THE SEMI-INFINITE INTERSECTION COHOMOLOGY SHEAF-II: THE RAN SPACE VERSION

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1 THE SEM-NFNTE NTERSECTON COHOMOLOGY SHEAF-: THE RAN SPACE VERSON DENNS GATSGORY Abstract. This paper is a sequel to [Ga1]. We study the semi-infinite category on the version of the affine Grassmannian, and study a particular object in it, denoted C 2, which we call the semi-infinite intersection cohomology sheaf. Unlike the situation of [Ga1], this C 2 is defined as the middle of extension of the constant (more precisely, dualizing) sheaf on the basic stratum, in a certain t-structure. We give several explicit description and characterizations of C 2 : we describe its!- and *- stalks; we present it explicitly as a colimit; we relate it to the C sheaf of Drinfeld s relative compactification Bun N ; we describe C 2 via the Drinfeld-Plucker formalism. Contents ntroduction What are trying to do? What is done in this paper? Organization Background, conventions and notation Acknowledgements 6 1. The version of the semi-infinite category The Grassmannian The semi-infinite category Stratification Structure of the semi-infinite category An aside: the ULA property An application of Braden s theorem The t-structure and the semi-infinite C sheaf The t-structure on the semi-infinite category Definition of the semi-infinite C sheaf Digression: from commutative algebras to factorization algebras Restriction of C 2 to strata Digression: categories over the space Presentation of C 2 as a colimit Presentation of C 2 as a colimit Description of the *-restriction to strata Proof of coconnectivity The semi-infinite C sheaf and Drinfeld s compactification Drinfeld s compactification The global semi-infinite category Local vs global compatibility for the semi-infinite C sheaf The local vs global compatibility for the semi-infinite category The key isomorphism Computation of fibers 30 1

2 2 DENNS GATSGORY 3.7. Proof of Proposition Proof of Proposition The Hecke property of the semi-infinite C sheaf Pointwise Hecke property Categories over the space, continued Digression: right-lax central structures Hecke and Drinfeld-Plücker structures The Hecke property enhanced statement Local vs global compatibility of the Hecke structure The relative version of the Grassmannian Hecke property in the global setting Local vs global compatibility Proof of Theorem Appendix A. Proof of Theorem A.1. The space of G-bundles with a generic reduction 45 A.2. Towards Theorem A A.3. Proof of Theorem A A.4. Proof of Theorem A.3.3 for H reductive 49 References What are trying to do? ntroduction This paper is a sequel of [Ga1]. n loc. cit. an attempt was made to construct a certain object, denoted C 2, in the (derived) category Shv(GrG) of sheaves on the affine Grassmannian, whose existence had been predicted by G. Lusztig. Notionally, C 2 is supposed to be the intersection cohomology complex on the closure S 0 of the unit N((t))-orbit S 0 Gr G. ts stalks are supposed to be given by periodic Kazhdan-Lusztig polynomials. deally, one would want the construction of C 2 to have the following properties: t should be local, i.e., only depend on the formal disc, where we are thinking of Gr G as G((t))/G[[t]]; When our formal disc is the formal neighborhood of a point x in a global curve X, then C 2 should be the pullback along the map S 0 Bun N of the intersection cohomology sheaf of Bun N, where the latter is Drinfeld s relative compactification of the stack of G-bundles equipped with a reduction to N (which is an algebraic stack locally of finite type, so C BunN is well defined). The construction in [Ga1] indeed produced such an object, but with the following substantial drawback: in loc. cit., C 2 was given by an ad hoc procedure; namely, it was written as a certain explicit direct limit. n particular, C 2 was not the middle extension of the constant 1 sheaf on S 0 with respect to the natural t-structure on Shv(Gr G) (however, C 2 does belong to the heart of this t-structure) n the present paper we will construct a variant of C 2, denoted C 2, closely related to C 2, that is actually given by the procedure of middle extension in a certain t-structure. Namely, instead of the single copy of the affine Grassmannian Gr G, we will consider its space version, denoted Gr G,. We will equip the corresponding category Shv(Gr G,) with a t-structure, and we will define C 2 as the middle extension of the dualizing sheaf on S 0 Gr G,. 1 Technically, not constant but rather dualizing.

3 THE SEM-NFNTE C SHEAF- 3 Remark Technically, the space is attached to a smooth (but not necessarily complete) curve X, and one may think that this compromises the locality property of the construction of C 2. However, if one day a formalism becomes available for working with the space of a formal disc, the construction of C 2 will become purely local For a fixed point x X, we have the embedding Gr G {x} Gr G, Gr G,, (X) and we will show that the restriction of C 2 along this map recovers C 2 from [Ga1] Our C 2 retains the relation to C BunN. Namely, we have a natural map S 0 Bun N and we will prove that C 2 identifies with the pullback of C BunN n particular, this implies the isomorphism C 2 CBunN S 0, which had been established in [Ga1] by a different method. along this map To summarize, we can say that we still do not know how to intrinsically characterize C 2 on an individual Gr G as an intersection cohomology sheaf, but we can do it, once we allow the point of the curve to move along the space. But ce n est pas grave: as was argued in [Ga1, Sect. 0.4], our C 2, equipped with its factorization structure, is perhaps a more fundamental object than the original C What is done in this paper? The main constructions and results of this paper can be summarized as follows: We define the semi-infinite category on the version of the affine Grassmannian, denoted S, and equip it with a t-structure. This is largely parallel to [Ga1]. We define C 2 S as the middle extension of the dualizing sheaf on stratum S 0 Gr G,. (We will also show that the corresponding!- and *- extensions both belong to the heart of the t- structure, see Proposition 2.1.7; this contrasts with the situation for C 2, see [Ga1, Theorem 1.5.5]). We describe explicitly the!- and *-restrictions of C 2 to the strata S λ S 0 Gr G, (here λ is an element of Λ neg, the negative span of positive simple coroots), see Theorem These descriptions are given in terms of the combinatorics of the Langlands dual Lie algebra; more precisely, in terms, of the factorization algebras attached to O(Ň) and U(ň ). We give an explicit presentation of C 2 as a colimit (parallel to the definition of C 2 in [Ga1]), see Theorem This implies the identification C 2 GrG C 2, where C 2 Shv(GrG) is the object constructed in [Ga1] We show that C 2 identifies canonically (up to a cohomological shift by [d], d = dim(bun N )) with the pullback of C BunN along the map (0.1) S 0 Bun N, see Theorem n fact, we show that the above pullback functor is t-exact (up to the shift by [d]), when restricted to the subcategory S 0 glob Shv(BunN ) that consists of objects equivariant with respect to the action of the adelic N, see Corollary The proof of this t-exactness property is based on applying Braden s theorem to Gr G, and the Zastava space.

4 4 DENNS GATSGORY We note that, unlike [Ga1], the resulting proof of the isomorphism (0.2) C BunN S 0 [d] C 2 does not use the computation of C BunN from [BFGM], but rather reproves it. As an aside we prove an important geometric fact that the map (0.1) is universally homologically contractible (=the pullback functor along any base change of this map is fully faithful), see Theorem We show that C 2 has an eigen-property with respect to the action of the Hecke functors for G and T, see Theorem n the course of the proof of this theorem, we give yet another characterization of C 2 (which works for C 2 as well): We show that the δ-function δ 1Gr, Shv(Gr G,) on the unit section Gr G, possesses a natural Drinfeld-Plucker structure with respect to the Hecke actions of G and T (see Sect. 4.4 for what this means), and that C 2 can be obtained from δ 1Gr, by applying the functor from the Drinfeld- Plücker category to the graded Hecke category, left adjoint to the tautological forgetful functor (see Sect. 4.5). Finally, we establish the compatibility of the isomorphism (0.2) with the Hecke eigen-structures on C 2 and C BunN, respectively (see Theorem 5.3.5) Organization n Sect. 1 we recall the definition of the space (X), the version of the affine Grassmannian Gr G,, and the stratification of the closure S 0 of the adelic N-orbit S 0 by locally closed substacks S λ. We define the semi-infinite category S and study the standard functors that link it to the corresponding categories on the strata n Sect. 2 we define the t-structure on S 0 and our main object of study, C 2. We state Theorem that describes the *- and!- restrictions of C 2 to the strata S λ. The proof of the statement concerning *-restrictions will be given in this same section (it will result from Theorem mentioned below). This proof of the statement concerning!-restrictions will be given in Sect. 3. We state and prove Theorem that gives a presentation of C 2 as a colimit n Sect. 3, we recall the definition of Drinfeld s relative compactification Bun N. We define the global semi-infinite category S 0 glob Shv(BunN ). We prove that the pullback functor along (0.1), viewed as a functor S 0 glob S 0, is t-exact (up to the shift by [d]). From here we deduce the identification (0.2), which is Theorem We also state Theorem 3.4.4, whose proof is given in Sect. A n Sect. 4 we establish the Hecke eigen-property of C 2. n the process of doing so we discuss the formalism of lax central objects and Drinfeld-Plücker structures, and their relation to the Hecke eigen-structures. n Sect. 5 we prove the compatibility between the eigen-property of C 2 and that of C BunN Background, conventions and notation. The notations and conventions in this follow closely those of [Ga1]. Here is a summary:

5 THE SEM-NFNTE C SHEAF This paper uses higher category theory. t appears already in the definition of our basic object of study, the category of sheaves on the Grassmannian, Gr G,. Thus, we will assume that the reader is familiar with the basics of higher categories and higher algebra. The fundamental reference is [Lu1, Lu2], but shorter expositions (or user guides) exist as well, for example, the first chapter of [GR] Our algebraic geometry happens over an arbitrary algebraically closed ground field k. Our geometric objects are classical (i.e., this paper does not need derived algebraic geometry). We let Sch aff ft denote the category of (classical) affine schemes of finite type over k. By a prestack (locally of finite type) we mean an arbitrary functor (0.3) (Sch aff ft ) op Groupoids (we de not need to consider higher groupoids). We let PreSk lft denote the category of such prestacks. t contains Sch aff ft via the Yoneda embedding. All other types of geometric objects (schemes, algebraic stacks, ind-schemes) are prestacks with some specific properties (but not additional pieces of structure) We let G be a connected reductive group over k. We fix a Borel subgroup B G and the opposite Borel B G. Let N B and N B denote their respective unipotent radicals. Set T = B B ; this is a Cartan subgroup of G. We use it to identify the quotients B/N T B /N. We let Λ denote the coweight lattice of G, i.e., the lattice of cocharacters of T. We let Λ pos Λ denote the sub-monoid consisting of linear combinations of positive simple roots with non-negative integral coefficients. We let Λ + Λ denote the sub-monoid of dominant coweights While our geometry happens over a field k, the representation-theoretic categories that we study are DG categories over another field, denoted e (assumed algebraically closed and of characteristic 0). For a crash course on DG categories, the reader is referred to [GR, Chapter 1, Sect. 10]. All our DG categories are assumed presentable. When considering functors, we will only consider functors that preserve colimits. We denote the -category of DG categories by DGCat. t carries a symmetric monoidal structure (i.e., one can consider tensor products of DG categories). The unit object is the DG category of complexes of e-vector spaces, denoted Vect. We will use the notion of t-structure on a DG category. Given a t-structure on C, we will denote by C 0 the corresponding subcategory of cohomologically connective objects, and by C >0 its right orthogonal. We let C denote the heart C 0 C The source of DG categories will be a sheaf theory, which is a functor Shv : (Sch aff ft ) op DGCat, S Shv(S). For a morphism of affine schemes f : S 0 S 1, the corresponding functor is the!-pullback f!. The main examples of sheaf theories are: Shv(S 1) Shv(S 0) (i) We take e = Q l and we take Shv(S) to be the ind-completion of the (small) DG category of constructible Q l -sheaves. (ii) When k = C and e arbitrary, we take Shv(S) to be the ind-completion of the (small) DG category of constructible e-sheaves on S(C) in the analytic topology. (iii) f k has characteristic 0, we take e = k and we take Shv(S) to be the DG category of D-modules on S.

6 6 DENNS GATSGORY n examples (i) and (ii), the functor f! always has a left adjoint, denoted f!. n example (iii) this is not the case. However, the partially defined left adjoint f! is defined on holonomic objects. t is defined on the entire category if f is proper Sheaves on prestacks. We apply the procedure of right Kan extension along the embedding (Sch aff ft ) op (PreStk lft ) op to the functor Shv, and thus obtain a functor (denoted by the same symbol) By definition, for Y PreStk lft we have Shv : (PreStk lft ) op DGCat. (0.4) Shv(Y) = lim Shv(S),,y:S Y S Sch aff ft where the transition functors in the formation of the limit are the!-pullbacks 2. For a map of prestacks f : Y 0 Y 1 we thus have a well-defined pullback functor f! : Shv(Y 1) Shv(Y 0). We denote by ω Y the dualizing sheaf on Y, i.e., the pullback of along the tautological map Y pt. e Vect Shv(pt) This paper is closely related to the geometric Langlands theory, and the geometry of the Langlands dual group Ǧ makes it appearance. By definition, Ǧ is a reductive group over e and geometric objects constructed out of Ǧ give rise to e-linear DG categories by considering quasi-coherent (resp., ind-coherent) sheaves on them. The most basic example of such a category is QCoh(pt /Ǧ) =: Rep(Ǧ) Acknowledgements. The author would like to thank S. Raskin for his suggestion to consider the formalism of Drinfeld-Plücker structures, as well as numerous stimulating discussions. The author is grateful to M. Finkelberg for igniting his interest in the semi-infinite C sheaf, which has been on author s mind for some 20 years now. The author is supported by NSF grant DMS The version of the semi-infinite category n this section we extend the definition of the semi-infinite category given in [Ga1] from the affine Grassmannian Gr G,x corresponding to a fixed point x X to the version, denoted Gr G, The Grassmannian. 2 Note that even though the index category (i.e., (Sch aff ft ) /Y) is ordinary, the above limit is formed in the -category DGCat. This is how -categories appear in this paper.

7 THE SEM-NFNTE C SHEAF We recall that the space of X, denoted (X), is the prestack that assigns to an affine test scheme Y the set of finite non-empty subsets Hom(Y, X). One can explicitly exhibit (X) as a colimit (in PreStk) of schemes: (X) colim X, where the colimit is taken over the category opposite to the category Fin surj of finite non-empty sets and surjective maps, where to a map φ : J we assign the corresponding diagonal embedding X J φ X We can consider the version of the affine Grassmannian, denoted Gr G,, defined as follows. t assigns to an affine test scheme Y, the set of triples (, P G, α), where is an Y -point of (X), P G is a G-bundle on Y X, and α is a trivialization of P G on the open subset of Y X equal to the complement of the union Γ of the graphs of the maps Y X that comprise. We also consider the versions of the loop and arc groups (ind)-schemes, denoted L + (G) L(G). The Grassmannian Gr G, identifies with the étale (equivalently, fppf) sheafification of the prestack quotient L(G) /L + (G) For a fixed finite non-empty set, we denote Gr G, := X Gr G,, L(G) := X L(G), L + (G) := X L + (G). (X) (X) (X) For a map of finite sets φ : J, we will denote by φ the corresponding map Gr G,J Gr G,, so that we have the Cartesian square: Gr G,J X J φ Gr G, φ X We introduce also the following closed (resp., locally closed) subfunctors S 0 S 0 Gr G,. Namely, a Y -point (, P G, α) belongs to S 0 if for every dominant weight ˇλ, the composite meromorphic map of vector bundles on Y X (1.1) O Vˇλ P 0 G α Vˇλ PG is regular. n the above formula the notations are as follows: Vˇλ denotes the Weyl module over G with highest weight ˇλ; Vˇλ PG (resp., Vˇλ P 0 denotes the vector bundles associated with Vˇλ and the G-bundle P G (resp., G) the trivial G-bundle P 0 G); O Vˇλ P 0 is the map coming from the highest weight vector in Vˇλ. G A point as above belongs to S 0 if the above composite map is a bundle map (i.e., has no zeroes) The semi-infinite category.

8 8 DENNS GATSGORY Since Gr G, a prestack locally of finite type, we have a well-defined category Shv(Gr G,). We have Shv(Gr G,) := lim Shv(Gr G,), where the limit is formed using the!-pullback functors Although the group ind-scheme L(N) is not locally of finite type, we have a well-defined full subcategory S := Shv(Gr G,) L(N) Shv(Gr G,). Namely, for every fixed finite non-empty set, we consider the full subcategory S := Shv(Gr G,) L(N) Shv(Gr G,), defined as in [Ga1, Sect. 1.2]. We say that the object of Shv(Gr G,) belongs to Shv(Gr G,) L(N) if its restriction to any Gr G, yields an object of Shv(Gr G,) L(N). By construction, we have an equivalence S := lim S Let S 0 S be the full subcategory consisting of objects supported on S0..e., S 0 = S Shv(S0 ), and Shv(S 0 ) lim Shv(S 0 ) We are going to introduce a t-structure on S 0 and describe a certain object in its heart, called the semi-infinite intersection cohomology sheaf. n order to do so we will first need to describe the corresponding stratification of S 0, whose open stratum will be S Stratification For λ Λ neg, let X λ denote the corresponding partially symmetrized power of X..e., if λ = Σ ( n i i) α i, n i 0 where α i are the simple positive coroots, then X λ = Π X (ni). i n other words, S-points of X λ are effective Λ neg -valued divisors on X. For λ = 0 we by definition have X 0 = pt.

9 THE SEM-NFNTE C SHEAF Let ((X) X λ ) (X) X λ be the closed subfunctor, whose S-points are pairs (, D) for which the support of the divisor D is set-theoretically supported on the union of the graphs of the maps S X that comprise. where n other words, is the correspondence incidence subvariety. For future use we note: Lemma The map is universally homologically contractible. ((X) X λ ) = colim (X X λ ), (X X λ ) X X λ pr λ : ((X) X λ ) X λ The proof will be given in Sect. A.2.8 (we refer the reader to [Ga2, Sect. universal local contractibility). Corollary The pullback functor is fully faithful. (pr λ )! : Shv(X λ ) Shv(((X) X λ ) ) We let S λ ((X) X λ ) (X) Gr G, be the closed subfunctor defined by the following condition: 2.5] for the notion of An S-point (, D, P G, α) of the fiber product ((X) X λ ) Gr G, belongs to S λ if for (X) every dominant weight ˇλ the map (1.1) extends to a regular map (1.2) O(ˇλ(D)) Vˇλ PG. We denote by i λ the composite map S λ ((X) X λ ) Gr G, Gr G,. (X) This map is proper, and its image is contained in S 0. Note that for λ = 0, the map i 0 is the identity map on S 0. Let p λ denote the projection S λ ((X) X λ ) We define the open subfunctor S λ S λ to correspond to those quadruples (, D, P G, α) for which the map (1.2) is a bundle map. The projection (1.3) p λ := p λ S λ : S λ ((X) X λ ) admits a canonically defined section (1.4) s λ : ((X) X λ ) S λ. Namely, it sends (, D) to the quadruple (, D, P G, α), where P G is the G-bundle induced from the T -bundle P T := P 0 T (D), and α is the trivialization of P G induced by the tautological trivialization of P T away from the support of D.

10 10 DENNS GATSGORY We let j λ : S λ S λ, denote the resulting maps. i λ = i λ j λ : S λ Gr G, For a fixed finite non-empty set, we obtain the corresponding subfunctors S λ (X X λ ) Gr G, X and S λ S λ, and maps, denoted by the same symbols j λ, i λ, i λ. Let p λ (resp., p λ ) denote the corresponding map from S λ (resp., S λ ) to (X X λ ) The following results easily from the definitions: Lemma The maps i λ : S λ S 0 and S λ S 0 are locally closed embeddings. Every field-valued point of S 0 (resp., S 0 ) belongs to the image of exactly one such map Structure of the semi-infinite category As in the case of Gr G,, we can consider the full subcategories S λ Shv(Sλ ) and S =λ Shv(S λ ), obtained by imposing the equivariance condition with respect to the corresponding group ind-scheme. The maps j λ, i λ and i λ induce functors (i λ )! := (i λ ) : S λ S 0, (i λ )! : S 0 S λ ; (j λ ) := (j λ )! : S λ S=λ ; (j λ ) : S =λ S λ ; (i λ )! (j λ )! (i λ )! : S 0 S=λ ; (i λ ) (i λ ) (j λ ) : S =λ S 0. The above discussion applies to replaced by X for a fixed finite non-empty set n Sect. 1.6 we will prove: Proposition (a) For a fixed finite set, the partially defined left adjoint (i λ ) of (i λ ) : Shv(S λ ) Shv(S 0 ) is defined on the full subcategory S 0 Shv(S 0 ), and takes values in S =λ Shv(S λ ). (b) For F Shv(S 0 ) and F Shv(X ), the map 3 (i λ ) ((p 0 )! (F )! F) (p λ )! (F (X X λ ) )! (i λ ) (F) is an isomorphism, provided that F S 0. (c) For a map of finite sets φ : J, the natural transformation (i λ ) ( φ )! ( φ )! (i λ ), Shv(S 0 ) Shv(S λ J ) is an isomorphism, when evaluated on objects of S 0 Shv(S0 ). From here, by a formal argument, we obtain: 3 n the formula below (X X λ ) denotes the!-restriction along the projection (X X λ ) X.

11 THE SEM-NFNTE C SHEAF- 11 Corollary (a) For a fixed finite set, the partially defined left adjoint (i λ )! of is defined on the full subcategory S =λ (b) For F Shv(S λ ) and F Shv(X ), the map is an isomorphism. (i λ )! : Shv(S 0 ) Shv(S λ ) Shv(S λ ), and takes values in S 0 (i λ )! ((p λ )! (F (X X λ ) ))! F) (p 0 )! (F )! (i λ )! (F) (c) For a map of finite sets φ : J, the natural transformation (i λ )! ( φ )! ( φ )! (i λ )!, Shv(S λ J ) Shv(S 0 ) is an isomorphism, when evaluated on objects of S =λ Shv(S λ ). As a corollary we obtain: Corollary (a) The partially defined left adjoint (i λ ) of (i λ ) : Shv(S λ ) Shv(S 0 ) Shv(S 0 ). is defined on the full subcategory S 0 Shv(S0 ), and takes values in S =λ Shv(S λ ). (a ) The partially defined left adjoint (i λ )! of (i λ )! : Shv(S 0 ) Shv(S λ ) is defined on the full subcategory S =λ Shv(S λ ), and takes values in S 0 Shv(S0 ). (b) For F Shv(S 0 ) and F Shv((X)), the map (i λ ) ((p 0 )! (F )! F) (p λ )! (F ((X) X λ ) ))! (i λ ) (F) is an isomorphism, provided that F S 0. (b ) For F Shv(S λ ) and F Shv((X)), the map is an isomorphism. (i λ )! ((p λ )! (F ((X) X λ ) ))! F) (p 0 )! (F )! (i λ )! (F) Remark A slight variation of the proof of Proposition shows that the assertions remain valid for i λ replaced by i λ. Similarly, the assertion of Corollary remains valid for i λ replaced by j λ, and the same is true for their variants We have the following explicit description of the category on each stratum separately: Lemma Pullback along the map p λ of (1.3) defines an equivalence Shv(((X) X λ ) ) S =λ. The inverse equivalence is given by restriction to the section s λ of (1.4), and similarly for (X) replaced by X for an individual. Proof. Follows from the fact that the action of the group ind-scheme ((X) X λ ) L(G) (X) on S λ transitive along the fibers of the map (1.3), with the stabilizer of the section s λ being a pro-unipotent group-scheme over ((X) X λ ).

12 12 DENNS GATSGORY 1.5. An aside: the ULA property. Consider the object (j 0 )! (ω S 0 ) Shv(S 0 ). We will now formulate a certain strong acyclicity property that it enjoys with respect to the projection p 0 : S 0 X Let Y be a scheme, and let C be a DG category equipped with an action of the Shv(Y ), viewed as a monoidal category with respect to!. We shall say that an object c C is ULA with respect to Y if for any compact F Shv(Y ) c, and any c C, the map is an isomorphism. Hom C (F! c, c ) Hom(D(F)! F! c, D(F)! c ) Hom(e Y! c, D(F)! c ) n the above formula, D( ) denotes the Verdier duality anti-equivalence of Shv(Y ) c, and e Y is the constant sheaf on Y, i.e., e Y := D(ω Y ). Note that when Y is smooth of dimension d, we have e Y ω Y [ 2d] We regard Shv(S 0 ) as tensored over Shv(X ) via We claim: F Shv(X ), F Shv(S 0 ) (p 0 )! (F)! F. Proposition The object (j 0 )! (ω S 0 ) Shv(S 0 ) is ULA with respect to X. Proof. For F Shv(X ) and F Shv(S 0 ), we have a commutative square Hom((p 0 )! (F), (j 0 )! (F )) Hom((p 0 )! (e X ), (p 0 )! (D(F))! (j 0 )! (F )) Hom((j 0 )! (p 0 )! (F), F ) Hom((j 0 )! (p 0 )! (e X ), (p 0 )! (D(F))! F ) Hom((p 0 )! (F)! (j 0 )! (ω S 0 ), F ) Hom((p 0 )! (e X )! (j 0 )! (ω S 0 ), (p 0 )! (D(F))! F ). n this diagram the lower vertical arrows are isomorphisms by Corollary 1.4.4(b). The top horizontal arrow is an isomorphism because S 0 can be exhibited as a union of closed subschemes, each being smooth over X. Hence, the bottom horozontal arrow is an isomorphism, as required An application of Braden s theorem. n this subsection we will prove Proposition Let S,λ j,λ S,λ i λ Gr G, be the objects defined in the same way as their counterparts but where we replace N by N. S λ j λ S λ i λ Gr G,, Choose a regular dominant coweight G m T. t gives rise to an action of G m on S 0 fibers of the projection p 0. We have: along the

13 THE SEM-NFNTE C SHEAF- 13 Lemma The attracting (resp., repelling) locus of the above G m action identifies with respectively. The fixed point locus identifies with λ Λ neg Sλ and λ Λ neg S,λ, λ Λ neg sλ (X ) Let us prove point (a) of Proposition A priori, the functor (i λ ) applied to objects from S 0 maps to the pro-completion of S =λ, viewed as a full subcategory in the pro-completion of Shv(S λ ). Now, using Lemma 1.4.8, it suffices to show that for F S(G) 0, the object actually belongs to Shv(X ). (s λ )! (i λ ) Pro(Shv(X )) t is easy to see that every object in Shv(S 0 ) is G m-monodromic. Using Braden s theorem (see [DrGa]), combined with Lemma 1.6.2, we obtain that (s λ )! (i λ ) indeed belongs to Shv(X ), as required Point (b) of Proposition also follows from Braden s theorem: we identify and the assertion follows by base change. Point (c) is a formal corollary of point (b). (s λ )! (i λ ) (p,λ ) (i,λ )!, 2. The t-structure and the semi-infinite C sheaf n this section we define a t-structure on S 0, and define the main object of study in this paper the version of the semi-infinite intersection cohomology sheaf, denoted C 2. We will also give a presentation of C 2 as a colimit, and describe explicitly its *- and!-restrictions to the strata S λ The t-structure on the semi-infinite category We introduce a t-structure on the category Shv(((X) X λ ) ) as follows. Note that the projection (2.1) pr λ : ((X) X λ ) X λ is pseudo-proper (see [Ga2, Sect. 1.5] for what this means). n particular, the functor left adjoint to (pr λ )!, is well-defined. (pr λ )! : Shv(((X) X λ ) Shv(X λ ), We declare an object F Shv(((X) X λ ) ) to be connective if Hom(F, (pr λ )! (F )) = 0 for all F Shv(X λ ) that are strictly coconnective (in the perverse t-structure). By construction, the functor is left t-exact. However, we claim: (pr λ )! : Shv(X λ ) Shv(((X) X λ ) ) Proposition The functor (pr λ )! : Shv(X λ ) Shv(((X) X λ ) ) is t-exact.

14 14 DENNS GATSGORY Proof. We need to show that the composition (pr λ )! (pr λ )! is right t-exact. However, we claim that the above composition is actually isomorphic to the identity functor. ndeed, the latter assertion is equivalent to the fact that the functor (pr λ )! is fully faithful, which is the assertion of Corollary to be connec We define a t-structure on S =λ as follows. We declare an object F S =λ tive/coconnective if (s λ )! (F)[ λ, 2ˇρ ] Shv(((X) X λ ) ) is connective/coconnective. n other words, this t-structure is transferred from Shv(((X) X λ ) ) via the equivalences (s λ )! : S =λ Shv(((X) X λ ) ) : (p λ )! of Lemma 1.4.8, up to the cohomological shift [ λ, 2ˇρ ] We define a t-structure on S 0 by declaring that an object F is coconnective if is coconnective in the above t-structure. (i λ )! (F) S =λ By construction, the subcategory of connective objects in S 0 is generated under colimits by objects of the form (2.2) (i λ )! (p λ )! (F)[ λ, 2ˇρ ], λ Λ neg where F is a connective object of Shv(((X) X λ ) ). We claim: Lemma An object F S 0 every λ Λ neg. is connective if and only if (iλ ) (F) S λ is connective for Proof. t is clear that for objects of the form (2.2), their *-pullback to any S λ is connective. Hence, the same is true for any connective object of S 0. Vice versa, let 0 F be a strictly coconnective object of S 0. We need to show that if all (iλ ) (F) are connective, then F = 0. Let λ be the largest element such that (i λ )! (F) 0. On the one hand, since F is strictly coconnective, (i λ )! (F) is strictly coconnective. On the other hand, by the maximality of λ, we have (i λ )! (F) (i λ ) (F), and the assertion follows By construction, the object (i λ )! (ω S λ )[ λ, 2ˇρ ] (resp., (i λ ) (ω S λ )[ λ, 2ˇρ ]) of S 0 is connective (resp., coconnective). However, in Sect we will prove: Proposition The objects belong to (S 0 ) Definition of the semi-infinite C sheaf. (i λ )! (ω S λ )[ λ, 2ˇρ ] and (i λ ) (ω S λ )[ λ, 2ˇρ ]

15 THE SEM-NFNTE C SHEAF Consider the canonical map (i λ )! (ω S λ )[ λ, 2ˇρ ] (i λ ) (ω S λ )[ λ, 2ˇρ ]. According to Proposition both sides belong to (S 0 ). We let denote the image of this map. C 2 (S 0 ) The above object is the main character of this paper Our goal is to describe C 2 as explicitly as possible. Specifically, we will do the following: We will describe the!- and *- restrictions of C 2 to the strata S λ ; We will exhibit the values of C 2 in Shv(Gr G,) explicitly as colimits; We will relate C 2 to the intersection cohomology sheaf of Drinfeld s compactification Bun N Digression: from commutative algebras to factorization algebras. Let A be a commutative algebra, graded by Λ neg with A(0) e. Then to A we can attach an object Fact alg (A) X λ Shv(X λ ), characterized by the property that its!-fiber at a divisor Σ k λ k x k X λ, 0 λ k Λ neg, Σ k λ k = λ, k k x k x k equals k A(λ k ). n the present subsection we recall this construction Consider the category TwArr λ whose objects are diagrams (2.3) Λ neg 0 λ φ J, Σ λ(i) = λ, i where and J are finite non-empty sets. A morphism between two such objects is a diagram Λ neg 0 λ 1 J1 φ 1 K 1 id ψ J ψ K Λ neg 0 where: The right square commutes; The maps ψ J and ψ K are surjective; λ 2 (j 2) = Σ j 1 ψ 1 J (j 2) λ 1 (j 1) The algebra A defines a functor constructed as follows. λ 2 φ 2 J2 K 2, TwArr(A) : TwArr λ Shv(X λ ), For an object (2.3), let K,λ denote the map X K X λ that sends a point {x k, k K} X K to the divisor Σ Σ λ(j)) x k X λ. ( k K j φ 1 (k) Then the value of TwArr(A) on (2.3) is ( K,λ ) (ω X K ) ( ) A(λ j), j J where λ j = λ(j).

16 16 DENNS GATSGORY The structure of functor on TwArr(A) is provided by the commutative algebra structure on A We set Fact alg (A) X λ := colim TwArr λ TwArr(A) Shv(X λ ) An example. Let V be a Λ neg -graded vector space with V (0) = 0. Let us take A = Sym(V ). n this case Fact alg (A) X λ, can be explicitly described as follows: t is the direct sum over all ways to write λ as a sum λ = Σ k n k λ k, n k > 0, λ k Λ neg 0 of the direct images of along the corresponding maps ω Πk X (n k ) k Sym n k (V (λ k )) Π k X (n k) X λ, {D k X (n k) } Σ k λ k D k Dually, if A is a co-commutative co-algebra, graded by Λ neg with A(0) e, then to A we attach an object Fact coalg (A) X λ Shv(X λ ) characterized by the property that its *-fiber at a divisor Σ k λ k x k X λ, 0 λ k Λ neg, Σ k λ k = λ, k k x k x k equals k A(λ k ). f all the graded components of A are finite-dimensional, we can view the dual A of A as a Λ neg - graded algebra, and we have 2.4. Restriction of C 2 to strata. D(Fact coalg (A) X λ) Fact alg (A ) X λ We apply the above construction to A being the (classical) algebra U(ň )). Thus, we obtain the objects O(Ň) (resp., co-algebra Fact alg (O(Ň)) X λ and Factcoalg (U(ň )) X λ in Shv(X λ ). Note also that U(ň ) is the graded dual of O(Ň), and so the objects Factalg (O(Ň)) X λ Fact coalg (U(ň )) X λ are Verdier dual to each other. and From the construction it follows that for λ 0, and hence Fact alg (O(Ň)) X λ Shv(Xλ ) <0, Fact coalg (U(ň )) X λ Shv(X λ ) >0. Remark Note that by the PBW theorem, when viewed as a co-commuatative co-algebra, U(ň ) is canonically identified with Sym(ň ); in this paper we will not use the algebra structure on U(ň ) that allows to distinguish it from Sym(ň ). So Fact alg (O(Ň)) X λ falls into the paradigm of Example Similarly, when viewed just as a commutative algebra (i.e., ignoring the Hopf algebra structure), O(Ň) is canonically identified with Sym(ň ).

17 THE SEM-NFNTE C SHEAF n Sect. 3.6 we will prove: Theorem The objects (i λ )! (C 2 ) and (i λ ) (C 2 ) of Shv(S λ ) identify with the!-pullback along S λ p λ ((X) X λ ) prλ X λ of Fact coalg (U(ň )) X λ[ λ, 2ˇρ ] and Fact alg (O(Ň)) Xλ[ λ, 2ˇρ ], respectively, 2.5. Digression: categories over the space. We will now discuss a variant of the construction in Sect. 2.3 that attaches to a symmetric monoidal category A a category spread over the space, denoted Fact alg (A) Consider the category TwArr whose objects are (2.4) φ J, where and J are finite non-empty sets. A morphism between two such objects is is a commutative diagram (2.5) ψ J J 1 φ 1 K 1 ψ K where the maps ψ J and ψ K are surjective To A we attach a functor φ J 2 2 K 2, TwArr(A) : TwArr DGCat by sending an object (2.4) to Shv(X K ) A J, and a morphism (2.5) to a functor comprised of and the functor given by the symmetric monoidal structure on A We set Shv(X K 1 ) ( ψ J ) Shv(X K 2 ) A J 1 A J 2, (2.6) Fact alg (A) := colim TwArr(A) DGCat. TwArr Let us consider some examples. (i) Let A = Vect. Then Fact alg (A) Shv(). (ii) Let A be the (non-unital) symmetric monoidal category consisting of vector spaces graded by the semi-group Λ neg 0. We have a canonical functor (2.7) Fact alg (A) Shv( λ Λ neg 0 Xλ ), and it follows from [Ga2, Lemma (d)] that this functor is an equivalence Similarly, if A is a symmetric co-monoidal category, we can form the limit of the corresponding functor TwArr(A) : TwArr op DGCat, and obtain a category that we denote by Fact coalg (A).

18 18 DENNS GATSGORY Recall that whenever we have a diagram of categories indexed by some category T, then t C t colim t T Ct is canonically equivalent to lim Ct, t T op where the transition functors are given by the right adjoints of those in the original family Let A be again a symmetric monoidal category. Applying the observation of Sect to the colimit (2.6), and obtain that Fact alg (A) can also be written as a limit. Assume now that A is such that the functor A A A, right adjoint to the tensor product operation, is continuous. n this case, the above tensor co-product operation makes A into a symmetric co-monoidal category, and we can form Fact coalg (A). We note however, that the limit presentation of Fact alg (A) tautologically coincides with the limit defining Fact coalg (A)..e., we have a canonical equivalence: Fact alg (A) Fact coalg (A). Hence, in what follows we will sometimes write simply Fact(A), having both of the above realizations in mind Let be a fixed finite non-empty set. The above constructions have a variant, where instead of TwArr we use its variant, denoted TwArr /, whose objects are commutative diagrams and whose morphisms are commutative diagrams id J φ K, φ J 1 1 K 1 ψ J ψ K φ J 2 2 K 2, We denote the resulting category by Fact alg (A) For a surjective morphism 1 2, we have the corresponding functor which induces a functor TwArr 2 / TwArr 1 /, Fact alg (A) 2 Fact alg (A) 1. An easy cofinality argument shows that the resulting functor (2.8) colim is an equivalence. Fact alg (A) Fact alg (A) When we think of Fact(A) as a limit, the functor TwArr 2 / TwArr 1 / defines a functor Fact(A) 1 Fact(A) 2. The corresponding functor Fact(A) lim Fact(A) is an equivalence obtained from (2.8) by the procedure of Sect Presentation of C 2 as a colimit.

19 THE SEM-NFNTE C SHEAF Denote Sph G, := Shv(L + (G) \Gr G,) and Sph G, := Shv(L + (G) \Gr G,). Consider the symmetric monoial category Rep(Ǧ). Geometric Satake defines functors Sat G, : Fact(Rep(Ǧ)) Sph G, that glue to a functor Sat G, : Fact(Rep(Ǧ)) Sph G, For a fixed finite non-empty set and a map λ : Λ +, we consider the following object of Fact(Rep(Ǧ)), denoted V λ. nformally, V λ is designed so its!-fiber at a point X, = k k, k x k, x k x k is V λ k Rep(Ǧ) k. k t terms of the presentation of Fact(Rep(Ǧ)) as a colimit Fact(Rep(Ǧ)) = colim TwArr(Rep(Ǧ)), TwArr / the object V λ corresponds to the colimit over TwArr / of the functor that sends TwArr / Fact(Rep(Ǧ)) (2.9) ( J K) TwArr / V λ J K Shv(XK ) Rep(Ǧ) J Fact(Rep(Ǧ)), where V λ J K = ω ( ) X K V λ j, λ j = Σ λ(i). j J i,i j The structure of a functor TwArr / Fact(Rep(Ǧ)) on (2.9) is given by the Plücker maps Denote V λ i V λ, λ = Σ λ i i i. Sph G, := Shv(L + (G) \Gr G,) and Sph G, := Shv(L + (G) \Gr G,). Consider the symmetric monoial category Rep(Ǧ). Geometric Satake defines functors Sat G, : Fact(Rep(Ǧ)) Sph G, that glue to a functor Consider the object Sat G, : Fact(Rep(Ǧ)) Sph G,. Sat G,(V λ ) Sph G,. The element λ gives rise to a section Denote Consider the object s λ : X Gr G,. δ λ := (s λ )!(ω X ) Shv(Gr G,). δ λ Sat G,(V λ )[ λ, 2ˇρ ] Shv(Gr G,). n the above formula, λ = Σ i λ(i), and denotes the (right) convolution action of Sph G, on Shv(Gr G,).

20 20 DENNS GATSGORY Consider now the set Maps(, Λ + ) of maps λ : Λ +. We equip it with a partial order by declaring λ 1 λ 2 λ 2 (i) λ 1 (i) Λ +, i. The assignment λ δ λ Sat G,(V λ )[ λ, 2ˇρ ] Shv(Gr G,) has a structure of a functor Maps(, Λ + ) Shv(Gr G,), see Sects and Set C 2 := colim δ λ Sat G,(V λ )[ λ, 2ˇρ ] Shv(Gr G,). λ Maps(,Λ + ) As in [Ga1, Proposition 2.3.7(a,b,c)] one shows: Lemma The object C 2 (a) t is supported on S 0 ; has the following properties: (b) t belongs to S 0 = Shv(S 0 ) L(N) Shv(S 0 ); (c) ts restriction to S 0 is identified with ω S For a surjective map φ : 2 1 and the corresponding map φ : Gr G,1 Gr G,2, we have a canonical identification ( φ )! (C 2 2 ) C 2 1. One endows this system of isomorphisms with a homotopy-coherent system of compatibilities, thus making the assignment into an object of S 0, see Sect C 2 We denote this object by C 2. By Lemma 2.6.7(c), the restriction of C 2 to S 0 identifies canonically with ω S Fix a point x X, and consider the restriction of C 2 along the map Gr G,x {x} Gr G, Gr G,. (X) t follows from the construction, that this restriction identifies canonically with the object C 2 x Shv(Gr G,x), constructed in [Ga1, Sect. 2.3] Presentation of C 2 as a colimit.

21 THE SEM-NFNTE C SHEAF We will prove: Theorem There exists a canonical isomorphism C 2 C 2. The rest of this section is devoted to the proof of Theorem t amounts to the combination of the following two assertions: Proposition For µ Λ neg, the object identifies canonically with the!-pullback along of Fact alg (O(Ň)) Xλ[ µ, 2ˇρ ]. S µ Proposition For 0 µ Λ neg, the object (i µ ) ( C 2 ) S =µ p µ ((X) X µ ) prµ X µ (i µ )! ( C 2 )[ µ, 2ˇρ ] S =µ is a pullback along pr µ pµ of an object of Shv(Xµ ) that is strictly coconnective. Remark Note that Theorem and Proposition imply the assertion of Theorem about the *-fibers As a corollary of Theorem and Sect we obtain: Corollary The restriction of C 2 along the map Gr G,x {x} Gr G, Gr G, (X) identifies canonically with the object C 2 x Shv(Gr G,x) of [Ga1, Sect. 2.3] Note that by construction, C 2 has the following factorization property with respect to : Let ((X) (X)) disj denote the disjoint locus..e., for an affine test scheme Y, Hom(Y, ((X) (X)) disj ) Hom(Y, (X)) Hom(Y, (X)) consists of those pairs 1, 2 Hom(Y, X), for which for every i 1 1 and i 2 2, the corresponding two maps Y X have non-intersecting images. t is well-known that we have a canonical isimorphism (2.10) (Gr G, Gr G,) (X) (X) ((X) (X)) disj Gr G, (X) ((X) (X)) disj, where is the map ((X) (X)) disj (X) (X) (X) 1, Then, in terms of the identification (2.10), we have a canonical isomorphism (2.11) ( C 2 C 2 ) (GrG, Gr G, ) ((X) (X)) disj (X) (X) The rest of this section is devoted to the proof of Theorem C 2 Gr G, ((X) (X)) disj. (X) 2.8. Description of the *-restriction to strata. The goal of this subsection is to prove Proposition

22 22 DENNS GATSGORY We will compute (i µ ) ( C 2 ) S 0 for each individual finite non-empty set, and obtain the!-pullback of (pr µ pµ )! (Fact alg (O(Ň)) X λ) along S µ Sµ. By Lemma 1.4.8, we need to construct an identification (2.12) (p µ )! (i µ ) ( C 2 ) (pr µ )! (Fact alg (O(Ň)) X λ), where pr µ denotes the map (X X µ ) X µ We will compute (2.13) (p µ )! (i µ ) (δ λ Sat G,(V λ ))[ λ, 2ˇρ ] Shv((X X µ ) ) for each individual λ : Λ +. Namely, we will show that (2.12) identifies with the following object, denoted described below. V λ (λ + µ) Shv((X X µ ) ), Before we give the definition of V λ (λ + µ), let us describe what its!-fibers are. Fix a point of (X X µ ). By definition, a datum of such a point consists of: A partition = k k ; A collection of distinct points x k of x; As assignment x k µ k Λ neg, so that Σ k µ k = µ. Then the!-fiber of V λ (λ + µ) at a such a point is where λ k = V λ k (λ k + µ k ), k Σ i k λ(i), and where V (ν) denotes the ν-weight space in a Ǧ-representation V Consider the category, denoted TwArr µ,/, whose objects are commutative diagrams υ J φ J ψ K φ K φ J J Λ neg µ J ψ K φ K ψ K, where the maps υ, ψ, ψ, ψ, φ J, φ K are surjective (but φ J and φ K are not necessarily so), and Σ j J µ( j ) = µ. Morphisms in this category are defined by the same principle as in TwArr µ and TwArr / introduced earlier, i.e., the sets J, J, J map forward and the sets K, K, K map backwards. Let K,,λ denote the map comprised of X K X X µ, φk ψ υ : X K X

23 THE SEM-NFNTE C SHEAF- 23 and X K φ K X K K,µ X µ. We let V λ (λ + µ) be the colimit over TwArr µ,/ of the objects ( K,,λ ) (ω X K ) ( ) V λ j (λ j + µ j ), j J where λ j = Σ i,i j λ(i) and µ j = Σ j J, j j µ( j ) Applying Sect. 1.6, we obtain a canonical isomorphism (p µ )! (i µ ) (δ λ Sat G,(V λ )) (p,µ ) (i,µ )! (δ λ Sat G,(V λ )). Now, the properties of the geometric Satake functor Sat G, imply a canonical isomorphism giving rise to the desired expression for (2.13) Finally, it is not difficult to see that (p,µ ) (i,µ )! (δ λ Sat G,(V λ ))[ λ, 2ˇρ ] V λ (λ + µ), colim V λ (λ + µ) λ Maps(,Λ + ) identifies canonically with (pr µ )! (Fact alg (O(Ň)) X λ). ndeed, this follows from the fact that we have a canonical identification where Λ + is endowed with the order relation colim λ Λ + V λ (λ + µ) O(Ň)(µ), λ 1 λ 2 λ 2 λ 1 Λ Proof of coconnectivity. n this subsection we will prove Proposition 2.7.4, thereby completing the proof of Theorem Consider the diagonal stratification of X µ. For each parameter β of the stratification, let X µ β let denote the corresponding stratum, and denote by ((X) X µ β ) := X µ β X µ ((X) Xµ ) ) ι β ((X) X µ ) and the resulting maps. Let F µ Shv(((X) X µ ) ) be such that ((X) X µ prµ β ),β X µ β (i µ )! ( C 2 ) (p µ )! (F µ ). To prove Proposition 2.7.4, its suffices to show that each is of the form (ι β )! F µ Shv(((X) X µ β ) ) (pr µ,β )! (F µ β ), where F µ β Shv(Xµ β ) is such that Fµ β [ µ, 2ˇρ ] is strictly coconnective.

24 24 DENNS GATSGORY By the factorization property of C 2 (see (2.11)), it suffices to prove the above assertion for β corresponding to the main diagonal X X µ. Denote the corresponding stratum in ((X) X µ ) by ((X) X). Denote the corresponding map pr µ,β by Denote the restriction of the section to this stratum by s µ ((X) X). We claim that pr µ ((X) X) : ((X) X) X. s µ : ((X) Xµ ) S µ (s µ ((X) X) )! ( C 2 ) (pr µ ((X) X) )! (ω X) Sym(ň [ 2])(µ)[ µ, 2ˇρ ]. This would prove the required estimate (the important thing is that Sym(ň [ 2])(µ) lives in cohomological degrees 2) n fact, we claim that for every, we have: where (s µ (X X) )! (C 2 ) (pr µ (X X) )! (ω X) Sym(ň [ 2])(µ)[ µ, 2ˇρ ], pr µ (X X) := pr µ ((X) X) (X X) ndeed, it follows from the definitions that for any λ : Λ +, (s µ (X X) )! (δ λ Sat G,(V λ ))[ λ, 2ˇρ ] (pr µ (X X) )! (ω X) W λ,µ [ µ, 2ˇρ ], where W λ,µ is the cohomogically graded vector space such that Sat(V λ ) Gr λ+µ G C Gr λ+µ G W λ,µ, where means!-restriction (again, the important thing is that W λ,µ Vect is cohomologically 2). Now, it follows (see [Ga1, Sect ]) that colim W λ Λ + λ,µ Sym(ň [ 2])(µ). 3. The semi-infinite C sheaf and Drinfeld s compactification n this section we will express C 2 in terms of an actual intersection cohomology sheaf, i.e., one arising in finite-dimensional algebraic geometry (technically, on an algebraic stack locally of finite type) Drinfeld s compactification Let Bun B Drinfeld s relative compactification of the stack Bun B along the fibers of the map Bun B Bun G..e., Bun B is the algebraic stack that classifies triples (P G, P T, κ), where: (i) P G is a G-bundle on X; (ii) P T is a T -bundle on X; (iii) κ is a Plücker data, i.e., a system of non-zero maps κˇλ : ˇλ(P T ) Vˇλ PG,

25 THE SEM-NFNTE C SHEAF- 25 (here Vˇλ denotes the Weyl module with highest weight ˇλ ˇΛ + ) that satisfy Plücker relations, i.e., for ˇλ 1 and ˇλ 2 the diagram ˇλ 1(P T ) ˇλ 2(P T ) κˇλ 1 κˇλ2 Vˇλ 1 P G 2 Vˇλ P G must commute. (ˇλ 1 + ˇλ 2)(P T ) κˇλ 1 +ˇλ 2 Vˇλ 1 +ˇλ 2 P G For λ Λ neg we let i λ glob denote the map given by Bun λ B := Bun B X λ Bun B, (P G, P T, κ, D) (P G, P T, κ ) with P G = P G, P T = P T (D) and κ given by precomposing κ with the natural maps ˇλ(P T ) = ˇλ(P T )(ˇλ(D)) ˇλ(P T ). t is known that i λ glob is a finite morphism Let j λ glob denote the open embedding Denote Bun =λ N =: Bun B X λ Bun B X λ =: Bun λ N. i λ glob = i λ glob j λ glob. Note that by definition i 0 glob = j 0 glob; this is the embedding of the open substack The following is known: Bun B Bun B. Lemma The maps i λ glob are locally closed embeddings. Every field-valued point of Bun B belongs to the image of exactly one such map The global semi-infinite category Denote where pt Bun T n particular, where X λ Bun T Bun N := Bun B pt, Bun λ N := Bun λ B pt, Bun =λ N := Bun =λ B pt, Bun T Bun T Bun T corresponds to the trivial bundle. is the Abel-Jacobi map. Bun =λ N Bun B Bun T X λ, We will denote by the same symbols the corresponding maps i λ glob : Bun λ N Bun N, j λ glob : Bun =λ N Bun λ N, i λ glob : Bun =λ N Bun N. Denote by p λ glob the projection Bun =λ N X λ.

26 26 DENNS GATSGORY We define (3.1) S 0 glob Shv(BunN ) to be the full subcategory defined by the following condition: An object F Shv(Bun N ) belongs to S 0 glob if and only if for every λ Λneg, the object (i λ glob)! (F) Shv(Bun =λ N ) belongs to the full subcategory S =λ glob Shv(Bun =λ N ), equal by definition to the essential image of the pullback functor (p λ glob)! : Shv(X λ ) Shv(Bun =λ N ). We note that the above pullback functor is fully faithful, since the map p λ glob, being a base change of Bun B Bun T, is smooth with homologically contractible fibers We note that the full subcategory (3.1) can also be defined by an equivariance condition with respect to a certain pro-unipotent groupoid, see [Ga3, Sect ]. n particular, this fully faithful embedding admits a right adjoint 4. and We have the naturally defined functors (i λ glob)! : S 0 glob S=λ glob (i λ glob) : S =λ glob S 0 glob. n addition, one shows that the partially defined functor (i λ glob)! left adjoint to (i λ glob)!, is defined on glob Shv(Bun =λ N ) and takes values in S 0 Shv(BunN ). S =λ glob t follows formally that the partially defined functor (i λ glob), left adjoint to (i λ glob), is defined on S 0 glob Shv(BunN ) and takes values in S=λ glob Shv(Bun =λ N ) The embeddings glob Shv(Bun =λ N ) and S 0 S =λ glob Shv(BunN ) are compatible with the t-structure on the target categories. This follows from the fact that the right adjoints to these embeddings are right t-exact. Hence, the categories S =λ glob and S 0 glob acquire t-structures. By construction, an object F S 0 glob is connective (resp., coconnective) if and only if (i λ glob) (F) (resp., (i λ glob)! (F) is connective (resp., coconnective) for every λ Λ neg We will denote by C 2 glob (S 0 glob ) the minimal extension of C BunN (S =0 glob) along j 0 glob Local vs global compatibility for the semi-infinite C sheaf For every finite set we have a canonically defined map Together these maps combine to a map π : S 0 Bun N. π : S 0 Bun N. 4 The corresponding assertion would be false for the corresponding embedding S 0 Shv(S0 ).