locally compact associative/commutative/lie algebras in D(Ran(X, Λ pos with respect to the tensor structure and the corresponding co-algebras in D(Ran(X, Λ neg. Moreover, Koszul duaility in any of the contexts: associative/commutative/lie sends locally compact objects to locally compact ones, and we have: D KD KD D Let A be a factorization algebra which is locally compact. It is clear that D(A has a natural structure of factorization algebra. The following is evident from the definitions: Lemma Let A be a factorization algebra which is locally compact. Then the structure on it of commutative/co-commutative factorization algebra is equivalent to the structure of cocommutative/commutative factorization algebra on D(A 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 , we obtain that the objects Υ(L, Ω(L, U(L Ran and U (L Ran are all locally compact. Moreover, Lemma 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 10 D. GAITSGORY 3. Eisenstein series, take I (a summary of [BG2] 3.1. Action of Ran(X, Λ pos on Bun T We assume now that X is compact and that Λ pos maps to the lattice Λ of coweights of some torus T. Consider the category D(Bun T. We denote the action of D(Ran(X, Λ pos as a monoidal category on D(Bun T by (F := {F λ }, S F S := (π λ id BunT (π λ! (F λ! mult! λ(s, λ where mult λ is the natural map where is the Abel-Jacobi map. X λ AJ Bun λ id T Bun T Bun T mult Bun T, AJ λ : X λ Bun T Similarly, we define an action of D(Ran(X, Λ neg. Since the maps mult λ are smooth and proper, we have: Lemma For a compact object F D(Ran(X, Λ pos, the functors are both left and right adjoint of one another. S F S and S D(F S 3.2. Action on Drinfeld s compactifications For λ Λ pos denote by ι λ : X λ Bun B Bun B the corresponding map obtained by adding zeros. This is a finite map. We let ι λ denote its restriction to the open substack X λ Bun B. The above procedure defines an action of D(Ran(X, Λ pos as a monoidal category on D(Bun B by (F := {F λ }, S F S := ι λ (F λ S. λ Let p, q denote the projections: by Bun G We define the functor of Eisenstein series The following is a diagram chase: p q Bun B Bun T. Eis : D(Bun B D(Bun T D(Bun G T, F p (T! q! (F. Proposition For S D(Ran(X, Λ pos there exists a canonical isomorphism: Eis(S T, F Eis(T, S F A factorization algebra attached to ň.

11 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES Consider the Lie algebra ň ; consider the constant Lie-* algebra ň X := ň C X, which is graded by Λ neg, and its Verdier dual (ň X (ň C X [2] which is graded by Λ pos. Consider the corresponding commutative algebra Ω(ň X in D(Ran(X, Λpos with respect to. λ. By Sect , Ω λ (ň X is a D-module (i.e., belongs to the heart of the t-structure for every Let j denote the open embedding Bun B Bun B, and consider the object j! (IC BunB D(Bun B. NB: Since Bun B is smooth, IC BunB is isomorphic to C BunB [dim(bun B ], where we apply the cohomological shift by the corresponding amount on each connected component. The following has been established in [BG2]: Theorem The object j! (IC BunB D(Bun B is naturally a Ω(ň X -module, with respect to the above action of D(Ran(X, Λ pos on D(Bun B. Moreover, the map arising by adjunction from identifies Ω λ (ň X IC Bun B ι λ! j! (IC BunB, ι λ! ( Ω λ (ň X IC Bun B ι λ! ( Ω λ (ň X j!(ic BunB j! (C BunB, Ω λ (ň X IC Bun B H 0 ( ι λ! (j! (IC BunB Let Eis! denote the functor D(Bun T D(Bun G defined as Eis(j! (IC BunB,, i.e., (! Eis! (F = p! IC BunB q! (F. As a corollary of Theorem we obtain: Corollary There exists a natural transformation Eis! (Ω(ň X F Eis!(F, compatible with the algebra structure on Ω(ň X Intermediate Eisenstein series Let Ω(ň X -mod(d(bun T denote the category of Ω(ň X -modules in D(Bun T. We have the natural forgetful functor which admits a left adjoint, denoted ind Ω(ň X, Corollary implies: Ω(ň X -mod(d(bun T D(Bun T, F Ω(ň X F. Corollary The functor Eis! canonically extends to a functor so that Eis! Eis int! ind Ω(ň X. Eis int! : Ω(ň X -mod(d(bun T D(Bun G,

12 12 D. GAITSGORY Let s give an interpretation of the category Ω(ň X -mod(d(bun T in terms of geometric Langlands. Recall that we have an equivalence (Fourier-Mukai transform: Φ T : D(Bun T QCoh(LocSysŤ. The following results from deformation theory: Proposition There exists a canonical equivalence making the diagrams Φ B : Ω(ň X -mod(d(bun T QCoh(LocSys ˇB, Ω(ň X -mod(d(bun T D(Bun T (where the left vertical arrow is the forgetful functor and commute. Φ B QCoh(LocSys ˇB q spec Φ T QCoh(LocSysŤ, Φ B Ω(ň X -mod(d(bun T QCoh(LocSys ˇB ind Ω(ň X q spec D(Bun T Φ T QCoh(LocSysŤ The above achieves the stated goal 2 mentioned in the introduction: The functor Eis int! Φ B can be interpreted as a functor which is the thought-for Ψ G p spec. The composition QCoh(LocSys ˇB D(Bun G, Eis int! Φ B q spec : QCoh(LocSysŤ D(Bun G identifies with Eis! Φ T and is supposed to be isomorphic to the composition QCoh(LocSysŤ Eisspec QCoh N (LocSysǦ Φ G D(Bun G Compactified Eisenstein series Let s recall the second main result of [BG2]. Consider the Bar-construction (3.1 Bar ( Ω(ň X, j!(ic BunB D(Bun B. Theorem There exists a canonical isomorphism in D(Bun B : Bar ( Ω(ň X, j!(ic BunB IC BunB Let triv Ω(ň X : D(Bun T Ω(ň X -mod(d(bun T be the functor that associates to F D(Bun T the same object endowed with the trivial action of Ω(ň X. From Theorem we obtain: Corollary There exists a canonical isomorphism of functors D(Bun T D(Bun G : Eis! Eis int! triv Ω(ň X.

13 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES By Sect , the object (3.1 acquires a natural co-action of the co-associative coalgebra U (ň X (again, with respect to the above action of D(Ran(X, Λpos on D(Bun B. As a corollary of Theorem 3.5.2, we obtain: Corollary The object IC BunB D(Bun B is naturally a U (ň X -comodule, or, equivalently, a co-module for the Lie co-algebra (ň X D(Ran(X, Λ pos We define the functor of compactified Eisenstein series Eis! : D(Bun T D(Bun G as Eis(IC BunB,, i.e., (! Eis! (F = p IC BunB q! (F. From Corollary 3.5.6, we obtain: Corollary There exists a natural transformation Eis! (F Eis! (U (ň X F compatible with the co-algebra structure on U (ň X. From Lemma 3.1.3, we obtain: Corollary There exists a natural transformation Eis! (U(ň X F Eis!(F compatible with the algebra structure on U(ň X Let (ň X -comod(d(bun T = ň X -mod(d(bun T denote the category of (ň X -comodules or (which is equivalent by Lemma of ň X -modules in D(Bun T. Let indň X : D(BunT ň X -mod(d(bun T denote the left adjoint to the forgetful functor. From Corollary we obtain: Corollary The functor Eis! canonically extends to a functor so that Eis! Eis int! indň X. Eis int! : ň X -mod(d(bun T D(Bun G, 4. Relationship between two kinds of Eisenstein series and ɛ-factors 4.1. The renormalized categories Let A be an associative algebra in D(Ran(X, Λ pos, which is locally compact as an object of this category. Let A be the Verdier dual co-algebra in D(Ran(X, Λ neg. Note that the forgetful functor A-mod(D(Bun T D(Bun T does not send compact objects to compact ones. We define the renormalized version of category A-mod(D(Bun T, denoted A-mod(D(Bun T ren to be the ind-completion of the full subcategory of A-mod(D(Bun T consisting of objects whose image under the forgetful functor in D(Bun T is compact.

14 14 D. GAITSGORY We have a tautological functor By construction, the forgetful functor Ξ A : A-mod(D(Bun T ren A-mod(D(Bun T. Ξ A : A-mod(D(Bun T ren D(Bun T sends compact objects to compact ones. Hence, it admits a right adjoint that commutes with direct sums, which we will denote by coind A. Explicitly, where we regard A := D(A as an A-module. coind A (F A F, In addition to the above functor Ξ A, we have a functor defined so that Let triv A denote the functor Ψ A : A-mod(D(Bun T A-mod(D(Bun T ren Ψ A ind A coind A. D(Bun T A-mod(D(Bun T that attaches to an object of D(Bun T the same object endowed with the trivial action of A. This functor factors naturally as Ξ A triv A,ren. The resulting functor triv A,ren : D(Bun T A-mod(D(Bun T ren has the property that it sends compact objects to compact ones. Hence, it admits a right adjoint that commutes with direct sums. We ll denote this right adjoint by inv A Koszul dualities For A D(Ran(X, Λ pos as above, let B D(Ran(X, Λ pos be the Koszul dual co-algebra. By the local compactness assumption, B is also locally compact. Let B D(Ran(X, Λ neg be its Verdier dual algebra. We have the following Koszul duality type result: Proposition (1 The functor canonically factors through a functor followed by the forgetful functor. inv A : A-mod(D(Bun T ren D(Bun T inv A enh : A-mod(D(Bun T ren B-mod(D(Bun T, (2 The functor inv A enh is an equivalence. Its inverse is the functor coinv B enh : B-mod(D(Bun T A-mod(D(Bun T ren, whose composition with the forgetful functor is the functor left adjoint to triv B. coinv B : B-mod(D(Bun T D(Bun T,

15 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES 15 (3 The following diagram of functors is commutative: (4.1 A-mod(D(Bun T Ξ A B-mod(D(Bun T ren Ψ B A-mod(D(Bun T ren B-mod(D(Bun T Let s apply the above discussion to A being Ω(ň X. In this case B = U(ň X. We obtain: Corollary (1 The functor canonically factors through a functor followed by the forgetful functor. (2 The functor inv Ω(ň X enh inv Ω(ň X : Ω(ň X -mod(d(bun T ren D(Bun T inv Ω(ň X enh : Ω(ň X -mod(d(bun T ren ň X -mod(d(bun T, is an equivalence. Its inverse is the functor coinvň X enh : ň X -mod(d(bun T Ω(ň X -mod(d(bun T ren, whose composition with the forgetful functor is the functor coinvň X : ň X -mod(d(bun T D(Bun T, left adjoint to trivň X. (3 The functor invň X canonically factors through a functor followed by the forgetful functor. : ň X -mod(d(bun T ren D(Bun T invň X enh : ň X -mod(d(bun T ren Ω(ň X -mod(d(bun T, (4 The functor invň X enh is an equivalence. Its inverse is the functor coinv Ω(ň X enh : Ω(ň X -mod(d(bun T ň X -mod(d(bun T ren, whose composition with the forgetful functor is the functor left adjoint to triv Ω(ň X. coinv Ω(ň X : Ω(ň X -mod(d(bun T D(Bun T, (5 The above equivalences make the following diagrams commutative: (4.2 Ω(ň X -mod(d(bun T Ξ Ω(ň X ň X -mod(d(bun T ren Ψň X Ω(ň X -mod(d(bun T ren ň X -mod(d(bun T

16 16 D. GAITSGORY and (4.3 Ω(ň X -mod(d(bun T Ψ Ω(ň X ň X -mod(d(bun T ren Ξň X Ω(ň X -mod(d(bun T ren ň X -mod(d(bun T 4.3. Implications for Eisenstein series Consider the functor ň X -mod(d(bun T Ω(ň X -mod(d(bun T given by (either of the two circuits of the diagram (4.2. From Corollary 3.5.4, we obtain: Corollary The following diagram of functors commutes: ň X -mod(d(bun T Ω(ň X -mod(d(bun T Eis int Eis int! D(Bun G Id D(Bun G On the other hand, we should point out that the diagram involving the functors Eis int! and Eis int!, and the diagram (4.3 will not commute. However, by Corollary 4.3.2, the calculation of the resulting functor Ω(ň X -mod(d(bun T ň X -mod(d(bun T Eisint! D(Bun G boils down to the calculation of the composite functor (4.4 Ξ Ω(ň X Ψ Ω(ň X : Ω(ň X -mod(d(bun T Ω(ň X -mod(d(bun T.! 4.4. ɛ-factors In this subsection we ll study the functor (4.4, introduced above, and the corresponding functor (4.5 Ξň X Ψ ň X : ň X -mod(d(bun T ň X -mod(d(bun T. Note that by Corollary 4.2.4, the study of the above functors is equivalent to that of the composed functors and Ψ Ω(ň X Ξ Ω(ň X : Ω(ň X -mod(d(bun T ren Ω(ň X -mod(d(bun T ren Ψň X Ξ ň X : ň X -mod(d(bun T ren ň X -mod(d(bun T ren. We ll calculate these compositions on certain subcategories in both cases.

17 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES For every coroot α consider the functor D(Bun T D(Bun T given by D(Bun T mult! (id iα! D(Bun T Bun T where D(Bun T Pic(X D(Bun T Pic (X pr D(Bun T, i α : Pic(X Bun T is the map induced by α; Pic (X Pic(X is the coarse moduli space (i.e., the Picard scheme. Let D reg (Bun T D(Bun T denote the full subcategory spanned by objects annihilated by the above functors for all coroots α. I.e., this is the right orthogonal of the category generated by the images of the pull-push functors corresponding to the diagram D(Bun T α D(Bun T D(Bun T, where T α is the quotient torus of T by the corresponding copy of G m, and for a torus T, Bun T denotes the corresponding coarse moduli space. Thus, the embedding D reg (Bun T D(Bun T admits a right adjoint, so we can think of D reg (Bun T as a localization of D reg (Bun T Let Ω(ň X -mod(d reg(bun T, Ω(ň X -mod(d reg(bun T ren, ň X -mod(d reg(bun T and ň X -mod(d reg(bun T ren be the preimages of in the corresponding categories of D reg (Bun T D(Bun T under the forgetful functors. We ll prove: Proposition (1 The vertical functors in the diagram Ω(ň X -mod(d reg(bun T Ψ Ω(ň X ň X -mod(d reg(bun T ren Ξň X Ω(ň X -mod(d reg(bun T ren ň X -mod(d reg(bun T are localizations, and the vertical functors in the diagram are fully faithful. Ω(ň X -mod(d reg(bun T Ξ Ω(ň X ň X -mod(d reg(bun T ren Ψň X Ω(ň X -mod(d reg(bun T ren ň X -mod(d reg(bun T (2 The composition (4.5 restricted to ň X -mod(d reg(bun T, is isomorphic to the shift functor F ( 2ρ(ω X F[(2g 2 dim(n ], where 2ρ(ω X is the point of Bun T induced from ω X Pic(X, using the cocharacter 2ρ. Te rest of this section is devoted to the proof of this proposition.

18 18 D. GAITSGORY Let us apply the Fourier-Mukai equivalence Ψ T : D(Bun T QCoh(LocSysŤ. Under this equivalence, ň X -mod(d(bun T corresponds to quasi-coherent sheaves, endowed with an action of the sheaf of DG Lie-algebras ň univ : E Γ(X, ň E. The category ň X -mod(d(bun T ren is the ind-completion of the full subcategory of ň X -mod(d(bun T, conisiting of objects that are compact as objects of QCoh(LocSysŤ. Note also that under Fourier-Mukai, D reg (Bun T corresponds to localization on the open substack QCoh reg (LocSysŤ QCoh(LocSysŤ consisting of Ť -local systems E, such that the 1-dimensional local system α(e is non-trivial for every coroot α of G (=root α of Ǧ. Over this substack, the sheaf ň univ is concentrated in cohomological degree 1. This implies that Ξň X : ň X -mod(d reg(bun T ren ň univ -mod(qcoh reg(locsysť ren is a localization. ň univ -mod(qcoh reg(locsysť ren ň X -mod(d reg(bun T The composition Ξň X Ψ ň X is given by the ň univ -module equal to U(ň univ, where F F denotes the natural duality on QCoh(LocSysŤ. Since ň univ [1] is a locally free coherent sheaf, we have a canonical isomorphism of ň X -modules: U(ň univ U(ň univ det(ň univ [1] [rk(ň univ ]. By Riemann-Roch, rk(ň univ = (2g 2 dim(n. Finally, we need to identify the line bundle det(ň univ [1] with the line bundle where, is the Weil pairing. E E, 2ρ(ω X Π α α(e, ω X, The required identification follows from the next general observation: Lemma Let E be a non-trivial 1-dimensional local system on X. canonical isomorphism det(h 1 (X, E E, ω X. Then we have a 5.1. The dual categories. 5. Verdier dualily and the functional equation Let us return to the general setting of Sect For an associative algebra A + D(Ran(X, Λ pos, let s denote by A D(Ran(X, Λ neg the algebra obtained from the tautological map (5.1 λ λ : Λ neg Λ pos.

19 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES Note that Verdier duality on D(Bun T gives rise to a canonical identification ( A + -mod(d(bun T A -mod(d(bun T, in such a way that the following diagram is commutative (D(Bun T c ind A+ D (D(Bun T c ind A (A + -mod(d(bun T c D (A -mod(d(bun T c, where we use the notation D to denote the canonical anti-equivalence C c (C c for a compactly generated category C and its subcategory C c of compact objects. In other words, the dual of the forgetful functor A + -mod(d(bun T D(Bun T is the induction functor ind A, and vice versa In addition, we have a canonical identification ( A + -mod(d(bun T ren A -mod(d(bun T ren, in such a way that the diagram (D(Bun T c D (D(Bun T c (A + -mod(d(bun T ren c D (A -mod(d(bun T ren c, where the vertical arrows are the forgetful functors. In other words, the functor dual to the forgetful functor A + -mod(d(bun T ren D(Bun T is the co-induction functor coind A, and vice versa. The next assertion follows from the definitions: Lemma The dual of the functor is and vice versa. Ξ A+ : A + -mod(d(bun T ren A + -mod(d(bun T Ψ A : A -mod(d(bun T A -mod(d(bun T ren, 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 are compatible with those above, in the sense that the duals of the functors are coinv A+ enh : A + -mod(d(bun T B + -mod(d(bun T ren : inv B+ enh inv B enh : B -mod(d(bun T ren A -mod(d(bun T : coinv B+ enh Verdier duality on Bun B.

20 20 D. GAITSGORY Theorems and have the following Verdier dual cousins. Consider the Λ pos -graded Lie-* algebra ň + X, and the corresponding co-commutative co-algebra Υ(ň+ X in D(Ran(X, Λ pos with respect to. Theorem The object j (IC BunB D(Bun B is naturally a co-module with respect to Υ(ň + X. Th corresponding map ι λ j (IC BunB Υ λ (ň + X IC Bun B identifies the latter with H 0 of the former Consider now the object (5.2 cobar ( Υ(ň + X, j (IC BunB D(Bun B. NB: The formation of the co-bar complex involves a limit, so such is the case in forming the object (5.2. However, the corresponding inverse system is easily seen to stabilize when restricted to every open substack of Bun B of finite type. Theorem There exists a canonical isomorphism in D(Bun B : cobar ( Υ(ň + X, j (IC BunB IC BunB. Corollary The object IC BunB D(Bun B is naturally a U(ň + X -module, or, equivalently, a module for the Lie algebra ň + X D(Ran(X, Λneg We observe the following phenomenon: the object IC BunB has a structure of module with respect to ň + X and co-module with respect to (ň X. It is natural to ask: Question Can one formulate in what sense IC BunB carries an action of the entire ǧ? This is closely related to our goal 3 stated in the introduction. In fact, we ll consider on object (an algebra in D(Ran(X, Λ pos, which is richer than Ω(ň X, which acts on j!(ic BunB, and such as this action encodes the required structure Verdier dual picture for Eisenstein series In addition to D(Bun G, we can consider its dual category, D(Bun G. We will distinguish them notationally, by denoting the former by D(Bun G! and the latter by D(Bun G. The functor Eis(j (IC BunB, is naturally a functor Eis : D(Bun T D(Bun G. As in Corollary 3.4.2, from Theorem 5.2.2, we obtain: Corollary The functor Eis canonically extends to a functor so that Eis Eis int ind Ω(ň+ X. Eis int : Ω(ň + X -mod(d(bun T D(Bun G,

21 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES The following has been established in [Eis]: Proposition The functor Eis! : D(Bun T D(Bun G! sends compact objects to compacts. For a compact F D(Bun T we have: Corollary The functors D Eis! (F Eis (D(F. Eis int! : Ω(ň X -mod(d(bun T D(Bun G! and Eis int : Ω(ň + X -mod(d(bun T D(Bun G both send compact objects to compacts, and for a compact F Ω(ň X -mod(d(bun T we have: D Eis int! (F Eis int (D(F In addition to Eis int, one would wish to use Theorem to define a functor so that where Eis! is a functor Eis int! : U(ň + X -mod(d(bun T D(Bun G, Eis int! indň+ X Eis!, D(Bun T D(Bun G, defined using the kernel IC BunB. According to Theorem 5.2.4, this functor should also make the following diagram commutative: (5.3 Ω(ň + X -mod(d(bun T Ψ Ω(ň+ X Ω(ň + X -mod(d(bun T ren Eis int D(Bun G Eis int U(ň + X -mod(d(bun T. Unfortunately, we do not know how to define Eis! (making sure that it commutes with direct sums. So, we do not know how to define Eis int! either. However, we can define the functor Eis int! (and, hence, Eis! on the subcategories respectively. U(ň + X -mod(d reg(bun T U(ň + X -mod(d(bun T and D reg (Bun T D(Bun T, Namely, we have the following assertion: Lemma The functor Eis int : Ω(ň + X -mod(d(bun T D(Bun G, when restricted to U(ň + X -mod(d reg(bun T factors through the localization 5.4. Functional equation. U(ň + X -mod(d reg(bun T U(ň + X -mod(d reg(bun T ren Recall that we have the functor with the following property: F : D(Bun G D(Bun G! (5.4 F Eis Eis! w 0, where w 0 : D(Bun T D(Bun T denotes the functor corresponding to the action of w 0 on T.!

22 22 D. GAITSGORY Note that the action of w 0 on D(Bun T extends to an equivalence w 0 : Ω(ň + X -mod(d(bun T Ω(ň X -mod(d(bun T. Thus, from (5.4, we obtain the following form of the functional equation: Corollary There exists a canonical isomorphism F Eis int Eis int! w We shall now establish another form of the functional equation, this time for the functor Eis!. Namely, we claim: Proposition For F D reg (Bun T, we have a canonical isomorphism F Eis! (F Eis! (w 0 (F, where F is the shift of F equal to ( 2ρ(ω X F[(2g 2 dim(n ]. Proof. By Corollary 5.4.3, the RHS is isomorphic to F Eis int ( triv Ω(ň+ X (F Hence, the assertion follows from Proposition New algebras.. 6. Eisenstein series, take II We ll change the framework of Sect. 2 slightly. Instead of D(Ran(X, Λ pos, we ll consider the category D(Ran(X, Λ pos Bun T. The only difference is that the operation gets replaced by a twisted one, namely, F λ D(X λ, F µ D(X µ F λ F µ D(X λ X µ F λ D(X λ Bun T, F µ D(X µ Bun T F λ F µ := (id X λ mult µ! (F λ! (π λ id BunT! (F µ D(X λ X µ Bun T. This makes D(Ran(X, Λ pos Bun T into a monoidal category by means of As such it acts on D(Bun T via where and similarly for D(Bun B. F λ, F µ F λ F µ := (add λ,µ id BunT (F λ F µ. F λ, S F λ S := mult λ (F λ S, F λ S := F λ! (π λ id BunT! (S, The discussion of Koszul dualities (in the associative/co-associative setting goes through without change. We can also talk about factorization algebras in D(Ran(X, Λ pos Bun T. By this we mean an object A := {A λ D(X λ Bun T } endowed with an isomorphism (6.1 A λ A µ (X λ X µ disj Bun T add! λ,µ(a λ+µ (X λ X µ disj Bun T, satisfying a natural associativity condition.

23 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES 23 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 A λ A µ add! λ,µ(a λ+µ and add λ,µ(a λ+µ A λ A µ corresponding to the algebra (resp., co-algebra structure restrict to the map (6.1 on the open part (X λ X µ disj Bun T We introduce an associative algebra Ω(ň X in D(Ran(X, Λpos Bun T with a compatible factorization structure as follows: Proposition-Construction There exists a canonically defined factorization algebra Ω(ň X equipped with a structure of associative algebra in D(Ran(X, Λpos Bun T such that ι λ! (j! (IC BunB Ω(ň X λ IC BunB := (id X λ q! ( Ω(ň X λ! (π λ id BunB! (IC BunB, and the resulting map is an algebra action. Ω(ň X j!(ic BunB j! (IC BunB Let Ω(ň X -mod(d(bun T denote the category of Ω(ň X -modules in D(Bun T. We obtain that the functor Eis! can be canonically extended to a functor such that Eis ult! : Ω(ň X -mod(d(bun T D(Bun G!, Eis ult! ind Ω(ň X Eis! Let Υ(ň X denote the Verdier dual of Ω(ň X ; this is a factorization algebra in D(Ran(X, Λ neg with a compatible structure of co-associative co-algebra. Let Υ(ň + X (resp., Ω(ň + X be the corresponding objects obtained via (5.1. Applying Verdier duality to Proposition 6.1.3, we obtain: Corollary The object j (IC BunB D(Bun B is naturally a Υ(ň + X -comodule Consider the corresponding category Υ(ň + X -comod(d(bun T Ω(ň + X -mod(d(bun T. As in Sect. 5.1, we have a canonical equivalence ( Ω(ň X -mod(d(bun T + Ω(ň X -mod(d(bun T. As in Sect. 5.3, we can canonically extend the functor Eis : D(Bun T D(Bun G to a functor Eis ult : Ω(ň + X -mod(d(bun T D(Bun G, such that Eis ult ind Ω(ň X Eis. The functors Eis ult! and Eis ult are Verdier conjugate of each other in the same sense as in Corollary Finally, the action of w 0 W defines an equivalence Ω(ň + X -mod(d(bun T Ω(ň X -mod(d(bun T, and as Corollary we have the functional equation: (6.2 F Eis ult Eis ult! w 0.

24 24 D. GAITSGORY 6.2. Koszul duality for Ω A crucial observation concerning the quadruple of (co-algebras Ω(ň + X, Ω(ň X, + Υ(ň X, Υ(ň X is that it possesses an extra symmetry with respect to Koszul duality: Proposition We have a canonical isomorphism of co-associative co-algebras: (6.3 Bar( Ω(ň X Υ(ň + X, and of resulting co-modules in D(Bun B (6.4 Bar ( Ω(ň X, j!(ic BunB j (IC BunB. Proof. Consider the object Bar ( Ω(ň X, j!(ic BunB D(Bun B. We claim that it is isomorphic to j (IC BunB. Indeed, it suffices to show that when we apply ι λ! to it for λ ( 0, we obtain 0. However, ι λ! Bar ( Ω(ň X, j!(ic BunB Bar ( Ω(ň X, Ω(ň X λ IC BunB = 0. Now, applying ι λ to both sides of (6.4, we obtain: Bar( Ω(ň X λ IC BunB ι λ (j (IC BunB, Applying Verdier duality to Proposition 6.1.3, we obtain that whence the identification ( Let and ι λ (j (IC BunB Υ(ň + X IC Bun B, Ξ Ω(ň X : Ω(ň X -mod(d(bun T ren Ω(ň X -mod(d(bun T : Ψ Ω(ň X Ξ Ω(ň + X : Ω(ň + X -mod(d(bun T ren Ω(ň + X -mod(d(bun T : Ψ Ω(ň + X be the corresponding categories and functors as in Sect We have the corresponding commutative diagram of functors with the rows being mutually inverse equivalences: and Ω(ň X -mod(d(bun T Ψ Ω(ň X Ω(ň X -mod(d(bun T ren Ω(ň X -mod(d(bun T Ξ Ω(ň + X coinv Ω(ň X enh Ω(ň + X -mod(d(bun T ren Ξ Ω(ň + X inv Ω(ň X enh Ω(ň + X -mod(d(bun T inv Ω(ň + X enh + Ω(ň coinv X Ω(ň + X -mod(d(bun T ren Ψ Ω(ň + X Ω(ň X -mod(d(bun T ren enh Ω(ň + X -mod(d(bun T.

25 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES Recall now that in addition to the functor F, there exists the tautological functor T : D(Bun G D(Bun G!. We have: Proposition The following diagram of functors is commutative: Ω(ň + X -mod(d(bun T Eis ult D(Bun G Ξ Ω(ň X coinv Ω(ň + X enh Ω(ň X -mod(d(bun T Eis ult T D(Bun G!.! Proof. The statement of the proposition is equivalent to the following one: j (IC BunB Bar ( Ω(ň X, j!(ic BunB in a way compatible with the co-action of Υ(ň + X. However, this follows from Proposition Summary of the functional equation. We can now summarize what we have obtained so far regarding the functional equation (our goal 4: For non-compactified Eisenstein series we have the isomorphisms: F Eis Eis! w 0, F Eis int D BunG Eis Eis! D BunT, D BunG Eis int and T Eis ult Eis int! w 0, and Eis ult Eis uld! w 0, Eis int! D BunT, and D BunG Eis ult Eis ult! D BunT, ( Eis ult! Ξ Ω(ň X coinv Ω(ň + X enh. For compactified Eisenstein series for F D reg (Bun T we have: F Eis! (F Eis! w 0 shift(f, where shift is the shift functor on Bun T by 2ρ(ω X. Combined from the real functional equation of [BG1]: Eis! (F Eis! w 0 shift(f for F D reg (Bun T, 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 26 D. GAITSGORY Recall that the functor Eis! : D(Bun T D(Bun G, has a right adjoint, known as the constant term functor CT : D(Bun G D(Bun T, given by F D(Bun G q p! (F, up to a cohomological shift by 2 dim(bun T dim(bun B, the latter being different for each connected component Bun λ B. Therefore, we have a monad Γ := CT Eis! acting on D(Bun T such that there exists a canonical isomorphism Hom D(BunG (Eis! (F 1, Eis! (F 2 Hom D(BunT (F 1, Γ(F Recall, however, that according to [Eis], the functor CT! : D(Bun G! D(Bun T given by q! p is also well-defined, and there is a canonical isomorphism: (7.1 w 0 CT! CT. The latter observation allows to calculate the composition Γ explicitly Consider the Cartesian product Bun B Bun G Bun B. Let p and p denote its projections to Bun B and Bun B, respectively: Bun B Bun G Bun B p p Bun B Bun B q p p q Bun T Bun G Bun T 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 w 0 CT! Eis! (q p! ( p q, up to the cohomological shift by dim(bun B dim(bun B, which again depends on the connected component of Bun B and Bun B.

27 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES Bruhat decomposition defines a decomposition of this stack into locally closed substacks according to the relative position of the two reductions at the generic point of the curve. For w W let ( Bun B Bun B Bun G w denote the corresponding locally closed substack; our conventions are such that w = 1 is the open stratum, which we also denote by Z, the Zastava space. Note also that for w = w 0, we have ( w0 Bun B Bun B Bun B. Bun G ( Let p w and p w denote the projections from Bun B Bun B to Bun B and Bun B, Bun G w respectively. Let r w := q p w and r w := q p w. We obtain that the functor Γ admits a canonical filtration indexed by w, with the w-th subquotient denoted Γ w given by (7.2 Γ w := r w! r w, again up to the cohomological shift by dim(bun B dim(bun B Relation to Ω. Our current goal is to establish the following: Proposition The functor Γ 1 is canonically isomorphic to F Ω(ň X F. Remark. With a little extra work one can show that the resulting map Ω(ň X F Γ(F is a map of monads. The rest of this section is devoted to the proof of this proposition For λ Λ pos let Z λ be the union of connected components of Z equal to ( Bun λ1 B Bun λ2 B. Bun G 1 λ 1 λ 2=λ There exists a natural projection (defect of transversality: such that the composition identifies with r 1. p λ : Z λ X λ, Z λ pλ r 1 X λ mult Bun λ T BunT Hence, by the projection formula, Γ 1 is given by F mult λ! ( (p λ r 1! (C Z λ (π λ id BunT (F [2 λ ], where C Z λ denotes the D-module corresponding to the constant sheaf on Z λ, and 2 λ appears as the difference dim(bun λ2 B dim(bun λ1 B. On the other hand, the functor F Ω(ň X λ F can be written as F mult λ! ( Ω(ň X λ (π λ id X λ (F [ 2 dim(bun T ].

28 28 D. GAITSGORY This is due to the fact that Ω(ň X is ULA with respect to the projection π λ id BunT : X λ Bun T Bun T. Thus, the required isomorphism would follow from the following one: (7.3 (p λ r 1! (C Z λ[2 λ ] Ω(ň X λ [ 2 dim(bun T ] The proof of (7.3 follows from the usual contraction picture for Zastava spaces: Let (7.4 Z λ j Z Z λ be the partial compactification of Z λ, i.e., the open substack ( Bun B Bun B Bun B Bun B. Bun G Bun G The corresponding morphism 1 p λ : Z λ X λ Bun T admits a section, denoted s λ. Moreover, there is a G m -action on Z λ that contracts it on the image of s λ. Hence, (p λ r 1! (C Z λ (p λ r 1! j Z! (C Z λ s λ! j Z! (C Z λ. The assertion follows now from the fact that the pair in (7.4 is smoothly equivalent to the pair and dim(z λ = 2 λ. Bun B j Bun B, 8. The space of rational reductions to B The rest of the paper is devoted to addressing our goal The category What follows is an attempt to realize Drinfeld s idea of the space of G-bundles with a rational reduction to B. Unfortunately, we won t be able to construct the space itself, but rather the category of D-modules on it. Our approach will be very naive: we ll start with Bun B and we ll contract the strata under π λ : X λ Bun B Bun B. ι λ : X λ Bun B Bun B

29 WHAT ACTS ON GEOMETRIC EISENSTEIN SERIES Let H be the closed substack of Bun B Bun G Bun B corresponding to G-bundles with a pair of generalized B-reductions which agree at the generic point of X. This is a groupoid over Bun B, and let p 1, p 2 : H Bun B denote its two projections to Bun B. The maps p i are proper, since Bun B is proper over Bun G. Let D(Bun rat B denote the category of H-equivariant objects on D(Bun B. I.e., it consists of F D(Bun B, endowed with an isomorphism: p! 1(F p! 2(F, which satisfies a natural associativity condition. We have a natural forgetful functor D(Bun rat B D(Bun B, which admits a left adjoint denoted ind Bunrat B and given by Bun B F p 1 p! 2(F. Our current goal is to describe the category D(Bun rat B more explicitly. Note that from the definitions we obtain: Lemma The functor p! factors as D(Bun G prat! D(Bun rat B D(Bun B. By adjunction, the functor p : D(Bun B D(Bun G naturally actors as D(Bun B 8.2. A monoid(al approach. ind Bunrat B Bun B D(Bun rat B prat D(Bun G We define the category D(Bun rat B to consist of the data of F D(Bun B, endowed with isomorphisms α λ : π λ! (F ι λ! (F, for λ Λ pos, that are associative in the sense that for λ = λ 1 + λ 2 the two isomorpisms taking place on X λ1 X λ2 Bun B : and π λ1,λ2! (F (id X λ 1 π λ2! π λ1! (F αλ 1 (id X λ 1 π λ2! ι λ1! (F (id X λ 2 ι λ1! π λ2! (F αλ 2 (id X λ 2 ι λ1! ι λ2! (F ι λ1,λ2! (F, π λ1,λ2! (F (add λ1,λ 2 id BunB! π λ1+λ2! (F αα 1 +α 2 coincide, where (add λ1,λ 2 id BunB! ι λ1+λ2! (F ι λ1,λ2! (F π λ1,λ2 : X λ1 X λ2 Bun B Bun B and ι λ1,λ2 : X λ1 X λ2 Bun B Bun B are the natural maps.