Nearby cycles of Whittaker sheaves on Drinfeld s compactification

Transcription

1 Nearby cycles of Whittaker sheaves on Drinfeld s compactification Justin Campbell September 18, 2017 Abstract In this article we study the perverse sheaf on Drinfeld s compactification obtained by applying the geometric Jacquet functor (alias nearby cycles) to a nondegenerate Whittaker sheaf. Namely, we describe its restrictions along the defect stratification in terms of the Langlands dual Lie algebra, in particular showing that this nearby cycles sheaf is tilting. We also describe the subquotients of the monodromy filtration using the Picard-Lefschetz oscillators introduced by S. Schieder, including a proof that the subquotients are semisimple without reference to any theory of weights. This argument instead relies on the action of the Langlands dual Lie algebra on compactified Eisenstein series (or what is essentially the same, cohomology of quasimaps into the flag variety) constructed by Feigin, Finkelberg, Kuznetsov, and Mirković. 1 Introduction 1.1 In this article we study the degeneration of a Whittaker sheaf on Drinfeld s compactification to an object of the principal series category (for us sheaf will mean D-module, but our results carry over mutatis mutandi to l-adic sheaves in characteristic p > 0 for l p). In [1] this degeneration is implemented by a gluing functor, meaning!-extension from the general locus followed by!- restriction to the special fiber. This operation produces a complex of sheaves, which the authors link to the constant term of Poincaré series. Nearby cycles provide a different, although closely related, method of degeneration. Notably, if we apply nearby cycles (more precisely, the geometric Jacquet functor of [7]) to the perverse cohomological shift of a nondegenerate Whittaker sheaf, the result is still a perverse sheaf. It comes equipped with a nilpotent endomorphism, which gives rise to the so-called monodromy filtration. We describe the restrictions of this nearby cycles sheaf to the defect strata in terms of the Langlands dual Lie algebra, showing in particular that they are perverse, i.e. ours is a geometric construction of a tilting sheaf on Drinfeld s compactification. We also describe the associated graded sheaf of nearby cycles with respect to the monodromy filtration, along with its Lefschetz sl 2 -action, in terms of the Picard-Lefschetz oscillators, which are certain factorizable perverse sheaves with sl 2 -action introduced in [14]. In [5], the author introduced a similar construction in the finite-dimensional situation of a flag variety. Namely, we showed that the nearby cycles of a one-parameter family of nondegenerate Whittaker sheaves on a flag variety is the big projective sheaf, which is isomorphic to the tilting extension of the constant perverse sheaf on the big cell. Thus in both cases taking nearby cycles of Whittaker sheaves produces tilting sheaves. We also remark that both of these constructions have the feature that they degenerate a D-module with irregular singularities to one with regular singularities. 1

2 1.2 Fix an algebraically closed field k of characteristic zero and a smooth connected curve X over k. Let G be a connected reductive group over k. We assume for simplicity that the derived subgroup [G, G] is simply connected. We write I for the set of vertices of the Dynkin diagram of G, and Z G for the center. Fix a Borel subgroup B with unipotent radical N, and put T := B/N. We write D(Y ) for the unbounded derived category of D-modules on an algebraic stack Y and denote by D hol (Y ) the full subcategory consisting of bounded complexes with holonomic cohomologies. The reader is free to view D(Y ) as a triangulated category for most of the paper, with the exception of Section 3, where the DG enhancement of D(Y ) is used. We denote by Bun N ω Drinfeld s compactification of the moduli stack Bun N ω of canonically twisted N-bundles. A choice of isomorphism N/[N, N] = G I a gives rise to a map ev : Bun N ω G a, constructed for example in Section 4.1 of [9]. Let χ be a nontrivial exponential D-module on G a. As in [9], one shows that ev χ extends cleanly to a perverse sheaf W 1 on Bun N ω, where ev denotes cohomologically normalized inverse image along the smooth morphism ev. The sheaf W 1 has irregular singularities because χ does, so by perverse sheaf we mean any holonomic D-module. Choose a dominant regular cocharacter γ : G m T, which determines an action a γ : G m Bun N ω Bun N ω. We will denote by W the perverse sheaf a γ W 1 on G m Bun N ω. Consider the embeddings Bun N ω i A 1 Bun N ω j G m Bun N ω. The G m -equivariant object i! j! W of D(Bun N ω) is studied in [1]. We will consider instead a closely related perverse sheaf on Bun N ω, namely the nearby cycles Ψ(W ) with respect to the projection A 1 Bun N ω A 1. This sheaf is unipotently G m -monodromic, and (the logarithm of) the monodromy endomorphism of nearby cycles agrees with the obstruction to G m -equivariance, a nilpotent endomorphism. The composition Ψ a γ for a G m -equivariant oen-parameter family is studied in [7] under the name geometric Jacquet functor. The object i! j! W can be recovered from Ψ(W ) as the derived invariants of monodromy, and conversely Ψ(W ) is the derived coinvariants of the natural H (G m )-action on i! j! W (for the latter see Definition and Proposition 3.2.2). One can summarize by saying that nearby cycles and the gluing functor i! j! are related by Koszul duality. This motivates the study of Ψ(W ), since these gluing functors (in the extended Whittaker situation) play a role in the proof of the geometric Langlands equivalence sketched in [10]. Nearby cycles has the advantage of being t-exact, and therefore Ψ(W ) is amenable to more concrete description, e.g. it has a composition series which is finite over any quasi-compact open substack of Bun N ω. We also remark that up to isomorphism Ψ(W ) does not depend on the choice of χ. In particular Ψ(W ) is Verdier self-dual, since Ψ commutes with Verdier duality and the Verdier dual of W is the clean extension of ev χ The stack Bun N ω has a stratification by defect of the generalized N-bundle. Our first main theorem is a description of the restrictions of Ψ(W ) to the strata. In particular we will prove that this sheaf is tilting with respect to the defect stratification, meaning its!- and -restrictions to the strata are perverse (although generally not lisse). 2

3 Write Λ for the lattice of cocharacters of T, and denote by Λ pos Λ the positive coweights with respect to B. The defect stratification of Bun N ω is indexed by Λ pos, and for each µ Λ pos we denote the locally closed stratum embedding by j =µ : Bun N ω,=µ Bun N ω. If µ = i I n iα i for some nonnegative integers n i (here α i is the simple coroot corresponding to i) then the corresponding configuration space of points in X is defined by X µ := i I X (n i), and there is a smooth surjection m µ : Bun N ω,=µ X µ. In [4] the authors introduced certain factorizable perverse sheaves Ω µ on the configuration spaces X µ. They can be characterized as follows: the!-fiber of Ω µ at a point i µ ix i is C (ň) µ i, i where ň is the unipotent radical of a Borel subalgebra in the Langlands dual Lie algebra ǧ, and C (ň) is its Ť -graded cohomological Chevalley complex. Theorem For any µ Λ pos there is an isomorphism j! =µψ(w ) m µ Ω µ. In fact, Ω µ is indecomposable and m µ has contractible fibers, so the isomorphism in the theorem is automatically unique up to scaling. Since Ψ(W ) is Verdier self-dual, Theorem implies that it is tilting with respect to the defect stratification. 1.4 Recall that the monodromy filtration on Ψ(W ) is the unique filtration by perverse sheaves F m F m 1 F m = Ψ(W ) such that for all 1 i n, the i th power of the monodromy endomorphism induces an isomorphism F i /F i 1 F i /F i 1. The associated graded sheaf gr Ψ(W ) has an action of the so-called Lefschetz sl 2 such that the lowering operator is induced by the monodromy endomorphism and F i /F i 1 has weight i for the Cartan operator. We now formulate a description of gr Ψ(W ) in terms of certain factorizable perverse sheaves with sl 2 -action, called the Picard-Lefschetz oscillators after [14] and [15] (see also Section 3.2 of loc. cit.). First let us define the Picard-Lefschetz oscillators on X (n). Let std denote the standard 2- dimensional representation of the Lefschetz sl 2, and write sgn for the sign character of the symmetric group Σ n. Now the Σ n sl 2 -representation sgn std n (here Σ n also permutes the std factors) and the Σ n -torsor Xdisj n X(n) disj give rise to a local system on X(n) disj with sl 2-action. Then P n is defined as the intermediate (i.e. Goresky-MacPherson) extension to X (n) of the perverse cohomological shift of this local system. The perverse sheaf P n carries an sl 2 -action by functoriality, and is evidently semisimple. Recall that a Kostant partition of µ Λ pos is an expression of the form µ = β R + n ββ, where the n β are nonnegative integers and R + denotes the set of positive coroots. Denote by Kost(µ) the 3

4 set of Kostant partitions of µ. To any k Kost(µ) given by µ = β R + n ββ we attach a partially symmetrized power X k := β R + X (nβ), which is equipped with a canonical finite map ι k : X k X µ. The Picard-Lefschetz oscillator on X µ is defined by the formula P µ := ι k ( P n β R + β ). k Kost(µ) In particular, we have P nα = P n for α a simple coroot. By construction, P µ is a semisimple perverse sheaf with sl 2 -action. Theorem There is an sl 2 -equivariant isomorphism gr Ψ(W ) j =µ,! m µ P µ. (1.4.1) µ Λ pos In particular gr Ψ(W ) is semisimple. Using the theorem we can compute the kernel of the monodromy operator, whose simple subquotients are the lowest weight sheaves for the Lefschetz sl 2 -action. Corollary The canonical morphism j =0,! IC BunN ω kernel of the monodromy operator. Ψ(W ) is an isomorphism onto the 1.5 A key step in the proof of Theorem is establishing that gr Ψ(W ) is semisimple. This can be shown using Mochizuki s theory of weights for holonomic D-modules, developed in [12]. We also present a proof of Theorem which does not use weights for irregular D-modules, but instead relies Proposition 4.9 in [8]. We now state a conjecture containing this proposition as a special case, which we hope to return to in future work. In particular we will explain how this conjecture can be deduced from the (conjectural) geometric Langlands equivalence and its postulated compatibility with compactified Eisenstein series. Write p : Bun B Bun G and q : Bun B Bun T for the canonical morphisms. The functor of compactified Eisenstein series Eis! : D(Bun T ) D(Bun G ), introduced in [3], is defined by Eis! = p (IC BunB! q! ( )). Recall that geometric class field theory assigns to a Ť -local system E Ť on X a character sheaf (i.e. multiplicative line bundle with connection) L (EŤ ) on Bun T. We write ǧ EŤ for the local system on X attached to EŤ and the Ť -representation ǧ. Conjecture The DG Lie algebra H (X, ǧ EŤ ) acts on Eis! L (EŤ ). For the trivial Ť -local system, which is the only case we will use, the conjecture says that ǧ H (X) acts on p IC BunB. In Section 6.3 we specify the action of certain generators of ǧ H (X). In [8] the authors verify the necessary relations for ǧ H 0 (X), which suffices for our application. The derivedness of the Lie algebra ǧ H (X), or more generally H (X, ǧ EŤ ), makes this checking of relations difficult to do by hand, and in any case a more conceptual approach to the conjecture is desirable. 4

5 1.6 Acknowledgments: I thank my doctoral advisor Dennis Gaitsgory for suggesting that I study this particular nearby cycles sheaf, and for many invaluable conversations full of technical assistance and motivation. The ways in which he has helped and guided me through this project are too numerous to list, but I am especially grateful to him for explaining the statement of Conjecture and its relevance to the semisimplicity assertion of Theorem I thank Sam Raskin for many illuminating discussions about this material, which included helping me to formulate Lemma and explaining Definition I am grateful to Simon Schieder for explaining to me the definition of the Picard-Lefschetz oscillators for an arbitrary reductive group. 2 Drinfeld compactifications and Zastava spaces 2.1 Let 2ρ : G m T denote the sum of the simple coroots and fix a square root ω 1 2 X define ρ(ω X ) := 2ρ(ω 1 2 X ) Bun T (k). By definition, Bun N ω is the fiber product of ω X. We Bun N ω Bun B ρ(ω X ) Spec k Bun T. The T -bundle ρ(ω X ) and the action of T on N give rise to a group scheme N ω over X, and the open stratum Bun N ω Bun N ω is identified with the moduli stack of N ω -bundles, as the notation suggests. By construction, T acts on Bun N ω in such a way that Bun N ω Bun B factors through a closed embedding Bun N ω/t Bun B. For each µ Λ pos the corresponding stratum Bun N ω,=µ fits into a fiber square Bun N ω,=µ Bun B X µ Bun T, where the lower horizontal morphism is the twisted Abel-Jacobi map D ρ(ω X )(D). We write Bun N ω, µ := µ µ Bun N ω,=µ and j µ for the corresponding open embedding. The embedding j =µ of the stratum, which is known to be affine, extends to a finite map where Bun N ω, µ := X µ BunT Bun B. j µ : Bun N ω, µ Bun N ω, 5

6 2.2 Now we introduce the Zastava spaces, which are factorizable local models for Bun N ω. There are several versions of Zastava space, and notations vary significantly within the literature (ours is similar to [1]). Define the Zastava space Z to be the open locus in Bun N ω BunG Bun B where the generalized N-reduction and B -reduction are transverse generically on X. It is well-known that Z is a scheme, with connected components Z λ := Z (Bun N ω BunG Bun λ+deg ρ(ω X) B ) indexed by λ Λ pos. The map p : Bun B Bun G gives rise to p : Z Bun N ω. It is shown in [3] that p,λ and therefore p,λ are smooth for λ sufficiently dominant. Moreover, given a quasicompact open U Bun N ω, for λ sufficiently dominant the image of Z λ Bun N ω contains U. For each 0 µ λ we have the corresponding stratum Z=µ λ := Z λ Bun BunN ω N ω,=µ with locally closed embedding j λ =µ : Z=µ λ Z λ. Define Z λ := Z λ =0. Similarly, we have Zλ µ := Zλ BunN ω Bun N ω, µ with the open embedding j λ µ : Z λ µ Z λ. Put Z λ µ := Z Bun N ω Bun N ω, µ, so that j λ =µ extends to the finite map j λ µ : Z λ µ Z λ. 2.3 Let us recall the factorization structure on Z. For every λ Λ pos there is a canonical map π λ : Z λ X λ, which is well-known to be affine. We write (X λ 1 X λ 2 ) disj X λ 1 X λ 2 for the open locus where the two divisors are disjoint. Similarly, put (Z λ 1 Z λ 2 ) disj := (Z λ 1 Z λ 2 ) X λ 1 X λ 2 (X λ 1 X λ 2 ) disj. The factorization structure is a canonical morphism (Z λ 1 Z λ 2 ) disj Z λ 1+λ 2 fiber square (Z λ 1 Z λ 2 ) disj Z λ 1+λ 2 which fits into a (X λ 1 X λ 2 ) disj X λ 1+λ 2. The factorization structure is compatible with the defect stratification in the following sense. The factorization structure on the strata consists of, for each decomposition µ 1 + µ 2 = µ satisfying 0 µ 1 λ 1 and 0 µ 2 λ 2, a morphism (Z λ 1 =µ 1 Z λ 2 =µ 2 ) disj Z λ 1+λ 2 =µ which fits into a fiber square µ 1 +µ 2 =µ (Zλ 1 =µ 1 Z λ 2 =µ 2 ) disj Z λ 1+λ 2 =µ (X λ 1 X λ 2 ) disj X λ 1+λ 2. One has similar factorization structures on Z µ λ and Zλ µ. Moreover, these factorization structures are compatible with j λ =µ, j λ µ, etc. 6

7 2.4 We will also need the compactified Zastava space Z, which is the open locus in Bun N ω BunG Bun B where the generalized N- and B -reductions are generically transverse. In particular there is an open embedding j : Z Z obtained from j : Bun B Bun B by base change. For any ν Λ pos we put =νz := Z Bun BunB B,=ν, and similarly for ν Z and ν Z. The projections π λ extend to proper morphisms π λ : Z λ X λ. The factorization structure on Z extends to Z in a way compatible with both defect stratifications on Z. 3 Nearby cycles and adelic invariance 3.1 We now reinterpret Beilinson s construction of nearby cycles from [2], which will help to streamline our first proof of Theorem In this section we use the language of DG categories in the sense of Lurie, meaning k-linear stable -categories: see Chapter I.1 in [11] for a summary of the theory. Let Y be a scheme of finite type equipped with an action of G m and a G m -equivariant morphism f : Y A 1. Write Y 0 for the fiber of Y over 0 and Y for the preimage of G m A 1. We temporarily denote the embeddings by Y 0 i Y j Y. The projection Y G m gives rise to an action of D(G m ), viewed as a symmetric monoidal DG category under!-tensor product, on the DG category D( Y ). We claim that this naturally induces an action of the DG algebra H (G m ) on the identity functor of D( Y ). Namely, we view the action of D(G m ) on D( Y ) as a monoidal functor D(G m ) Fun(D( Y ), D( Y )). Since H (G m ) is the algebra of endomorphisms of the monoidal unit in D(G m ), this monoidal functor induces the desired action. In particular, for any holonomic D-module F on Y, we have an action of H (G m ) on i! j! F by functoriality. The point 1 G m induces an augmentation H (G m ) k, and since H (G m ) is generated by a single element in cohomological degree 1, the endomorphism algebra End H (G m)-mod(k) is canonically isomorphic to the polynomial ring k[t]. Here H (G m )-mod is the DG category of modules over the DG algebra H (G m ). Definition We define unipotent nearby cycles with respect to f to be the functor given on objects by the formula Ψ : D hol ( Y ) D(Y 0 ) Ψ(F ) = k i! j! F. H (G m) In particular End H (G m)-mod(k) = k[t] acts on Ψ, and we call the action of the generator the monodromy endomorphism of Ψ. More precisely, the action of H (G m ) on the identity functor of D( Y ) defines a lift of the latter to a functor D( Y ) H (G m )-mod(d( Y )), 7

8 and likewise on the holonomic subcategory. Then Ψ is by definition the composition D hol ( Y ) H (G m )-mod(d hol ( Y )) i! j! H (G m )-mod(d(y 0 )) D(Y 0 ), where the last functor is tensor product with the augmentation H (G m )-module. Since H (G m ) is isomorphic to Sym(k[ 1]), an action of H (G m ) is the same as an action of the abelian DG Lie algebra k[ 1]. Tensor product with the augmentation H (G m )-module corresponds to (homotopy) k[ 1]-coinvariants. 3.2 The following lemma will be used to compare our construction of Ψ with Beilinson s. It says that the homotopy coinvariants of H 1 (G m )[ 1] acting on the constant sheaf k Gm is the infinite Jordan block. For any a 1 let L a be the shifted D-module on G m corresponding to the local system whose monodromy is a unipotent Jordan block of rank a. There are canonical injections L a L a+1 (in the heart of the constructible t-structure), and we put L := colim a 1 Observe that L has a canonical injective shift endomorphism with cokernel k Gm. Lemma There is a canonical isomorphism k H (G m) k Gm L a. L in D(G m ), which identifies the action of the generator in k[t] = End H (G m)-mod(k) with the canonical endomorphism of L. Proof. First, observe that since any self-extension of L splits, the action of H 1 (G m )[ 1] on L is trivial. Thus the canonical inclusion k Gm = L 1 L factors through a map k k Gm L. (3.2.1) H (G m) To prove that this map is an isomorphism, we use a Koszul-type resolution of the augmentation H (G m )-module. Namely, put M 1 := H (G m ), and let M 2 := cofib(h (G m )[ 1] H (G m )) be the cofiber (i.e. mapping cone) of the action of a generator of H 1 (G m ). Proceeding inductively, we define M a+1 := cofib(h (G m )[ a] M a ), so there are canonical maps M a M a+1. We define M to be the colimit of the M a. Since the module M a has cohomology k k[ a], the canonical map M k to the augmentation H (G m )-module is an isomorphism. In particular k[t] acts on M, and it is not hard to see that the generator acts by the canonical shift map M M with cofiber M 1. Thus we may replace the left hand side of (3.2.1) with M k Gm. A straightforward inductive argument shows that M a k Gm H (G m) H (G m) maps isomorphically onto L a, which proves that (3.2.1) is an isomorphism which moreover preserves the filtrations on both sides. The action of k[t] on both sides agrees by inspection: both endomorphisms shift the filtration by 1. 8

9 Proposition The functor Ψ has the following properties. (i) It coincides with the construction in [2] and in particular preserves holonomicity. (ii) If F is G m -equivariant, then Ψ(F ) is unipotently G m -monodromic, and the monodromy endomorphism agrees with the obstruction to G m -equivariance. Proof. Let F be a holonomic D-module on Y. Then Beilinson s nearby cycles is given by the formula Ψ(F ) := colim i! j! (F f L a ). a 1 Moreover, we have Ψ(F ) = H 0 i! j! (F f L a ) for large a, which is evidently holonomic. The functor D hol (G m ) D(Y 0 ) given on objects by M i! j! (F f M ) admits a unique colimit-preserving extension to the category of ind-holonomic sheaves on G m, so one can write Ψ(F ) = i! j! (F f L ). The functor Ψ carries a canonical endomorphism induced by the shift endomorphism L L. Now it follows from Lemma that Ψ(F ) Ψ(F ), preserving the monodromy endomorphisms on both sides. For (ii), note that the G m -equivariance of i! j! F implies that Ψ(F ) = H 0 i! j! (F f L a ) is G m -monodromic (here a is large). By construction, the monodromy endomorphism is induced by the canonical endomorphism of L a with one-dimensional kernel and cokernel. But the latter is precisely the obstruction to G m -equivariance for L a, so the claim follows from the functoriality of this obstruction. It follows from part (i) of Proposition that Ψ enjoys the standard properties of the unipotent nearby cycles functor: it is t-exact, commutes with Verdier duality, and commutes with proper direct image and smooth inverse image. 3.3 Before proving Theorem 1.3.1, we will show that j! =µψ(w ) is pulled back from X µ for any µ Λ pos. This property is equivalent to invariance under the adelic N ω, as we now explain. For any x X, we define the open substack Bun x N ω Bun N ω to consist of those generalized N ω -bundles whose defect is disjoint from x. A point of the ind-algebraic stack HN x ω consists of two points of Bun x N ω together with an identification over X \ {x}. Note that H N x ω has the structure of a groupoid acting on Bun x N ω. The fibers of H N x ω over Bunx N ω Bunx N ω are isomorphic to ind-affine space colim n A n, which implies that the functor which forgets HN x ω-equivariance is fully faithful, i.e. HN x ω-equivariance is a property. We say that an object of D(Bun N ω) is N ω (A)-equivariant if, for every x X, its restriction to Bun x N ω is H N x ω-equivariant. Proposition An object F of D(Bun N ω) is N ω (A)-equivariant if and only if, for every µ Λ pos, the canonical morphism is an isomorphism. m µm µ, j! =µf j! =µf 9

10 For each x X, denote by O x the completed local ring of X at x, with fraction field K x. If R is a k-algebra, we denote by R ˆ O x and R ˆ K x the respective completed tensor products. The local Hecke stack H loc,x N is defined as follows: a Spec R-point of H loc,x ω N consists of two N ω - ω bundles over Spec(R ˆ O x ) equipped with an isomorphism over Spec(R ˆ K x ). There is a natural restriction map res x : HN x ω H loc,x N. ω Our choice of isomorphism N/[N, N] = G I a The projection N [N, N] induces a morphism induces a map H loc,x N ω G a in the following way. H loc,x N ω I H loc,x G, ω a where G ω a is the additive group scheme over X attached to the canonical line bundle ω X. Note that there is a canonical isomorphism of groupoids H loc,x G ω a so taking residues defines a morphism rsd x : H loc,x G ω a G ω a (O x )\G ω a (K x )/G ω a (O x ), G a. The composition rsd x ψ : H x N ω res x H loc,x N ω I H loc,x G ω a rsd x I G a add G a is an additive character, meaning it is a morphism of groupoids. It follows that χ x := rsd x,! ψ χ is a character sheaf on H N x ω, i.e. it is multiplicative for the groupoid structure. In particular we can speak of (HN x ω, χx )-equivariant sheaves on Bun x N ω, which form a full subcategory of D(Bun x N ω). Likewise, if a sheaf on Bun N ω is (H N x ω, χx )-equivariant for all x X we say that it is (N ω (A), χ)-equivariant. Although we will not use this fact, the category of (N ω (A), χ)-equivariant sheaves on Bun N ω is equivalent to the derived category of vector spaces, being generated by the object W 1 introduced above. Moreover (HN x ω, χx )-equivariance for a single x X implies (N ω (A), χ)-equivariance. Observe that there is a natural T -action on H loc,x N. Using the chosen dominant regular cocharacter γ : G m T, the resulting G m -action on H loc,x N ω is contracting. In particular, it extends to ω an action A 1 H loc,x N H loc,x ω N of the multiplicative monoid A 1. The!-pullback of χ along the ω composition A 1 HN x id A 1 res x ω A 1 H loc,x N H loc,x ω N G ω a defines an A 1 -family χ x ext of character sheaves on HN x ω. Its!-restriction to {1} H N x ω is χ x, and it is trivial along {0} HN x ω. Lemma The sheaf Ψ(W ) is N ω (A)-equivariant. Proof. Fix x X; we omit restriction to Bun x N ω from the notation in what follows. By construction W is χ x ext Gm H N x -equivariant. Since ω χx ext! {0} HN x is the trivial character sheaf, it follows from our construction of Ψ (or equivalently, Beilinson s) ω that Ψ(W ) is HN x ω-equivariant as desired. 4 Restriction to the strata 4.1 Now we give the first proof of Theorem by deducing it from Theorem in [1], which describes the restrictions to the strata of i!j! W in terms of the perverse sheaf Ω. Since we work 10

11 with a fixed dominant regular coweight γ rather than the entire torus T, it will be necessary to prove a slightly different formulation of the latter theorem. As in Section 7.2 of [1], the inclusion D(Bun N ω) N ω (A) D(Bun N ω) of N ω (A)-equivariant sheaves on Bun N ω admits a right adjoint, which we denote by Av N ω (A) : D(Bun N ω) D(Bun N ω) N ω (A). Let i and j be as in Section 1.2, and write p : A 1 Bun N ω Bun N ω for the projection. We will also abusively denote p j by p. Proposition There is a canonical isomorphism i! j! W Av N ω (A) p! W. Proof. See Section 10.3 of [1], where the claim is proved for the action of the entire torus T. The same proof applies mutatis mutandi to our claim, which involves only the G m -action. First proof of Theorem Since j! =µ preserves (homotopy) colimits, we see that By Proposition we have and Proposition implies that Now Theorem of [1] yields j! =µψ(w ) k j! =µi! j! W. H (G m) j! =µi! j! W j! =µ Av N ω (A) p! W, j! =µ Av N ω (A) p! W m µm µ, j! =µp! W. m µm µ, j! =µp! W It remains to show that under the composed isomorphism j! =µi! j! W m µ Ω µ H c (G m )[1]. (4.1.1) m µ Ω µ H c (G m )[1], the action of H (G m ) on the left hand side corresponds to the natural action on H c (G m ) on the right hand side. Since H c (G m ) = H (G m )[ 1] as H (G m )-modules, this will finish the proof. It is clear that j! =µi! j! W m µm µ, j! =µp! W intertwines the actions of H (G m ), since it is obtained by evaluating a morphism of functors on W. Tracing through the proof of Theorem in [1], we see that the isomorphism (4.1.1) is also obtained by evaluating a morphism of functors on W, with the appearance of H c (G m ) accounted for by the isomorphism a γ,! W W 1 H (G m )[ 1] W 1 H c (G m )[1]. The latter isomorphism intertwines the actions of H (G m ) as needed. 11

12 4.2 The rest of this subsection is devoted to the second proof of Theorem This proof applies Theorem in [13], which says that Ω can be realized as the twisted cohomology of Zastava space. Accordingly, we must formulate the analogue of Theorem on Zastava space. First, the Whittaker sheaf: we claim that W Z λ := (id Gm p,λ )! W [dim Bun N ω dim Z λ ] is perverse for any λ. For λ sufficiently dominant p,λ is smooth, so that W Z λ is the cohomologically normalized inverse image of the perverse sheaf W. If λ λ then we can pull back W Z λ along id Gm times the factorization morphism (Z λ Z λ λ ) disj Z λ, and it is not hard to see that we obtain the restriction of W Z λ W Zλ λ. Since the factorization map is étale, this implies that W Z λ is perverse as desired. Since the map Bun N ω Bun G is T -equivariant for the trivial action of T on Bun G, we obtain an action of T on Z which makes p a T -equivariant map. As with W, we also denote by W Z λ the corresponding perverse sheaves on Z λ /Z G and ch Z λ. In particular we have the perverse sheaf Ψ(W Z λ) on Z λ. Recall that in the introduction we chose a dominant regular cocharacter γ : G m T. Denote by a γ : G m Z Z the resulting action morphism. Then Ψ( a γ W Z λ) is canonically isomorphic to the previously constructed perverse sheaf Ψ(W Z λ), where in the former expression nearby cycles is taken with respect to the projection A 1 Z λ A 1. In particular, the choice of γ gives rise to a distinguished monodromy endomorphism for Ψ(W Z λ). Theorem For any µ Λ pos there is an isomorphism j! =µψ(w Z ) m µ Ω µ. We will need to use the factorization structure on Z in the following way. First, observe that W Z admits a natural factorization structure. Thus Ψ(W Z ) admits a factorization structure by the Künneth formula for nearby cycles. Although the Künneth formula holds for the total nearby cycles functor, in this case the total nearby cycles equals the unipotent nearby cycles because W Z is G m -equivariant. 4.3 In the second proof, Theorems and will be proved simultaneously by an inductive argument. The argument uses the following key lemma. Let f : X Y be a morphism of Artin stacks with Y smooth, and suppose we are given a function Y A 1. Let g : S Y be a morphism where S is an affine scheme and consider the cartesian square X Y S f g g X f S Y. Write i : S 0 S for the inclusion of the vanishing locus of the function S Y A 1. Lemma For any F D(X ) which is ULA over Y and any G D(S), there is a canonical isomorphism Ψ( g! (F )! f! (G )) g! (F )! f! (i! Ψ(G )). 12

13 Fix ν Λ pos. We will apply the lemma in the case X = A 1 Bun λ B, ν, F = IC A 1 j! (IC Bun λ ) B Bun λ, B, ν and Y = A 1 Bun G. Let us check that the ULA property holds when λ is sufficiently dominant relative to ν. According to Corollary 4.5 in [4], we have the following decomposition in the Grothendieck group: [j! (IC ) BunB Bun λ ] = B, ν η ν[j η,! (Ωη IC λ η Bun )]. B, ν η Observe that the diagram X η Bun λ η B j η Bunλ B Bun λ η B Bun G, commutes, where the left vertical arrow is projection onto the second factor. Since j η is proper it suffices to prove that, for λ sufficiently dominant, IC λ η Bun is ULA over Bun G for all η ν. B, ν η This follows from Corollary in [6]. Second proof of Theorems and Observe that Theorem is trivial on the open stratum, since Ψ(W ) BunN ω = Ψ(W Gm Bun N ω ) = Ψ(IC Gm IC BunN ω ) = IC BunN ω, and similarly for Theorem on Z. We begin by proving Theorem for the deepest strata, i.e. the closed embeddings j µ =µ : X µ Z µ. Recall that Ψ(W Z ) is G m -monodromic by construction, so the contraction principle says that j µ,! =µψ(w Z ) = π µ! Ψ(W Z). Write π µ := π µ j µ =0. Theorem in [13] implies that there is an isomorphism (id Gm π)! (W Z ) IC Gm Z G m Ω, compatible with the factorization structures. Since W Z is!-extended from G m Z, we obtain (id Gm π)! W Z IC Gm Ω. Since π = π j and Ψ commutes with proper pushforwards, we have π! Ψ((id Gm j )! W Z ) Ω. Therefore it suffices to prove that the canonical morphism j! Ψ(W Z ) Ψ((id Gm j )! W Z ) is an isomorphism. Since Ψ commutes with Verdier duality we can replace the!-pushforwards with -pushforwards. Fix S Bun N ω with S an affine scheme and apply Lemma with f = id A 1 p, g : A 1 S A 1 Bun N ω A 1 Bun G, 13

14 F = IC A 1 j (ω ) BunB Bun λ, and G = W! B A, ν 1 S. Then the lemma yields an isomorphism Ψ((id Gm j ) W Z ) νz λ j Ψ(W Z ) νz λ for λ sufficiently dominant. Changing λ if necessary so that λ µ, we can restrict this isomorphism along the map ( ν Z µ Z λ µ ) disj ν Z λ. By factorizability we obtain the desired isomorphism on ν Z µ. Since ν was arbitrary, Theorem is proved for the deepest strata. Now we prove Theorem Fix µ Λ pos and choose λ µ dominant enough that Z λ =µ surjects smoothly onto X µ BunT Bun B. Note that (X µ Z λ µ ) disj is one of the connected components of the fiber product (X µ X λ µ ) disj X λ Z λ =µ, and that the former surjects onto Bun N ω,=µ. Theorem for the deepest and open strata implies that the cohomologically normalized pullback of j! =µψ(w ) to (X µ Z λ µ ) disj is Ω µ IC Zλ µ. Theorem follows once we observe that the composition (X µ Z λ µ ) disj Bun N ω,=µ m µ X µ is the projection onto the first factor and apply Lemma The previous paragraph implies Theorem holds on the stratum Z=µ. λ change λ if necessary so that λ λ. By restricting along the morphism Let λ µ and (Z λ =µ Z λ λ ) disj Z λ =µ and invoking factorization, we obtain Theorem The remainder of Theorems and follows as in the first proof. 5 First proof of Theorem Like Theorem 1.3.1, we formulate the analogue of Theorem on Zastava space. Theorem For any λ Λ pos, there is an sl 2 -equivariant isomorphism of factorizable sheaves gr Ψ(W Z λ) j λ =µ,! m λ, µ P µ. (5.1.1) 0 µ λ Now we work out three of the simplest cases of Theorem Ψ := Ψ(W Z λ). For brevity, we will write Example Let α be a simple coroot. There is an isomorphism Z α = X A 1 under which Z α = X Å1, where Å1 := A 1 \ {0}. The canonical map Z α A 1 is given in these terms by (x, t) 1 t. It follows from Example 4.3 of [5] that Ψ is the cohomologically normalized pullback of the unique indecomposable tilting sheaf on A 1 which extends ICÅ1. Moreover, the monodromy filtration F 1 F 0 F 1 = Ψ satisfies F 1 = ICZ α =α, F 0 /F 1 = ICZ α, and F 1 /F 0 = ICZ α =α, where Z α =α = X {0} X A 1 = Z α. This confirms Theorem in the case λ = α. 14

15 Example Now consider the case λ = 2α. By Theorem for µ = 2α, we have a short exact sequence Ω 2α Ψ j 2α α, j 2α, α Ψ (recall that Ω 2α is the clean extension of the sign local system on X (2) Theorem for µ = α and 0 we obtain an exact triangle j 2α =α, IC Z 2α =α j 2α α, j 2α, α Ψ j 2α =0, IC Z2α, disj ). Similarly, applying where we used the fact that Ω α = IC X. Applying Verdier duality to the equation in Corollary 4.5 of [4] (or rather the analoguous equation on Zastava space), we have [ j 2α =0, IC Z2α] = [IC Z 2α] + [IC Z 2α =α ] + [Ω 2α ], where we identified Ω 2α with its Verdier dual. Finally, one computes the simple constituents of j 2α =α, IC Z 2α =α as follows: first, consider the short exact sequence IC Z 2α j 2α =α =α, IC Z 2α =α j 2α,! =2α IC [1]. Z=α 2α To compute the third term, observe that there is a Cartesian square X 2 Z 2α α j 2α j 2α α X (2) =2α Z 2α, and that the!-restriction of IC Z 2α along the top horizontal morphism is IC α X 2[ 1]. Since j 2α α is finite and birational onto its image, we can use base change to compute Summarizing, we have j 2α,! =2α IC [1] = j 2α,! Z=α 2α =2α j 2α α, IC Z 2α [1] = IC α Z 2α =2α Ω 2α. [Ψ] = [IC Z 2α] + 2[IC Z 2α =α ] + [IC Z 2α =2α ] + 3[Ω 2α ]. Now we will determine which graded component of gr Ψ each simple subquotient lies in. In what follows, weight refers to an eigenvalue of the Lefschetz Cartan operator. Since the monodromy filtration is compatible with the factorization structure, when we pull back gr Ψ along the factorization map (Z α Z α ) disj Z 2α we get (gr Ψ(W Z α)) 2 restricted to (Z α Z α ) disj. By the previous example, the latter sheaf with sl 2 -action is isomorphic to (std 2 IC X 2) (std IC X Z α) (std IC Z α X) (triv IC Z α Zα). (5.1.2) It follows immediately that IC Z 2α has weight 0 and that the two copies of IC Z 2α have weights ±1. =α Since std 2 = V2 triv, the three copies of Ω 2α have weights 2, 0, and 2, and IC Z 2α =2α has weight 0. In terms of the monodromy filtration F 2 F 1 F 0 F 1 F 2 = Ψ, 15

16 we have F 2 = Ω 2α = F2 /F 0, F 1 /F 2 = ICZ 2α = F1 /F 0, and F 0 /F 1 has simple constituents =α IC Z 2α, IC Z 2α =2α, and Ω 2α. So in order to prove Theorem in the case λ = 2α, it remains to show that F 0 /F 1 is semisimple. Its pullback along the factorization map is semisimple, and since semisimplicity is étale local, the restriction of F 0 /F 1 to Z 2α \ π 1 (X) is semisimple. But Ψ has no simple subquotients supported on π 1 (X), so F 0 /F 1 is the intermediate extension of its restriction to Z 2α \ π 1 (X) and therefore semisimple. Example Suppose λ = α + β is a coroot, where α and β are distinct simple coroots. Applying Theorem for µ = α + β, we obtain the short exact sequence Ω λ Ψ j λ <λ, j λ, <λ Ψ. Similar considerations yield the short exact sequence j λ =α, IC Z λ =α j λ =β, IC Z λ =β Applying Corollary 4.5 of [4], we have j λ <λ, j λ, <λ Ψ j λ =0, IC Zλ. [ j λ =0, IC Zλ] = [IC Z λ] + [IC Z λ =α ] + [IC Z λ =β ] + [Υ λ ], where Υ λ is the Verdier dual of Ω λ. According to Section in [1], in this case Ω λ is the -extension of IC X 2 to X λ = X 2, whence Υ λ is the!-extension. In particular [Ω λ ] = [Υ λ ] = disj [IC X 2] + [IC X ]. As for the remaining simple constituents, consider the short exact sequence IC Z λ =α j λ =α, IC Z λ =α j λ,! =α IC Z λ =α [1], and similarly for β. The third term is IC X 2, so finally we see that [Ψ] = [IC Z λ] + 2[IC Z λ =α ] + 2[IC Z λ =β ] + 4[IC X 2] + 2[IC X ]. Now we compute the weights of the simple subquotients of Ψ. morphism (Z α Z β ) disj Z α+β, We have the factorization and after pulling back gr Ψ the result is (5.1.2), up to relabeling β as α. As in Example 5.1.3, it follows that IC Z α+β has weight 0, the two copies of IC Z λ =α have weights ±1 and likewise for IC Z λ =β, and the four copies of IC X 2 have weights 2, 0, 0, and 2. We will see below that in any case where λ is a coroot, there are two simple subquotients of Ψ isomorphic to IC X, with weights ±1. In terms of the monodromy filtration, we have F 2 = ICX 2 = F 2 /F 1, F 1 /F 2 and F 1 /F 0 each have simple subquotients IC Z λ, IC =α Z λ, and IC X, and F 0 /F 1 has simple subquotients IC Z λ and IC X 2, the =β latter with multiplicity two. As in Example 5.1.3, one uses factorization to show that F 0 /F 1 is semisimple. To prove the semisimplicity of F 1 /F 2 and F 1 /F 0, it is enough to show that there are no extensions between IC X and IC Z λ =α. By Verdier duality, it suffices to prove that Ext 1 D(Z λ ) (IC X, IC Z λ =α ) = 0. We have j λ,! =λ IC Z λ =α = IC X 2[ 1], whence! IC Z λ =α = IC X [ 2]. Thus and in particular Ext 1 vanishes. RHom D(Z λ )(IC X, IC Z λ =α ) = H (X)[ 2], 16

17 5.2 The following lemma will be used in both proofs of Theorems and For any µ Λ pos write µ : X BunT Bun B Bun N ω for the finite birational map defined as the composition of j µ and the embedding induced by the diagonal map X X µ. X BunT Bun B = X X µ Bun N ω, µ Bun N ω, µ Lemma If µ is a coroot, then there is an indecomposable subquotient M of Ψ(W ) with a filtration M 1 M 0 M 1 = M such that M 1 = µ IC X BunT Bun B, M 0 /M 1 = ICBunN ω, and M /M 0 = µ IC X BunT Bun B. Proof. First, we claim the subsheaf j =0,! IC BunN ω of Ψ(W ) has a quotient M 0 of the form described above. Recall that j =0,! IC BunN ω has a descending filtration with subquotients j =ν,! r ν Ω ν, and in particular has j =µ,! Ω µ as a subquotient and IC BunN ω as a quotient. The former sheaf has µ IC X BunT Bun B as a quotient because µ is a coroot, so it suffices to show that that for any 0 < ν < µ we have Ext 1 ( µ IC X BunT Bun B, j =ν,! r ν Ω ν ) = 0. One computes using base change that µ,! j =ν,! r ν Ω ν =! Ω ν! µ,! IC BunN ω. Since µ,! IC is concentrated in cohomological degrees 1, and both BunN ω! Ω ν and µ,! IC BunN ω have lisse (actually constant) cohomology sheaves, their!-tensor product is concentrated in cohomological degrees 2. It follows that the Ext 1 above vanishes. We have shown that j =0,! IC BunN ω has a quotient M 0 which fits into a short exact sequence µ IC X BunT Bun B M 0 IC BunN ω. This sequence does not split because M 0 is the quotient of the indecomposable sheaf j =0,! IC BunN ω with simple cosocle IC. Dually, we obtain a subsheaf M /M BunN ω 1 of j =0, IC BunN ω of the desired form, from which follows the existence of M. We will first give a proof of Theorem under the assumption that gr Ψ(W ) is semisimple. The semisimplicity can be proved via Mochizuki s theory of weights for holonomic D-modules, since W is pure and (up to shift) the monodromy filtration on nearby cycles of a pure sheaf coincides with the weight filtration (see Corollary in [12]). First proof of Theorems and Both sides of the isomorphism (1.4.1) restrict to IC BunN ω. Suppose that we have constructed the isomorphism over Bun N ω,<µ. Then for λ µ sufficiently dominant, pulling back along p yields the isomorphism (5.1.1) over Z λ <µ. One obtains (5.1.1) on Z µ <µ by pullback along the factorization map (Z µ <µ Z λ µ ) disj Z λ <µ, 17

18 since the inverse images of both sides of the isomorphism factorize and are constant along the second component. The same argument yields (5.1.1) on Z <µ µ = Z µ for µ < µ. On the other hand, one can use factorization to obtain (5.1.1) on Z µ \ π 1 (X). Namely, for µ 1 + µ 2 = µ, µ 1, µ 2 < µ, the pullback of both sides of the isomorphism along (Z µ 1 Z µ 2 ) disj Z µ are identified. Since this factorization map is étale but not necessarily an embedding, we must argue that the isomorphism descends to its image. This immediately reduces to the case that µ = n α for some α. In this case, both sides of (5.1.1) are the intermediate extension of their restriction to π 1 (X µ disj ), so it suffices to show that the isomorphism over (Zα ) n disj descends to π 1 (X µ disj ) Zµ. Since (Z α ) n disj is a Σ n-torsor over π 1 (X µ disj ) and Σ n is generated by transpositions, the claim reduces to the case n = 2. But this was already done in Example Note that Z <µ µ (Z µ \π 1 (X)) = Z µ \ (X). The isomorphisms of the previous two paragraphs clearly agree on Z <µ µ (Z µ \ π 1 (X)), hence glue to an isomorphism away from the main diagonal. If µ is not a coroot, then we claim that Ψ(W Z µ) has no simple subquotients supported on the main diagonal, whence gr Ψ(W Z µ) is the intermediate extension of its restriction to Z µ \ (X). This is true for the right hand side of (5.1.1) by construction, so the claim implies that the isomorphism extends to Z µ in this case. j µ =ν, m µ, ν By Theorem there is a filtration of Ψ(W Z µ) by the sheaves Ω ν for 0 ν µ. Using Corollary 4.5 of [4], one can show that j µ =ν, m µ, ν Ω ν surjects onto j µ =µ, add (Ω ν Υ µ ν ), and that no subquotient of the kernel is supported on X µ. Now the claim follows, because out of the latter sheaves only Ω µ and Υ µ could have subquotients supported on the diagonal, and by Section 3.3 of loc. cit. this occurs if and only if µ is a coroot. Suppose that µ is a coroot. Then IC X appears as a subquotient of Ω µ and of Υ µ with multiplicity one. By the analysis in the previous paragraph IC X appears as a summand of gr Ψ(W Z µ) with multiplicity two, and there are no other subquotients supported on the main diagonal. Thus the isomorphism (5.1.1) extends to Z µ, and it remains to show that sl 2 acts on the summand IC 2 X of gr Ψ(W Z µ) as the standard representation. The only other possibility is that sl 2 acts on IC 2 X trivially, which would imply that the subquotient M from Lemma is a subquotient of F 0 /F 1. But M is indecomposable and F 0 /F 1 is semisimple, so this is impossible. Having constructed the isomorphism of Theorem over Z µ, we can complete the inductive step of Theorem by extending the isomorphism from Bun N ω,<µ to Bun N ω, µ. Choose λ µ dominant enough that Z µ λ surjects smoothly onto Bun N ω, µ. As in the proof of Theorem 1.3.1, note that (Z µ Z λ µ ) disj is one of the connected components of the fiber product (X µ X λ µ ) disj X λ Z λ µ, and that the former surjects onto Bun N ω, µ. By factorization, the cohomologically normalized pullback of gr Ψ(W ) to (Z µ Z λ µ ) disj is (gr Ψ(W Z µ)) IC Zλ µ. A factorization argument as in the proof of Theorem allows us to construct the isomorphism (5.1.1) over Z µ λ for arbitrary λ µ, which completes the proof of Theorem Proof of Corollary It suffices to prove the corresponding claim on Z λ for any λ Λ pos. The morphism j λ =0,! IC Bun N ω Ψ(W ) is injective because Ψ(W ) is tilting, so it suffices to show that j λ =0,! IC Bun N ω and the kernel of monodromy have the same class in the Grothendieck group. By factorization and induction this 18

19 holds away from the main diagonal, and Corollary 4.5 of [4] implies that the only subquotient of j λ =0,! IC Bun N ω supported on the main diagonal is IC X with multiplicity one. Theorem implies that the same is true for the kernel of the monodromy operator on Ψ(W ). 6 Second proof of Theorem In this section we will give a proof of Theorems and which does not use weights for irregular holonomic D-modules to prove the semisimplicity of gr Ψ(W ), but instead depends on Conjecture (but only the part proved in [8]). First we make the statement of the conjecture more precise in the case of a trivial Ť -local system by specifying the action of generators of ǧ H (X) on p IC BunB. We construct the action of ȟ H (X) as follows. Pullback along the evaluation map defines a homomorphism X Bun T pt/t Sym(h [ 2]) = H (pt/t ) H (X) H (Bun T ). By adjunction we obtain a morphism h H (X)[ 2] H (Bun T ). Identifying h = ȟ and H (X)[ 2] = H (X), the latter using Poincaré duality, we obtain a morphism ȟ H (X) H (Bun T ). Then the action of H (Bun T ) on ω BunT induces by functoriality the desired action of ȟ H (X) on Eis! ω BunT = p IC BunB. Next we construct the action of ň H (X). Denote by U (ň) the factorization algebra whose fiber at i µ ix i X µ is U(ň) µ i, i where the superscript µ i indicates the corresponding Ť -graded component. The following is part of Theorem 5.6 in [4], which itself is a restatement of results from [8]: for any µ Λ pos there is a canonical morphism j µ,! (U (ň) µ IC BunB ) IC BunB, which induces an isomorphism U (ň) µ IC BunB j! =µ IC BunB. Here we abuse notation slightly by denoting the maps j µ : X µ Bun B Bun B and j =µ : X µ Bun B Bun B by the same symbols we used in the case of Bun N ω. The latter are obtained from the former by base change along Bun N ω Bun B. Pushing forward to Bun G, we obtain a morphism H (X µ, U (ň) µ ) p IC BunB p IC BunB. The object U (ň) µ is concentrated in (perverse) cohomological degrees 1, and if µ is a coroot then we have H 1 (U (ň) µ ) = IC X. The resulting morphism k X U (ň) µ induces H (X) p IC BunB H (X µ, U (ň) µ ) p IC BunB p IC BunB, 19

20 which defines the action of ň µ H (X) = H (X) on p IC BunB. Dually, for any µ Λ pos there is a canonical morphism which induces an isomorphism IC BunB j µ, (U (ň ) µ IC BunB ), j =µ IC BunB U (ň ) µ IC BunB. Here U (ň ) is by definition the Verdier dual of U (ň ). Thus we obtain a morphism or by adjunction p IC BunB H (X µ, U (ň ) µ ) p IC BunB, H (X µ, U (ň ) µ ) p IC BunB p IC BunB. If µ is a coroot, then as before we have a morphism H (X) H (X µ, U (ň ) µ ), which defines the action of ň µ H (X) = H (X) on p IC BunB. 6.2 Fix a coroot µ. Recall the subquotient M of Ψ(W ) from Lemma The action of ǧ on p IC BunB relates to our problem through the following key lemma, whose proof will occupy this subsection. Lemma The sheaf M does not descend to Bun N ω/g m. First, observe that M 0 and M /M 1 descend to Bun N ω/t and hence to Bun N ω/g m, being subquotients of j =0,! IC BunN ω and j =0, IC BunN ω respectively. The obstruction to descent of M to Bun N ω/g m is the resulting composition µ IC (X BunT Bun B )/G m IC BunN ω /G m [1] µ IC (X BunT Bun B )/G m [2]. (6.2.1) Similarly, the obstruction to its descent to Bun N ω/t is the composition µ IC (X BunT Bun B )/T IC Bun N ω /T [1] µ IC (X BunT Bun B )/T [2]. (6.2.2) Denote by µ : X BunT Bun B Bun N ω the locally closed embedding given by composing µ with the open embedding X BunT Bun B X BunT Bun B. Composition with the canonical morphisms µ! IC (X BunT Bun B )/T µ IC (X BunT Bun B )/T and gives µ IC (X BunT Bun B )/T µ IC (X BunT Bun B )/T End( µ IC (X BunT Bun B )/T ) H (X pt/t ), (6.2.3) since the map (X BunT Bun B )/T X pt/t induces an isomorphism on cohomology. Lemma The image of the endomorphism (6.2.2) under (6.2.3) is 1 h µ H 0 (X) h H 2 (X pt/t ). 20

21 Proof. Theorem in [3] says that IC BunB is ULA over Bun T, which implies that the!-restriction of IC BunB to Bun N ω/t is IC BunN ω /T [dim T dim Bun T ]. It follows that the!-restriction of µ IC X BunB to Bun N ω/t is a shift of µ IC (X BunT Bun B )/T, where we abusively write µ : X Bun B Bun B for the similarly-defined finite map. This gives rise to a commutative square End( µ IC X BunB ) H (X Bun T ) End( µ IC (X BunT Bun B )/T ) H (X pt/t ), where the upper horizontal arrow is defined similarly to (6.2.3) and the right vertical arrow is id X times restriction along ρ(ω) : pt/t Bun T. The previous subsection implies that M 0 and M /M 1 extend to Bun B, giving rise to a morphism µ IC X BunB IC BunB [1] µ IC X BunB [2] (6.2.4) which restricts to (6.2.2) on Bun N ω/t. Thus it suffices to show that the image of (6.2.4) in H 2 (X Bun T ) restricts to 1 h µ along id X ρ(ω). Observe that so p induces a morphism End( µ IC X BunB ) = Hom( µ (k X IC BunB ), µ (ω X IC BunB ))[ 2], End( µ IC X BunB ) Hom(H (X) p IC BunB, H (X) p IC BunB )[ 2]. Composing with the canonical map k H (X) and its dual H (X) k, we obtain End( µ IC X BunB ) End(p IC BunB )[ 2]. (6.2.5) By construction, the image of (6.2.2) under (6.2.5) coincides with the action of f µ e µ U(ǧ). By composing with the morphisms p! IC BunB p IC BunB and p IC BunB p IC BunB, we obtain End(p IC BunB ) Hom(p! IC BunB, p IC BunB ). (6.2.6) Note that (6.2.6) annihilates the endomorphism of p IC BunB given by the action of e µ f µ, since it factors through a sheaf supported on the boundary. Proposition 4.9 in [8] says that the relation [e µ, f µ ] = h µ holds in End(p IC BunB ), which implies that the images of f µ e µ and h µ under (6.2.6) coincide. Now consider the commutative square (6.2.3) End( µ IC X BunB ) H (X Bun T ) (6.2.5) End(p IC BunB )[ 2] (6.2.6) Hom(p! IC BunB, p IC BunB )[ 2], where the right vertical morphism is the composition H (X Bun T ) = Hom( µ! IC X BunB, µ IC X BunB ) Hom(H (X) p! IC BunB, H (X) p IC BunB )[ 2] Hom(p! IC BunB, p IC BunB )[ 2]. 21