chiral principal series categories i: finite dimensional ...ย ยท 5this is compatible with the...
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL
CALCULATIONS
SAM RASKIN
Abstract. This paper begins a series studying ๐ท-modules on the Feigin-Frenkel semi-infinite flagvariety from the perspective of the Beilinson-Drinfeld factorization (or chiral) theory.
Here we calculate Whittaker-twisted cohomology groups of Zastava spaces, which are certainfinite-dimensional subvarieties of the affine Grassmannian. We show that such cohomology groupsrealize the nilradical of a Borel subalgebra for the Langlands dual group in a precise sense, follow-ing earlier work of Feigin-Finkelberg-Kuznetsov-Mirkovic and Braverman-Gaitsgory. Moreover, wecompare this geometric realization of the Langlands dual group to the standard one provided by(factorizable) geometric Satake.
Contents
1. Introduction 12. Review of Zastava spaces 153. Limiting case of the Casselman-Shalika formula 234. Identification of the Chevalley complex 285. Hecke functors: Zastava calculation over a point 326. Around factorizable Satake 397. Hecke functors: Zastava with moving points 53Appendix A. Proof of Lemma 6.18.1 65Appendix B. Universal local acyclicity 68References 72
1. Introduction
1.1. Semi-infinite flag variety. This paper begins a series concerning ๐ท-modules on the semi-infinite flag variety of Feigin-Frenkel.
Let ๐บ be a split reductive group over ๐ a field of characteristic zero. Let ๐ต be a Borel withradical ๐ and reductive quotient ๐ต๐ โ ๐ .
Let ๐ be a smooth curve. We let ๐ฅ P ๐ be a fixed ๐-point. Let ๐๐ฅ โ ๐rr๐ก๐ฅss and ๐พ๐ฅ โ ๐pp๐ก๐ฅqq be
the rings Taylor and Laurent series based at ๐ฅ. Let ๐๐ฅ and๐๐๐ฅ denote the spectra of these rings.
Informally, the semi-infinite flag variety should be a quotient Fl82๐ฅ :โ ๐บp๐พ๐ฅq๐p๐พ๐ฅq๐ p๐๐ฅq, but
this quotient is by an infinite-dimensional group and therefore leaves the realm of usual algebraicgeometry.
January 2015. Updated: September 5, 2018.
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2 SAM RASKIN
Still, we will explain in future work [Ras3] how to make precise sense of ๐ท-modules on Fl82๐ฅ ,
but we ask the reader to take on faith for this introduction that such a category makes sense.1
This category will not play any explicit role in the present paper, and will be carefully discussedin [Ras3]; however, it plays an important motivational role in this introduction.
1.2. Why semi-infinite flags? The desire for a theory of sheaves on the semi-infinite flag vari-ety stretches back to the early days of geometric representation theory: see [FF], [FM], [FFKM],[BFGM], and [ABB`]. Among these works, there are diverse goals and perspectives, showing the
rich representation theoretic nature of Fl82๐ฅ .
โ [FF] explains that the analogy between Wakimoto modules for an affine Kac-Moody al-gebra g๐ ,๐ฅ and Verma modules for the finite-dimensional algebra g should be understood
through the Beilinson-Bernstein localization picture, with Fl82๐ฅ playing the role of the finite-
dimensional ๐บ๐ต.โ [FM], [FFKM] and [ABB`] relate the semi-infinite flag variety to representations of Lusztigโs
small quantum group, following Finkelberg, Feigin-Frenkel and Lusztig.
โ As noted in [ABB`], ๐ทpFl82๐ฅ q โ ๐ทp๐บp๐พ๐ฅq๐p๐พ๐ฅq๐ p๐๐ฅqq plays the role of the universal
unramified principal series representation of ๐บp๐พ๐ฅq in the categorical setting of local geo-metric Langlands (see [FG2] and [Ber] for some modern discussion of this framework andits ambitions).
However, these references (except [FF], which is not rigorous on these points) uses ad hoc finite-dimensional models for the semi-infinite flag variety.
Remark 1.2.1. One of our principal motivations in this work and its sequels is to study ๐ทpFl82๐ฅ q from
the perspective of the geometric Langlands program, and then to use local to global methods toapply this to the study of geometric Eisenstein series in the global unramified geometric Langlandsprogram. But this present work is also closely2 connected to the above, earlier work, as we hope toexplore further in the future.
1.3. The present series of papers will introduce the whole category ๐ทpFl82๐ฅ q and study some inter-
esting parts of its representation theory: e.g., we will explain how to compute Exts between certainobjects in terms of the Langlands dual group.
Studying the whole of๐ทpFl82๐ฅ q was neglected by previous works (presumably) due to the technical,
infinite-dimensional nature of its construction.
1.4. The role of the present paper. Whatever the definition of ๐ทpFl82๐ฅ q is, it is not obvious how
to compute directly with it. The primary problem is that we do not have such a good theory ofperverse sheaves in the infinite type setting: the usual theory [BBD] of middle extensions โ whichis so crucial in connecting combinatorics (e.g., Langlands duality) and geometry โ does not existfor embeddings of infinite codimension.
Therefore, to study ๐ทpFl82๐ฅ q, it is necessary to reduce our computations to finite-dimensional
ones. This paper performs those computations, and this is the reason why the category ๐ทpFl82๐ฅ q
1For the overly curious reader: one takes the category ๐ท!p๐บp๐พ๐ฅqq from [Ber] (c.f. also [Ras2]) and imposes the
coinvariant condition with respect to the group indscheme ๐p๐พ๐ฅq๐ p๐๐ฅq.2But non-trivially, due to the ad hoc definitions in earlier works.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 3
does not explicitly appear here.3 (That said, the author finds these computations to be interestingin their own right.)
1.5. In the remainder of the introduction, we will discuss problems close to those to be consideredin [Ras3], and discuss the contents of the present paper and their connection to the above problems.
1.6. ๐ทpFl82๐ฅ q is boring. One can show4 that ๐ทpFl
82๐ฅ q is equivalent to the category ๐ทpFlaff๐บ,๐ฅq of
๐ท-modules on the affine flag variety ๐บp๐พ๐ฅq๐ผ (where ๐ผ is the Iwahori subgroup) in a ๐บp๐พ๐ฅq-equivariant way.5
At first pass, this means that essentially6 every question in local geometric Langlands about
๐ทpFl82๐ฅ q has either been answered in the exhaustive works of Bezrukavnikov and collaborators
(especially [AB], [ABG], and [Bez]), or else is completely out of reach (e.g., some conjectures from[FG2]).
Thus, it would appear that there is nothing new to say about ๐ทpFl82๐ฅ q.
1.7. ๐ทpFl82๐ฅ q is not boring (or: factorization). However, there is a significant difference between
the affine and semi-infinite flag varieties: the latter factorizes in the sense of Beilinson-Drinfeld [BD].We refer to the introduction to [Ras1] for an introduction to factorization. Modulo the non-
existence of Fl82๐ฅ , let us recall that this essentially means that for each finite set ๐ผ, we have a โsemi-
infinite flag varietyโ Fl82
๐๐ผ over ๐๐ผ whose fiber at a point p๐ฅ๐q๐P๐ผ P ๐๐ผ is the product
ล
t๐ฅ๐u๐P๐ผFl
82๐ฅ๐ .
Here t๐ฅ๐u๐P๐ผ is the unordered set in which we have forgotten the multiplicities with which pointsappear.
However, it is well-known that the Iwahori subgroup (unlike ๐บp๐๐ฅq) does not factorize.7
Remark 1.7.1. The methods of the Bezrukavnikov school do not readily adapt to studying Fl82๐ฅ
factorizably: they heavily rely on the ind-finite type and ind-proper nature of Flaff๐บ , which are notmanifested in the factorization setting.
1.8. But why is it not boring? (Or: why factorization?) As discussed in the introduction to[Ras1], there are several reasons to care about factorization structures.
โ Most imminently (from the perspective of Remark 1.2.1), the theory of chiral homology(c.f. [BD]) provides a way of constructing global invariants from factorizable local ones.Therefore, identifying spectral and geometric factorization categories allows us to compareglobally defined invariants as well.
3We hope the reader will benefit from this separation, and not merely suffer through an introduction some much ofwhose contents has little to do with the paper at hand.4This result will appear in [Ras3].5This is compatible with the analogy with ๐-adic representation theory: c.f. [Cas].6This is not completely true: for the study of Kac-Moody algebras, the semi-infinite flag variety has an interestingglobal sections functor. It differs from the global sections functor of the affine flag variety in as much as Wakimotomodules differ from Verma modules.7It is instructive to try and fail to define a factorization version of the Iwahori subgroup that lives over ๐2: a pointshould be a pair of points ๐ฅ1, ๐ฅ2 in ๐, ๐บ-bundle on ๐ with a trivialization away from ๐ฅ1 and ๐ฅ2, and with a reductionto the Borel ๐ต at the points ๐ฅ1 and ๐ฅ2. However, for this to define a scheme, we need to ask for a reduction to ๐ต atthe divisor-theoretic union of the points ๐ฅ1 and ๐ฅ2. Therefore, over a point ๐ฅ in the diagonal ๐ ฤ ๐2, we are askingfor a reduction to ๐ต on the first infinitesimal neighborhood of ๐ฅ, which defines a subgroup of ๐บp๐๐ฅq smaller than theIwahori group.
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4 SAM RASKIN
โ Factorization structures also play a key, if sometimes subtle, role in the purely local theory.Let us mention one manifestation of this: the localization theory [FG3] (at critical level)
for Flaff๐บ has to do with the structure of the Kac-Moody algebra pg๐๐๐๐ก as a bare Lie algebra.
A factorizable localization theory for Fl82๐ฅ would connect to the vertex algebra structure on
its vacuum representation.
โ In [FM], [FFKM], [ABB`], and [BFS], sheaves on Fl82๐ฅ are defined using factorization struc-
tures. We anticipate the eventual comparison between our category๐ทpFl82๐ฅ q and the previous
ones to pass through the factorization structure of Fl82๐ฅ .
1.9. Main conjecture. Our main conjecture is about Langlands duality for certain factorizationcategories: the geometric side concerns some ๐ท-modules on the semi-infinite flag variety, and thespectral side concerns coherent sheaves on certain spaces of local systems.
See below for a more evocative description of the two sides.
1.10. Let ๐ตยด be a Borel opposite to ๐ต, and let ๐ยด denote its unipotent radical.Recall that for any category C acted on by ๐บp๐พ๐ฅq in the sense of [Ber], we can form its Whittaker
subcategory, WhitpCq ฤ C consisting of objects equivariant against a non-degenerate character of๐ยดp๐พq.
Moreover, up to certain twists (which we ignore in this introduction: see S2.8 for their definitions),this makes sense factorizably.
For each finite set ๐ผ, there is therefore a category Whit82
๐๐ผ to be the of Whittaker equivariant
๐ท-modules on Fl82
๐๐ผ , and the assignment ๐ผ รร Whit82
๐๐ผ defines a factorization category in the senseof [Ras1]. This forms the geometric side of our conjecture.
1.11. For a point ๐ฅ P ๐ and an affine algebraic group ๐ค , let LocSys๐ค p๐๐๐ฅq denote the prestack of
de Rham local systems with structure group ๐ค on๐๐๐ฅ.
Formally: we have the indscheme Conn๐ค of Liep๐ค q-valued 1-forms (i.e., connection forms) on thepunctured disc, which is equipped with the usual gauge action of ๐ค p๐พ๐ฅq. We form the quotient and
denote this by LocSys๐ค p๐๐๐ฅq.
Remark 1.11.1. LocSys๐ค p๐๐๐ฅq is not an algebraic stack of any kind because we quotient by the loop
group ๐ค p๐พ๐ฅq, an indscheme of ind-infinite type. It might be considered as a kind of semi-infiniteArtin stack, the theory of which has unfortunately not been developed.
The assignment ๐ฅ รร LocSys๐ค p๐๐๐ฅq factorizes in an obvious way.
1.12. Recall that for a finite type (derived) scheme (or stack) ๐, [GR] has defined a DG categoryIndCohp๐q of ind-coherent sheaves on ๐.8
We would like to take as the spectral side of our equivalence the factorization category:
๐ฅ รร IndCoh`
LocSysp๐๐๐ฅq ห
LocSys๐ p๐๐๐ฅq
LocSys๐ p๐๐ฅqห
.
8For the reader unfamiliar with the theory of loc. cit., we recall that this sheaf theoretic framework is very closeto the more familiar QCoh, but is the natural setting for Grothendieckโs functor ๐ ! of exceptional inverse image (asopposed to the functor ๐ห, which is adapted to QCoh).
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 5
Here and everywhere, we use e.g. to refer to the reductive group Langlands dual to ๐บ, and ฤ to refer to the corresponding Borel subgroup, etc. (c.f. S1.41).
However, note that IndCoh has not been defined in this setting: the spaces of local systems onthe punctured disc are defined as the quotient of an indscheme of ind-infinite type by a group ofind-infinite type.
We ignore this problem in what follows, describing a substitute in S1.15 below.
1.13. We now formulate the following conjecture:
Main Conjecture. There is an equivalence of factorization categories:
Whit82ยปรร
ยด
๐ฅ รร IndCoh`
LocSysp๐๐๐ฅq ห
LocSys๐ p๐๐๐ฅq
LocSys๐ p๐๐ฅqห
ยฏ
. (1.13.1)
Remark 1.13.1. Identifying ๐ท-modules on the affine flag variety and on the semi-infinite flag variety,one can show that fiberwise, this conjecture recovers the main result of [AB]. However, as noted inRemark 1.7.1, the methods of loc. cit. are not amenable to the factorizable setting.
1.14. What is contained in this paper? In [FM], Finkelberg and Mirkovic argue that theirZastava spaces provide finite-dimensional models for the geometry of the semi-infinite flag variety.
In essence, we are using this model in the present paper: we compute some twisted cohomologygroups of Zastava spaces, and these computations will provide the main input for our later study[Ras3] of semi-infinite flag varieties.
In S1.15-1.21, we describe a certain factorization algebra ฮฅn and its role in the main conjecture(from S1.13). In S1.22-1.27, we recall some tactile aspects of the geometry of Zastava spaces. Finally,in S1.28-1.37, we formulate the main results of this text: these realize ฮฅn (and its modules) as twistedcohomology groups on Zastava space.
Remark 1.14.1. Some of the descriptions below may go a bit quickly for a reader who is a non-expert in this area. We hope that for such a reader, the material that follows helps to supplementwhat it is written more slowly in the body of the text.
1.15. The factorization algebra ฮฅn. To describe the main results of this paper, we need todescribe how we model the spectral side of the main conjecture i.e., the category of ind-coherentsheaves on the appropriate space of local systems.
We will do this using the graded factorization algebra ฮฅn, introduced in [BG2].After preliminary remarks about what graded factorization algebras are in S1.16, we introduce
ฮฅn in S1.17. Finally, in S1.20-1.21, we explain why factorization modules for ฮฅn are related to thespectral side of the main conjecture.
1.16. Let ฮ๐๐๐ ฤ ฮ :โ tcocharacters of ๐บu denote the Zฤ0-span of the simple coroots (relative to๐ต).
Let Divฮ๐๐๐
eff denote the space of ฮ๐๐๐ -valued divisors on ๐. I.e., its ๐-points are written:
รฟ
t๐ฅ๐uฤ๐ finite
๐ ยจ ๐ฅ๐ (1.16.1)
for ๐ P ฮ๐๐๐ , and for ๐บ of semi-simple rank 1, this space is the union of the symmetric powers of๐ (for general ๐บ, connected components are products of symmetric powers of ๐).
For P ฮ๐๐๐ , we let Diveff denote the connected component of Divฮ๐๐๐
eff of divisors of total degree
(i.e., in the above we haveล
๐ โ ).
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6 SAM RASKIN
A (ฮ๐๐๐ )-graded factorization algebra is the datum of ๐ท-modules:
A P ๐ทpDiveffq, P ฮ๐๐๐
plus symmetric and associative isomorphisms:
A`|rDiveff หDiveff s๐๐๐ ๐
ยป A bA|rDiveff หDiveff s๐๐๐ ๐
.
Here:
rDiveff หDiveff s๐๐๐ ๐ ฤ Diveff หDiveffdenotes the open locus of pairs of (colored) divisors with disjoint supports, which we consider
mapping to Div`eff through the map of addition of divisors (which is etale on this locus).
Remark 1.16.1. The theory of graded factorization algebras closely imitates the theory of factor-
ization algebras from [BD], with the above Divฮ๐๐๐
eff replacing the Ran space from loc. cit.
1.17. The ฮ๐๐๐ -graded Lie algebra n defines a Lie-ห algebra:9
n๐ :โ โ a coroot of ๐บ n bโห,๐๐ p๐๐q P ๐ทpDivฮ๐๐๐
eff q.
In this notation, for a finite type scheme ๐, ๐๐ denotes its (๐ท-module version of the) constant sheaf;n denotes the corresponding graded component of n; and โ : ๐ ร Diveff denotes the diagonalembedding.
As in [BD], we may form the chiral enveloping algebra of n๐ : we let ฮฅn denote the correspondingfactorization algebra. For the reader unfamiliar with [BD], we remind that ฮฅn is associated to n๐as a sort of Chevalley complex; in particular, the ห-fiber of ฮฅn at a point (1.16.1) is:
b๐๐ถโpnq
๐
where ๐ถโ denotes the (homological) Chevalley complex of a Lie algebra (i.e., the complex computingLie algebra homology).
1.18. Next, we recall that in the general setup of S1.16, to a graded factorization algebra A anda closed point ๐ฅ P ๐, we can associate a DG category Aโmodfact๐ฅ of its (ฮ-graded) factorizationmodules โat ๐ฅ P ๐.โ
First, let Divฮ๐๐๐ ,8ยจ๐ฅeff denote the indscheme of ฮ-valued divisors on ๐ that are ฮ๐๐๐ -valued on
๐z๐ฅ. So ๐-points of this space are sums:
ยจ ๐ฅ`รฟ
t๐ฅ๐uฤ๐z๐ฅ finite
๐ ยจ ๐ฅ๐
where P ฮ and ๐ P ฮ๐๐๐ (to see the indscheme structure, bound how small can be).
Then a factorization module for A is a ๐ท-module ๐ P ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅeff q equipped with an isomor-
phism:
add!p๐q|rDivฮ
๐๐๐ eff หDivฮ
๐๐๐ ,8ยจ๐ฅeff
ยป Ab๐ |rDivฮ
๐๐๐ eff หDivฮ
๐๐๐ ,8ยจ๐ฅeff
which is associative with respect to the factorization structure on A, where add is the map:
9Here the structure of Lie-ห algebra is defined in [BD] (see also [FG1], [Ras1] for derived versions). For the readerโssake, we simply note that this datum encodes the natural structure on n๐ inherited from the Lie bracket on n.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 7
Divฮ๐๐๐
eff หDivฮ๐๐๐ ,8ยจ๐ฅeff ร Divฮ๐๐๐ ,8ยจ๐ฅ
eff
of addition of divisors.Factorization modules form a DG category in the obvious way.
Remark 1.18.1. In what follows, we will need unital versions of the above, i.e., unital factorizationalgebras and unital modules. This is a technical requirement, and for the sake of brevity we do notspell it out here, referring to [BD] or [Ras1] for details. However, this is the reason that notations of
the form Aโmodfact๐ข๐,๐ฅ appear below instead of Aโmodfact๐ฅ . However, we remark that whatever theseunital structures are, chiral envelopes always carry them, and in particular ฮฅn does.
Remark 1.18.2. Note that the affine Grassmannian Gr๐,๐ฅ โ ๐ p๐พ๐ฅq๐ p๐๐ฅq with structure group
๐ embeds into Divฮ๐๐๐ ,8ยจ๐ฅeff as the locus of divisors supported at the point ๐ฅ. We remind that the
reduced scheme underlying Gr๐,๐ฅ is the discrete scheme ฮ.
1.19. The following provides the connection between ฮฅn and the main conjecture.
Principle. (1) There is a canonical equivalence:
ฮฅnโmodfact๐ข๐,๐ฅ ยป IndCoh`
LocSysp๐๐๐ฅq
^LocSys๐ p๐๐ฅq
ห
LocSys๐ p๐๐๐ฅq
LocSys๐ p๐๐ฅqห
(1.19.1)
where LocSysp๐๐๐ฅq
^LocSys๐ p๐๐ฅq
indicates the formal completion of LocSysp๐๐๐ฅq with respect
to the map from LocSys๐ p๐๐ฅq.10
(2) Under this equivalence, the functor:11
ฮฅnโmodfact๐ข๐,๐ฅOblvรรรร ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅ
eff q!ยดrestrictionรรรรรรรรร ๐ทpGr๐,๐ฅq ยป Repp๐ q ยป QCohpLocSys๐ p๐๐ฅqq
corresponds to the functor of !-restriction along the map:
LocSys๐ p๐๐ฅq ร LocSysp๐๐๐ฅq
^ ห
LocSys๐ p๐๐๐ฅq
LocSys๐ p๐๐ฅq.
(3) The above two facts generalize to the factorization setting, where ๐ฅ is replaced by severalpoints allowed to move and collide.
Remark 1.19.1. This is a principle and not a theorem because the right hand side of (1.19.1) isnot defined (we remind that this is because IndCoh is only defined in finite type situations, whileLocSys leaves this world). Therefore, the reader might take it simply as a definition.
For the reader familiar with derived deformation theory (as in [Lur2], [GR]) and [BD], we willexplain heuristically in S1.20-1.21 why we take this principle as given. However, the reader who isnot familiar with these subjects may safely skip this material, as it plays only a motivational rolefor us.
10For a fixed ๐-point ๐ฅ P ๐, LocSysp๐
๐๐ฅq^LocSys๐ p๐๐ฅq
is isomorphic to b^0 ^ยจ ๐ , so the whole fiber product is
isomorphic to n^0 ^๐ . Here ^ is the formal group for , i.e., the formal completion at the identity.
11Here and throughout the text, for an algebraic group ๐ค , Repp๐ค q denotes the derived (i.e., DG) category of itsrepresentations, i.e., QCohpB๐บq.
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8 SAM RASKIN
Remark 1.19.2. We note that (heuristically) ind-coherent sheaves on (1.19.1) should be a full sub-
category of IndCoh`
LocSysp๐๐๐ฅq ห
LocSys๐ p๐๐๐ฅq
LocSys๐ p๐๐ฅqห
.12
In [Ras3], we will use the computations of the present paper to construct a functor:
Whit82๐ฅ
ยปรร
ยด
๐ฅ รร IndCoh`
LocSysp๐๐๐ฅq
^ ห
LocSys๐ p๐๐๐ฅq
LocSys๐ p๐๐ฅqห
:โ ฮฅnโmodfact๐ข๐,๐ฅ
ยฏ
and identify a full subcategory of Whit82๐ฅ on which this functor is an equivalence. Moreover, this
equivalence is factorizable, and therefore gives the main conjecture (from S1.13) when restricted tothese full subcategories.
1.20. As stated above, the reader may safely skip S1.20-1.21, which are included to justify theprinciple of S1.19.
We briefly recall Lurieโs approach to deformation theory [Lur2].Suppose that X is a โnice enoughโ stack and ๐ฅ P X is a ๐-point, with the formal completion of
X at ๐ฅ denoted by X^๐ฅ . Then the fiber ๐X,๐ฅrยด1s of the shifted tangent complex of X at ๐ฅ identifieswith the Lie algebra of the (derived) automorphism group (alias: inertia) Aut๐ฅpXq :โ ๐ฅหX ๐ฅ of Xat ๐ฅ, and there is an identification of the DG category IndCohpX^๐ฅ q of ind-coherent sheaves on theformal completion of X at ๐ฅ with ๐X,๐ฅrยด1s-modules.
1.21. At the trivial local system, the fiber of the shifted tangent complex of LocSys p๐๐๐ฅq is the
(derived) Lie algebra ๐ปห๐๐ p๐๐๐ฅ, n b ๐q. The philosophy of [BD] indicates that modules for this Lie
algebra should be equivalent to factorization modules for the chiral envelope of the Lie-ห algebranb ๐๐ on ๐.
The ฮ-graded variant of thisโthat is, the version in the setting of S1.16 in which symmetricpowers of the curve replace the Ran space from [BD]โprovides the principle of S1.19.
1.22. Zastava spaces. Next, we describe the most salient features of Zastava spaces. We remarkthat this geometry is reviewed in detail in S2.
1.23. There are two Zastava spaces,๐๐ต and ๐ต, each fibered over Divฮ๐๐๐
eff : the relationship is that๐๐ต embeds into ๐ต as an open, and for this reason, we sometimes refer to ๐ต as Zastava space and
๐๐ต
as open Zastava space.For the purposes of this introduction, we content ourselves with a description of the fibers of the
maps:
๐๐ต //
๐๐ ""
๐ต
๐
Divฮ๐๐๐
eff .
To give this description, we will first recall the so-called central fibers of the Zastava spaces.
12This combines the facts that ind-coherent sheaves on a formal completion are a full subcategory of ind-coherentsheaves of the whole space, and the fact that ind-coherent sheaves on the classifying stack of the formal group of aunipotent group are a full subcategory of ind-coherent sheaves on the classifying stack of the group.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 9
1.24. Recall that e.g. Gr๐บ,๐ฅ denotes the affine Grassmannian ๐บp๐พ๐ฅq๐บp๐๐ฅq of ๐บ at ๐ฅ.
For ๐ฅ P ๐ a geometric point and P ฮ๐๐๐ , define the central fiber๐Z๐ฅ as the intersection:
Gr๐ยด,๐ฅXGr๐ต,๐ฅ โยด
๐ยดp๐พ๐ฅq๐บp๐๐ฅqฤ
๐p๐พ๐ฅqp๐ก๐ฅq๐บp๐๐ฅqยฏ
๐บp๐๐ฅq ฤ ๐บp๐พ๐ฅq๐บp๐๐ฅq โ Gr๐บ,๐ฅ
where ๐ก๐ฅ is any uniformizer at ๐ฅ. Here we recall that Gr๐ยด,๐ฅ โ ๐ยดp๐พ๐ฅq๐บp๐๐ฅq๐บp๐๐ฅq and Gr๐ต,๐ฅ โ
๐p๐พ๐ฅqp๐ก๐ฅq๐บp๐๐ฅq๐บp๐๐ฅq embed into Gr๐บ,๐ฅ as ind-locally closed subschemes (of infinite dimensionand codimension).13
A small miracle: the intersections๐Z๐ฅ are finite type, and equidimensional of dimension p๐, q.
Example 1.24.1. For โ a simple coroot, one has๐Z๐ฅ ยป A1zt0u.
1.25. Let Gr๐ต,๐ฅ denote the closure of Gr๐ต,๐ฅ in Gr๐บ,๐ฅ.14 We remind that Gr
๐ต,๐ฅ has an (infinite)
stratification by the ind-locally closed subschemes Grยด๐ต,๐ฅ for P ฮ๐๐๐ .
We then define Z๐ฅ as the corresponding intersection:
Gr๐ยด,๐ฅXGr๐ต,๐ฅ ฤ Gr๐บ,๐ฅ .
Again, this intersection is finite-dimensional, and equidimensional of dimension p๐, q.
Example 1.25.1. For โ a simple coroot, one has Z๐ฅ ยป A1.
1.26. Now, for a ๐-point (1.16.1) of Diveff (for :โล
๐), the corresponding fiber of๐๐ต along
๐๐
is:
ลบ ๐Z๐๐ฅ๐ (1.26.1)
and similarly for ๐ต.
Again,๐๐ต and ๐ต are equidimensional of dimension p2๐, q, and moreover,
๐๐ต is actually smooth.
1.27. Finally, there is a canonical map can : ๐ต ร G๐, which is constructed (fiberwise) as follows.First, define the map ๐ยดp๐พ๐ฅq ร G๐ by:
๐ยดp๐พ๐ฅq ร p๐ยดr๐ยด, ๐ยดsqp๐พ๐ฅq ยปลบ
simple roots
๐พ๐ฅsum over coordinatesรรรรรรรรรรรรรร ๐พ๐ฅ
๐ รรResp๐ ยจ๐๐ก๐ฅqรรรรรรรรร G๐
where Res denotes the residue map and ๐ก๐ฅ is a coordinate in ๐พ๐ฅ.
Remark 1.27.1. The twists we mentioned in S1.10 are included so that we do not have to choosea coordinate ๐ก๐ฅ, but rather have a canonical residue map to G๐. But we continue to ignore thesetwists, reminding simply that they are spelled out in S2.8.
It is clear that this map factors uniquely through the projection ๐ยดp๐พ๐ฅq ร Gr๐ยด .We now map (1.26.1) by embedding into the product of Gr๐ยด,๐ฅ๐ and summing the corresponding
maps to G๐ over the points ๐ฅ๐.
13The requirement that P ฮ๐๐๐ is included so that this intersection is non-empty.14As a moduli problem, Gr
๐ต,๐ฅ can be defined analogously to Drinfeldโs compactification of Bun
๐ต .
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10 SAM RASKIN
In what follows, we let ๐๐ต P ๐ทp๐ตq (resp. ๐ ๐๐ตP ๐ทp
๐๐ตq) denote the !-pullback of the Artin-Shreier
(i.e., exponential) ๐ท-module on G๐ (normalized to be in the same cohomological degree as thedualizing ๐ท-module of G๐).
Remark 1.27.2. The above map ๐ยดp๐พ๐ฅq ร G๐ is referred to as the Whittaker character, and werefer to sheaves constructed out of it the Artin-Shreier sheaf (e.g., ๐ ๐
๐ต, ๐๐ต) as Whittaker sheaves.
1.28. Formulation of the main results of this paper. Here is a rough overview of the mainresults of this paper, to be expanded upon below:
Roughly, the first main result of this paper, Theorem 4.6.1, identifies ฮฅn with certain Whittakercohomology groups on Zastava space; see loc. cit. for more details. This theorem, following [BG2]and [FFKM], provides passage from the group ๐บ to the dual group (via ฮฅn) which is differentfrom geometric Satake.
The second main result, Theorem 7.9.1 (see also Theorem 5.14.1) compares Theorem 4.6.1 withthe geometric Satake equivalence.
1.29. We now give a more precise description of the above theorems.Our first main result is the following.
Theorem (Thm. 4.6.1).๐๐ห,๐๐ p๐ ๐
๐ต
!b IC ๐
๐ตq P ๐ทpDivฮ๐๐๐
eff q is concentrated in cohomological degree
zero, and identifies canonically with ฮฅn. Here IC indicates the intersection cohomology sheaf15 (by
smoothness of the๐๐ต , this just effects cohomological shifts on the connected component of
๐๐ต).
Moreover, the factorization structure on Zastava spaces induces a factorization algebra structure
on๐๐ห,๐๐ p๐ ๐
๐ต
!b IC ๐
๐ตq, and the above equivalence upgrades to an equivalence of factorization algebras.
In words: the (Divฮ๐๐๐
eff -parametrized) cohomology of Zastava spaces twisted by the Whittakersheaf is ฮฅn.
Remark 1.29.1. We draw the readerโs attention to S1.35 below for a closely related result, but whichis less imminently related to the theme of semi-infinite flags.
1.30. Polar Zastava space. To formulate Theorem 5.14.1, we introduce a certain indscheme๐๐ต8ยจ๐ฅ
with a map๐๐8ยจ๐ฅ :
๐๐ต8ยจ๐ฅ ร Divฮ๐๐๐ ,8ยจ๐ฅ
eff , where the geometry is certainly analogous to๐๐ :
๐๐ต ร Divฮ
eff .
(Here we remind that Divฮ๐๐๐ ,8ยจ๐ฅeff parametrizes ฮ-valued divisors on๐ that are ฮ๐๐๐ -valued on๐z๐ฅ.)
As with๐๐ต, for this introduction we only describe the fibers of the map
๐๐8ยจ๐ฅ. Namely, at a point16
ยจ ๐ฅ`ล
t๐ฅ๐uฤ๐z๐ฅ finite ๐ ยจ ๐ฅ๐ of Divฮ๐๐๐ ,8ยจ๐ฅeff , the fiber is:
Gr๐ต,๐ฅหลบ
๐
๐Z๐๐ฅ๐ .
We refer the reader to S5 for more details on the definition.
15Since๐
๐ต is a union of the varieties๐
๐ต , we define this IC sheaf as the direct sum of the IC sheaves of the connectedcomponents.16We remind that this means that P ฮ and ๐ P ฮ๐๐๐ .
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 11
1.31. We will explain in S5 how geometric Satake produces a functor Reppq ร ๐ทp๐๐ต8ยจ๐ฅq.
Though this functor is not so complicated, giving its definition here would require further di-gressions, so we ask the reader to take this point on faith. Instead, for the purposes of an overview,we refer to S1.33, where we explain what is going on when we restrict to divisors supported at thepoint ๐ฅ, and certainly we refer to S5 where a detailed construction of this functor is given.
Example 1.31.1. The above functor sends the trivial representation to the ห-extension of ๐ ๐๐ต
under
the natural embedding๐๐ต รฃร
๐๐ต8ยจ๐ฅ.
We now obtain a functor:
Reppq ร ๐ทp๐๐ต8ยจ๐ฅq
๐๐8ยจ๐ฅห,๐๐ รรรร ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅ
eff q.
For geometric reasons explained in S5, Theorem 4.6.1 allows us to upgrade this construction toa functor:
Chevgeomn,๐ฅ : Reppq ร ฮฅnโmodfact๐ข๐,๐ฅ.
We now have the following compatibility between geometric Satake and Theorem 4.6.1.
Theorem (Thm. 5.14.1). The functor Chevgeomn,๐ฅ is canonically identified with the functor Chevspec
n,๐ฅ ,which by definition is the functor:
ReppqResรรร Reppq
Resรรร nโmodpRepp๐ qq
Ind๐โรรรร ฮฅnโmodfact๐ข๐,๐ฅ.
Here Ind๐โ is the chiral induction functor from Lie-* modules for nb ๐๐ to factorization modulesfor ฮฅn.
Remark 1.31.2. Here we remind the reader that chiral induction is introduced (abelian categori-cally) in [BD] S3.7.15. Like the chiral enveloping algebra operation used to define ฮฅn, chiral inductionis again a kind of homological Chevalley complex.
Example 1.31.3. For the trivial representation, Example 1.31.1 reduces Theorem 5.14.1 to The-orem 4.6.1. Here, the claim is that Chevgeom
n,๐ฅ of the trivial representation is the ๐ท-module on
๐ทpDivฮ๐๐๐ ,8ยจ๐ฅeff q obtained by pushforward from ฮฅn along Divฮ๐๐๐
eff รฃร Divฮ๐๐๐ ,8ยจ๐ฅeff , i.e., the so-called
vacuum representation of ฮฅn (at ๐ฅ).
1.32. Our last main result is the following, which we leave vague here.
Theorem (Thm. 7.9.1). A generalization of Theorem 5.14.1 holds when we work factorizably inthe variable ๐ฅ, i.e., working at several points at once, allowing them to move and to collide.
Somewhat more precisely, we define in S6 a DG category Reppq๐๐ผ โover ๐๐ผ๐๐ โ (i.e., with a
๐ทp๐๐ผq-module category structure) encoding the symmetric monoidal structure on Reppq๐๐ผ .17
Most of S6 is devoted to giving preliminary technical constructions that allow us to formulateTheorem 7.9.1.
17The construction of Reppq๐๐ผ is a categorification of the construction of [BD] that associated a factorization algebrawith a usual commutative algebra.
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12 SAM RASKIN
1.33. Interpretation in terms of Fl82๐ฅ . We now indicate briefly what e.g. Theorem 5.14.1 has to
do with Fl82๐ฅ . This section has nothing to do with the contents of the paper, and therefore can be
skipped; we include it only to make contact with our earlier motivation.Fix a closed point ๐ฅ P ๐, and consider the spherical Whittaker category Whit๐ ๐โ๐ฅ ฤ ๐ทpGr๐บ,๐ฅq,
which by definition is the Whittaker category (in the sense of S1.10) of ๐ทpGr๐บ,๐ฅq. There is acanonical object in this category (supported on Gr๐ยด,๐ฅ ฤ Gr๐บ,๐ฅ), and one can show (c.f. Theorem6.36.1) that the resulting functor:
Reppqgeometric Satakeรรรรรรรรรรร Sph๐บ,๐ฅ รWhit๐ ๐โ๐ฅ
is an equivalence, where Sph๐บ,๐ฅ :โ ๐ทpGr๐บ,๐ฅq๐บp๐q๐ฅ is the spherical Hecke category, and the latter
functor is convolution with this preferred object of Whit๐ ๐โ๐ฅ .
Let ๐82,! : ๐ทpFl
82๐ฅ q ร ๐ทpGr๐,๐ฅq denote the functor encoding !-restriction along:
๐82 : Gr๐,๐ฅ โ ๐ตp๐พ๐ฅq๐p๐พ๐ฅq๐ p๐๐ฅq รฃร ๐บp๐พ๐ฅq๐p๐พ๐ฅq๐ p๐๐ฅq โ Fl
82๐ฅ .
Consider the problem of computing the composite functor:
Reppq ยปWhit๐ ๐โ๐ฅpullbackรรรรรรWhitp๐บp๐พ๐ฅq๐ตp๐๐ฅqq
pushforwardรรรรรรรรWhit
82๐ฅ
๐82 ,!
รรร ๐ทpGr๐,๐ฅq ยป Repp๐ q.
By base-change, this amounts to computing pullback-pushforward of Whittaker18 sheaves along thecorrespondence:
๐บp๐พ๐ฅq๐ตp๐๐ฅq หFl82๐ฅ
Gr๐,๐ฅ
ww
Gr๐ต,๐ฅ
""Gr๐บ,๐ฅ Gr๐,๐ฅ .
One can see this is exactly the picture obtained by restricting the problem of Theorem 5.14.1
to Gr๐,๐ฅ ฤ Divฮ๐๐๐ ,8ยจ๐ฅeff , and therefore we obtain an answer in terms of factorization ฮฅn-modules.
Namely, this result says that the resulting functor:
Reppq ร Repp๐ q
is computed as Lie algebra homology along n.
Remark 1.33.1. The point of upgrading Theorem 5.14.1 to Theorem 7.9.1 is to allow a picture ofthis sort which is factorizable in terms of the point ๐ฅ, i.e., in which we replace the point ๐ฅ P ๐ bya variable point in ๐๐ผ for some finite set ๐ผ.
1.34. Methods. We now remark one what goes into the proofs of the above theorems.
1.35. Our key computational tool is the following result.
Theorem (Limiting case of the Casselman-Shalika formula, Thm. 3.4.1). The pushforward ๐ห,๐๐ p๐๐ต!b
IC๐ตq P ๐ทpDivฮ๐๐๐
eff q is the (one-dimensional) skyscraper sheaf at the zero divisor (concentrated incohomological degree zero).
In particular, the restriction of this pushforward to each Diveff with 0 โฐ P ฮ๐๐๐ vanishes.
18It is crucial here that our character be with respect to ๐ยด, not ๐ .
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 13
We prove this using reasonably standard methods (c.f. [BFGM]) for studying sheaves on Zastavaspaces.
1.36. Our other major tool is the study of ฮฅn given in [BG2], where ฮฅn is connected to theuntwisted cohomologies of Zastava spaces (in a less derived framework than in Theorem 4.6.1).
1.37. Finally, we remark that the proofs of Theorems 3.4.1, 4.6.1, and 5.14.1 are elementary:they use only standard perverse sheaf theory, and do not require the use of DG categories ornon-holonomic ๐ท-modules. (In particular, these theorems work in the โ-adic setting, with the usualArtin-Shreier sheaf replacing the exponential sheaf.) The reader uncomfortable with higher categorytheory should run into no difficulties here by replacing the words โDG categoryโ by โtriangulatedcategoryโ essentially everywhere (one exception: it is important that the definition of ฮฅnโmodfact๐ข๐,๐ฅ
be understood higher categorically).However, Theorem 7.9.1 is not elementary in this sense. This is the essential reason for the length
of S6: we are trying to construct an isomorphism of combinatorial nature in a higher categoricalsetting, and this is essentially impossible except in particularly fortuitous circumstances. We showin S6, S7 and Appendix B that the theory of ULA sheaves provides a suitable method for thisparticular problem.
1.38. Structure of the paper. S2 is a mostly self-contained review of the geometry of Zastavaspaces. In S3 and S4, we prove the limiting case of the Casselman-Shalika formula (Thm. 3.4.1) anduse it to realize ฮฅn in the geometry of Zastava spaces (Thm. 4.6.1). Then in S5, we give our firstcomparison (Thm. 5.14.1) between geometric Satake and the above construction of ฮฅn.
The remainder of the paper is dedicated to a generalization (Thm. 7.9.1) involving the fusionstructure from the geometric Satake theorem. In S6, we introduce prerequisite ideas and discussthe factorizable geometric Satake theorem; in particular, Theorem 6.36.1 proves a version of thefactorizable Cassleman-Shalika equivalence of [FGV], which is a folklore result in the subject. InS7, we use this language to formulate a comparison between geometric Satake and our constructionof ฮฅn using the factorizable structures on both sides.
There are two appendices. Appendix A proves a technical categorical lemma from S6. AppendixB introduces a general categorical language based on the theory of universally locally acyclic (ULA)sheaves, and which is suitable for general use in S6. The ULA methods are essential for S6-7.
1.39. Conventions. For the remainder of this introduction, we establish the conventions for theremainder of the text.
1.40. We fix a field ๐ of characteristic zero throughout the paper. All schemes, etc, are understoodto be defined over ๐.
1.41. Lie theory. We fix the following notations from Lie theory.Let ๐บ be a split reductive group over ๐, let ๐ต be a Borel subgroup of ๐บ with unipotent radical
๐ and let ๐ be the Cartan ๐ต๐ . Let ๐ตยด be a Borel opposite to ๐ต, i.e., ๐ตยด X ๐ตยปรร ๐ . Let ๐ยด
denote the unipotent radical of ๐ตยด.Let denote the corresponding Langlands dual group with corresponding Borel , who in turn
has unipotent radical and torus ๐ โ , and similarly for ยด and ยด.Let g, b, n, t, bยด, nยด, g, b, n, t, bยด and nยด denote the corresponding Lie algebras.Let ฮ denote the lattice of weights of ๐ and let ฮ denote the lattice of coweights. We let ฮ and
ฮ denote the weights and coweights of ๐บ. We let ฮ` (resp. ฮ`) denote the dominant weights (resp.coweights), and let ฮ๐๐๐ denote the Zฤ0-span of the simple coroots.
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14 SAM RASKIN
Let โ๐บ be the set of vertices in the Dynkin diagram of ๐บ. We recall that โ๐บ is canonicallyidentified with the set of simple positive roots and coroots of ๐บ. For ๐ P โ๐บ, we let ๐ผ๐ P ฮ (resp.๐ P ฮ) denote the corresponding root (resp. coroot).
Moreover, we fix a choice of Chevalley generators t๐๐u๐PI๐บ of nยด.Finally, we use the notation ๐ P ฮ for the half-sum of the positive roots of g, and similarly for
๐ P ฮ.
1.42. For an algebraic group ๐ค , let let B๐ค denote the classifying stack Specp๐q๐ค for ๐ค .
1.43. Let ๐ be a smooth projective curve.We let Bun๐บ denote the moduli stack of ๐บ-bundles on ๐. Recall that Bun๐บ is a smooth Artin
stack locally of finite type (though not quasi-compact).Similarly, we let Bun๐ต, Bun๐ , and Bun๐ denote the corresponding moduli stacks of bundles on
๐. However, we note that we will abuse notation in dealing specifically with bundles of structuregroup ๐ยด: we will systematically incorporate a twist discussed in detail in S2.8.
1.44. Categorical remarks. The ultimate result in this paper, Theorem 7.9.1, is about computinga certain factorization functor between factorization (DG) categories. This means that we need towork in a higher categorical framework (c.f. [Lur1], [Lur3]) at this point.
We will impose some notations and conventions below regarding this framework. With that said,the reader may read up to S5 essentially without ever worrying about higher categories.
1.45. We impose the convention that essentially everything is assumed derived. We will make thismore clear below, but first, we note the only exception: schemes can be understood as classicalschemes throughout the body of the paper, since we deal only with ๐ท-modules on them.
1.46. We find it convenient to assume higher category theory as the basic assumption in ourlanguage. That is, we will understand โcategoryโ and โ1-categoryโ to mean โp8, 1q-category,โโcolimitโ to (necessarily) mean โhomotopy colimit,โ โgroupoidโ to mean โ8-groupoidโ (aliases:homotopy type, space, etc.), and so on. We use the phrase โsetโ interchangeably with โdiscretegroupoid,โ i.e., a groupoid whose higher homotopy groups at any basepoint vanish.
When we need to refer to the more traditional notion of category, we use the term p1, 1q-category.As an example: we let Gpd denote the category (i.e.,8-category) of groupoids (i.e.,8-groupoids).
1.47. DG categories. By DG category, we mean an (accessible) stable (8-)category enriched over๐-vector spaces.
We denote the category of DG categories under ๐-linear exact functors by DGCat and the categoryof cocomplete19 DG categories under continuous20 ๐-linear functors by DGCat๐๐๐๐ก.
We consider DGCat๐๐๐๐ก as equipped with the symmetric monoidal structure b from [Lur3] S6.3.For C,D P DGCat๐๐๐๐ก and for F P C and G P D, we let F b G denote the induced object of C bD,since this notation is compatible with geometric settings.
For C an algebra in DGCat๐๐๐๐ก, we let Cโmod denote CโmodpDGCat๐๐๐๐กq: no other interpretationsof C-module category will be considered, and moreover, C should systematically be regarded as analgebra in DGCat๐๐๐๐ก.
19We actually mean presentable, which differs from cocomplete by a set-theoretic condition that will always be satisfiedfor us throughout this text.20There is some disagreement in the literature of the meaning of this word. By continuous functor, we mean a functorcommuting with filtered colimits. Similarly, by a cocomplete category, we mean one admitting all colimits.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 15
For C a DG category equipped with a ๐ก-structure, we let Cฤ0 denote the subcategory of cocon-nective objects, and Cฤ0 the subcategory of connective objects (i.e., the notation is the standardnotation for the convention of cohomological grading). We let C denote the heart of the ๐ก-structure.
We let Vect denote the DG category of ๐-vector spaces: this DG category has a ๐ก-structure withheart Vect the abelian category of ๐-vector spaces.
We use the material of the short note [Gai3] freely, taking for granted the readerโs comfort withthe ideas of loc. cit.
1.48. For a scheme ๐ locally of finite type, we let ๐ทp๐q denote its DG category of ๐ท-modules. Fora map ๐ : ๐ ร ๐ , we let ๐ ! : ๐ทp๐ q ร ๐ทp๐q and ๐ห,๐๐ : ๐ทp๐q ร ๐ทp๐ q denote the correspondingfunctors.
We always equip ๐ทp๐q with the perverse ๐ก-structure,21 i.e., the one for which IC๐ lies in the heartof the ๐ก-structure. In particular, if ๐ is smooth of dimension ๐, then the dualizing sheaf ๐๐ lies indegree ยด๐ and the constant sheaf ๐๐ lies in degree ๐. We sometimes refer to objects in the heart ofthis ๐ก-structure as perverse sheaves (especially if the object is holonomic), hoping this will not causeany confusion (since we do not assume ๐ โ C, we are in no position to apply the Riemann-Hilbertcorrespondence).
1.49. Finally, we use the notation Oblv throughout for various forgetful functors.
1.50. Acknowledgements. We warmly thank Dennis Gaitsgory for suggesting this project to usas his graduate student, and for his continuous support throughout its development. I have triedto acknowledge his specific ideas throughout the paper, but in truth, his influence on me and onthis project runs more deeply.
We further thank Dima Arinkin, Sasha Beilinson, David Ben-Zvi, Dario Beraldo, Roman Bezrukavnikov,Sasha Braverman, Vladimir Drinfeld, Sergey Lysenko, Ivan Mirkovic, Nick Rozenblyum, and SimonSchieder for their interest in this work and their influence upon it.
Within our gratitude, we especially single out our thanks to Dario Beraldo for conversations thatsignificantly shaped S6.
We thank MSRI for hosting us while this paper was in preparation.This material is based upon work supported by the National Science Foundation under Award
No. 1402003.
2. Review of Zastava spaces
2.1. In this section, we review the geometry of Zastava spaces, introduced in [FM] and [BFGM].Note that this section plays a purely expository role; our only hope is that by emphasizing
the role of local Zastava stacks, some of the basic geometry becomes more transparent than othertreatments.
2.2. Remarks on ๐บ. For simplicity, we assume throughout this section that ๐บ has a simply-connected derived group.
However, [ABB`] S4.1 (c.f. also [Sch] S7) explains how to remove this hypothesis, and the basicgeometry of Zastava spaces and Drinfeld compactifications remains exactly the same. The readermay therefore either assume ๐บ has simply-connected derived group for the rest of this text, or mayrefer to [Sch] for how to remove this hypothesis (we note that this applies just as well for citationsto [BG1], [BG2], and [BFGM]).
21Alias: the right (as opposed to left) ๐ก-structure. C.f. [BD] and [GR].
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16 SAM RASKIN
2.3. The basic affine space. Recall that the map:
๐บ๐ ร ๐บ๐ :โ Specp๐ป0pฮp๐บ๐,O๐บ๐ qqq โ SpecpFunp๐บq๐ q
is an open embedding. We call ๐บ๐ the basic affine space ๐บ๐ the affine closure of the basic affinespace.
The following result is direct from the Peter-Weyl theorem.
Lemma 2.3.1. For ๐ an affine test scheme,22 a map ๐ : ๐ ร ๐บ๐ with ๐ยด1p๐บ๐q dense in ๐ isequivalent to a โDrinfeld structureโ on the trivial ๐บ-bundle ๐บ ห ๐ ร ๐, i.e., a sequence of mapsfor ๐ P ฮ`.
๐๐ : โ๐ b๐O๐ ร ๐ ๐ b
๐O๐
are monomorphisms of quasi-coherent sheaves and satisfy the Plucker relations.
Remark 2.3.2. By dense, we mean scheme-theoretically, not topologically (e.g., for Noetherian ๐,the difference here is only apparent in the presence of associated points).
Example 2.3.3. For ๐บ โ SL2, ๐บ๐ identifies equivariantly with A2. The corresponding map SL2 ร
A2 here is given by:
ห
๐ ๐๐ ๐
ห
รร p๐, ๐q P A2.
2.4. Let ๐ be the closure of ๐ โ ๐ต๐ ฤ ๐บ๐ in ๐บ๐ .
Lemma 2.4.1. (1) ๐ is the toric variety Specp๐rฮ`sq (here ๐rฮ`s is the monoid algebra definedby the monoid ฮ`). Here the map ๐ โ Specp๐rฮsq ร ๐ corresponds to the embeddingฮ` ร ฮ and the map Funp๐บq๐ ร ๐rฮ`s realizes the latter as ๐ -coinvariants of the former.
(2) The action of ๐ on ๐บ๐ extends to an action of the monoid ๐ on ๐บ๐(where the coalgebrastructure on Funp๐ q โ ๐rฮ`s is the canonical one, that is, defined by the diagonal map forthe monoid ฮ`).
Here (1) follows again from the Peter-Weyl theorem and (2) follows similarly, noting that ๐ ๐ b
โ๐,_ ฤ Funp๐บq๐ โ Funp๐บ๐q has ฮ-grading (relative to the right action of ๐ on ๐บ๐) equal to๐ P ฮ`.
2.5. Note that (after the choice of opposite Borel) ๐ is canonically a retract of ๐บ๐ , i.e., the
embedding ๐ รฃร ๐บ๐ admits a canonical splitting:
๐บ๐ ร ๐ . (2.5.1)
Indeed, the retract corresponds to the map ๐rฮ`s ร Funp๐บq๐ sending ๐ to the canonical elementin:
โ๐ b โ๐,_ ฤ ๐ ๐ b ๐ ๐,_ ฤ Funp๐บq
(note that the embedding โ๐,_ รฃร ๐ ๐,_ uses the opposite Borel).
By construction, this map factors as ๐บ๐ ร ๐ยดzp๐บ๐q ร ๐ .
22It is important here that ๐ is a classical scheme, i.e., not DG.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 17
Let ๐ act on ๐บ๐ through the action induced by the adjoint action of ๐ on ๐บ. Choosing a
regular dominant coweight ๐0 P ฮ` we obtain a G๐-action on ๐บ๐ that contracts23 onto ๐ . The
induced map ๐บ๐ ร ๐ coincides with the one constructed above.
Warning 2.5.1. The induced map ๐บ๐ ร ๐ does not factor through ๐ . The inverse image in ๐บ๐of ๐ ฤ ๐ is the open Bruhat cell ๐ตยด๐๐ .
2.6. Define the stack B๐ต as ๐บz๐บ๐๐ . Note that B๐ต has canonical maps to B๐บ and B๐ .
2.7. Local Zastava stacks. Let๐๐ denote the stack ๐ตยดz๐บ๐ต โ B๐ตยดหB๐บB๐ต and and let ๐ denote
the stack ๐ตยดzp๐บ๐q๐ โ B๐ตยด หB๐บ B๐ต. We have the sequence of open embeddings:
B๐ รฃร๐๐ รฃร ๐
where B๐ embeds as the open Bruhat cell.The map B๐ รฃร ๐ factors as:
B๐ โ ๐ zp๐ ๐ q รฃร ๐ zp๐ ๐ q โ B๐ ห ๐ ๐ รฃร ๐. (2.7.1)
One immediately verifies that the retraction ๐บ๐ ร ๐ of (2.5.1) is ๐ตยด ห ๐ -equivariant, where
๐ตยด acts on the left on ๐บ๐ and ๐ acts on the right, and the action on ๐ is similar but is inducedby the ๐ ห ๐ -action and the homomorphism ๐ตยด ห ๐ ร ๐ ห ๐ . Therefore, we obtain a canonicalmap:
๐ โ ๐ตยดz๐บ๐๐ ร ๐ตยดz๐ ๐ ร ๐ z๐ ๐.
Moreover, up to the choice of ๐0 from loc. cit. this retraction realizes B๐ ห ๐ ๐ as a โdeformationretractโ of ๐.
We will identify ๐ z๐ ๐ with B๐ ห ๐ ๐ in what follows by writing the former as ๐ zp๐ ๐ q andnoting that ๐ acts trivially here on ๐ ๐ .
In particular, we obtain a canonical map:
๐ ร ๐ ๐. (2.7.2)
By Lemma 2.4.1 (2) we have an action of the monoid stack ๐ ๐ on ๐. The morphism ๐๐รร
B๐ ห ๐ ๐ ๐2รร ๐ ๐ is ๐ ๐ -equivariant.
Lemma 2.7.1. A map ๐ : ๐ ร ๐ ๐ with ๐ยด1pSpecp๐qq dense (where Specp๐q is realized as theopen point ๐ ๐ ) is canonically equivalent to a ฮ๐๐๐ :โ ยดฮ๐๐๐ -valued Cartier divisor on ๐.
First, we recall the following standard result.
Lemma 2.7.2. A map ๐ ร G๐zA1 with inverse image of the open point dense is equivalent to thedata of an effective Cartier divisor on ๐.
Proof. Tautologically, a map ๐ ร G๐zA1 is equivalent to a line bundle L on ๐ with a section๐ P ฮp๐,Lq.
We need to check that the morphism O๐๐ รร L is a monomorphism of quasi-coherent sheaves
under the density hypothesis. This is a local statement, so we can trivialize L. Now ๐ is a function
23We recall that a contracting G๐-action on an algebraic stack ๐ด is an action of the multiplicative monoid A1 on ๐ด.For schemes, this is a property of the underlying G๐ action, but for stacks it is not. Therefore, by the phrase โthatcontracts,โ we rather mean that it canonically admits the structure of contracting G๐-action. See [DG] for furtherdiscussion of these points.
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18 SAM RASKIN
๐ whose locus of non-vanishing is dense, and it is easy to see that this is equivalent to ๐ being anon-zero divisor.
Proof of Lemma 2.7.1. Let ๐บ1 ฤ ๐บ denote the derived subgroup r๐บ,๐บs of ๐บ and let ๐ 1 โ ๐ X ๐บ1
and ๐ 1 โ ๐ X ๐บ1. Then with ๐1
defined as the closure of ๐ 1 in the affine closure of ๐บ1๐ 1, theinduced map:
๐1๐ 1 ร ๐ ๐
is an isomorphism, reducing to the case ๐บ โ ๐บ1.Because the derived group (assumed to be equal to ๐บ now) is assumed simply-connected, we
have have canonical fundamental weights t๐๐u๐Pโ๐บ , ๐๐ P ฮ`. The mapล
๐Pโ๐บ ๐๐ : ๐ รล
๐Pโ๐บ G๐
extends to a map ๐ รล
๐Pโ๐บ A1 inducing an isomorphism:
๐ ๐ยปรร pA1G๐q
โ๐บ .
Because we use the right action of ๐ on ๐ , the functions on ๐ are graded negatively, and thereforewe obtain the desired result.
2.8. Twists. Fix an irreducible smooth projective curve ๐. We digress for a minute to normalizecertain twists.
Let ฮฉ๐ denote the sheaf of differentials on ๐. For an integer ๐, we will sometimes use thenotation ฮฉ๐
๐ for ฮฉb๐๐ , there being no risk for confusion with ๐-forms because ๐ is a curve.
We fix ฮฉ12๐ a square root of ฮฉ๐ . This choice extends the definition of ฮฉ๐
๐ to ๐ P 12Z. We obtain
the ๐ -bundle:
๐ซ๐๐๐๐ :โ ๐pฮฉยด1๐ q :โ 2๐pฮฉ
ยด 12
๐ q. (2.8.1)
We use the following notation:
Bun๐ยด :โ Bun๐ตยด หBun๐
t๐ซ๐๐๐๐ u
BunGยด๐ :โ BunG๐หG๐ หBunG๐
tฮฉยด 1
2๐ u.
Here G๐ หG๐ is the โnegativeโ Borel of PGL2.
Note that BunGยด๐ classifies extensions of ฮฉยด 1
2๐ by ฮฉ
12๐ and therefore there is a canonical map:
canGยด๐ : BunGยด๐ ร ๐ป1p๐,ฮฉ๐q โ G๐.
The choice of Chevalley generators t๐๐u๐Pโ๐บ of nยด defines a map:
๐ตยดr๐ยด, ๐ยดs รลบ
๐Pโ๐บ
pG๐ หG๐q.
By definition of ๐ซ๐๐๐๐ , this induces a map:
ลบ
๐Pโ๐บ
r๐ : Bun๐ยด รลบ
๐Pโ๐บ
BunGยด๐ .
We form the sequence:
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 19
Bun๐ยด รลบ
๐Pโ๐บ
BunGยด๐
ล
๐Pโ๐บcanGยด๐
รรรรรรรรรลบ
๐Pโ๐บ
G๐ ร G๐
and denote the composition by:
can : Bun๐ยด ร G๐. (2.8.2)
2.9. For a pointed stack p๐ด, ๐ฆ P ๐ดp๐qq and a test scheme ๐, we say that ๐ ห ๐ ร ๐ด is non-degenerate if there exists ๐ ฤ ๐ ห ๐ universally schematically dense relative to ๐ in the senseof [GAB`] Exp. XVIII, and such that the induced map ๐ ร ๐ด admits a factorization as ๐ ร
Specp๐q๐ฆรร ๐ด (so this is a property for a map, not a structure). We let Maps๐๐๐ยด๐๐๐๐๐.p๐,๐ดq denote
the open substack of Mapsp๐,๐ดq consisting of non-degenerate maps ๐ ร ๐ด.
We consider๐๐, ๐, and ๐ ๐ as openly pointed stacks in the obvious ways.
2.10. Zastava spaces. Observe that there is a canonical map:
๐ ร B๐ (2.10.1)
given as the composition:
๐ โ B๐ตยด หB๐บ
B๐ต ร B๐ตยด ร B๐.
Let ๐ต be the stack of ๐ซ๐๐๐๐ -twisted non-degenerate maps ๐ ร ๐, i.e., the fiber product:
Maps๐๐๐ยด๐๐๐๐๐.p๐, ๐q หBun๐
t๐ซ๐๐๐๐ u
where the map Maps๐๐๐ยด๐๐๐๐๐.p๐, ๐q ร Bun๐ is given by (2.10.1).
Let๐๐ต ฤ ๐ต be the open substack of ๐ซ๐๐๐
๐ -twisted non-degenerate maps ๐ ร๐๐. Note that ๐ต and
๐๐ต lie in Sch ฤ PreStk. We call ๐ต the Zastava space and
๐๐ต the open Zastava space. We let ๐ฅ :
๐๐ต ร ๐ต
denote the corresponding open embedding.We have a Cartesian square where all maps are open embeddings:
๐๐ต //
๐ต
Bun๐ยด ห
Bun๐บBun๐ต // Bun๐ยด ห
Bun๐บBun๐ต
The horizontal arrows realize the source as the subscheme of the target where the two reductionsare generically transverse.
2.11. Let Divฮ๐๐๐
eff โ Maps๐๐๐ยด๐๐๐๐๐.p๐,๐ ๐ q denote the scheme of ฮ๐๐๐ -divisors on ๐ (we include
the subscript โeffโ for emphasis that we are not taking ฮ-valued divisors).We have the canonical map:
deg : ๐0pDivฮ๐๐๐
eff q ร ฮ๐๐๐ .
For P ฮ๐๐๐ let Diveff denote the corresponding connected component of Divฮ๐๐๐
eff .
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20 SAM RASKIN
Remark 2.11.1. Writing โล
๐Pโ๐บ ๐๐๐ as a sum of simple coroots, we see that Diveff is a productล
๐Pโ๐บ Sym๐๐ ๐ of the corresponding symmetric powers of the curve.
Recall that we have the canonical map ๐ : ๐ ร B๐ ห๐ ๐ . For any non-degenerate map ๐ห๐ ร๐, Warning 2.5.1 implies that the induced map to ๐ ๐ (given by composing ๐ with the secondprojection) is non-degenerate as well.
Therefore we obtain the map:
๐ : ๐ต ร Divฮ๐๐๐
eff .
We let๐๐ denote the restriction of ๐ to
๐๐ต. It is well-known that the morphism ๐ is affine.
Let ๐ต (resp.๐๐ต ) denote the fiber of ๐ต (resp.
๐๐ต) over Diveff . We let ๐ (resp.
๐๐) denote the
restriction of ๐ to ๐ต (resp.๐๐ต ). We let ๐ฅ :
๐๐ต ร ๐ต denote the restriction of the open embedding
๐ฅ.
Note that ๐ admits a canonical section s : Divฮ๐๐๐
eff ร ๐ต, whose restriction to each Diveff we
denote by s. Note that up to a choice of regular dominant coweight, the situation is given bycontraction.
Each ๐ต is of finite type (and therefore the same holds for๐๐ต ). It is known (c.f. [BFGM] Corollary
3.8) that๐๐ต is a smooth variety.
For โ 0, we have๐๐ต0 โ ๐ต0 โ Div0
eff โ Specp๐q.We have a canonical (up to choice of Chevalley generators) map ๐ต ร G๐ defined as the compo-
sition ๐ต ร Bun๐ยดcanรรร G๐. For ๐ a positive simple coroot the induced map:
๐ต ๐ ร Div๐eff หG๐ โ ๐ หG๐ (2.11.1)
is an isomorphism that identifies๐๐ต ๐ with ๐ หG๐.
The dimension of ๐ต and๐๐ต is p2๐, q โ p๐, q`dim Diveff (this follows e.g. from the factorization
property discussed in S2.12 below and then by the realization discussed in S2.13 of the central fiberas an intersection of semi-infinite orbits in the Grassmannian, that are known by [BFGM] S6 to beequidimensional with dimension p๐, q).
Example 2.11.2. Let us explain in more detail the case of ๐บ โ SL2. In this case, tensoring with the
bundle ฮฉ12๐ identifies ๐ต with the moduli of commutative diagrams:
L
๐
!!
0 // ฮฉ12๐
//
๐_
E //
ฮฉยด 1
2๐
// 0
L_
in which the composition Lร L_ is zero and the morphism ๐ is non-zero. The open subscheme๐๐ต
is the locus where the induced map CokerpLร Eq ร L_ is an isomorphism. The associated divisor
of such a datum is defined by the injection L รฃร ฮฉยด 1
2๐ .
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 21
Over a point ๐ฅ P ๐, we have an identification of the fiber๐๐ต1๐ฅ of
๐๐ต1 over ๐ฅ P ๐ (considering
1 P Z โ ฮSL2 as the unique positive simple coroot) with G๐. Up to the twist by our square root
ฮฉ12๐ , the point 1 P G๐ corresponds to a canonical extension of O๐ by ฮฉ๐ associated to the point
๐ฅ, that can be constructed explicitly using the Atiyah sequence of the line bundle O๐p๐ฅq.Recall that for a vector bundle E, the Atiyah sequence (c.f. [Ati]) is a canonical short exact
sequence:
0 ร EndpEq ร AtpEq ร ๐๐ ร 0
whose splittings correspond to connections on E. For a line bundle L, we obtain a canonical extensionAtpLq b ฮฉ1
๐ of O๐ by ฮฉ1๐ . Taking L โ O๐p๐ฅq, we obtain the extension underlying the canonical
point of๐๐ต1๐ฅ.
Note that we have a canonical map L โ O๐p๐ฅq ร AtpO๐p๐ฅqq b ฮฉ1๐ that may be thought of
as a splitting of the Atiyah sequence with a pole of order 1, and this splitting corresponds to the
obvious connection on O๐p๐ฅq with a pole of order 1. This defines the corresponding point of๐๐ต1
completely.
2.12. Factorization. Now we recall the crucial factorization property of ๐ต.
Let add : Divฮ๐๐๐
eff หDivฮ๐๐๐
eff ร Divฮ๐๐๐
eff denote the addition map for the commutative monoid
structure defined by addition of divisors. For and fixed, we let add, denote the induced map
Diveff หDiveff ร Div`eff .Define:
rDivฮ๐๐๐
eff หDivฮ๐๐๐
eff s๐๐๐ ๐ ฤ Divฮ๐๐๐
eff หDivฮ๐๐๐
eff
as the moduli of pairs of disjoint ฮ๐๐๐ -divisors. Note that the restriction of add to this locus is etale.Then we have canonical โfactorizationโ isomorphisms:
๐ต หDivฮ
๐๐๐ eff
rDivฮ๐๐๐
eff หDivฮ๐๐๐
eff s๐๐๐ ๐ยปรร p๐ต ห ๐ตq ห
Divฮ๐๐๐
eff หDivฮ๐๐๐
eff
rDivฮ๐๐๐
eff หDivฮ๐๐๐
eff s๐๐๐ ๐
that are associative in the natural sense.The morphisms ๐ and s are compatible with the factorization structure.
2.13. The central fiber. By definition, the central fiber Z of the Zastava space ๐ต is the fiberproduct:
Z :โ ๐ต หDiveff
๐
where ๐ ร Diveff is the closed โdiagonalโ embedding, i.e., it is the closed subscheme where the
divisor is concentrated at a single point. We let๐Z denote the open in Z corresponding to
๐๐ต รฃร ๐ต .
Similarly, we let Z ฤ ๐ต be the closed corresponding to the union of the Z.
We let ๐ฝ (resp. 7) denote the closed embedding Z รฃร ๐ต (resp.๐Z รฃร
๐๐ต ).
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22 SAM RASKIN
2.14. Twisted affine Grassmannian. Let ๐ซ๐๐๐๐บ ,๐ซ๐๐๐
๐ต and ๐ซ๐๐๐๐ตยด be the torsors induced by the
๐ -torsor ๐ซ๐๐๐๐ under the embeddings of ๐ into each of these groups.
We let Gr๐บ,๐ denote the โ๐ซ๐๐๐๐บ -twisted Beilinson-Drinfeld affine Grassmannianโ classifying a
point ๐ฅ P ๐, a ๐บ-bundle ๐ซ๐บ on ๐, and an isomorphism ๐ซ๐๐๐๐บ |๐z๐ฅ ยป ๐ซ๐บ|๐z๐ฅ. More precisely, the
๐-points are:
๐ รร
#
๐ฅ : ๐ ร ๐, ๐ซ๐บ a ๐บ-bundle on ๐ ห ๐,๐ผ an isomorphism ๐ซ๐บ|๐ห๐zฮ๐ฅ
ยป ๐ซ๐๐๐๐บ |๐ห๐zฮ๐ฅ
+
.
Similarly for Gr๐ต,๐ , etc. We define Gr๐ยด,๐ :โ Gr๐ตยด,๐ หGr๐,๐๐, where the map ๐ ร Gr๐,๐
being the tautological section.Let Gr๐ต,๐ denote the โunion of closures of semi-infinite orbits,โ i.e., the indscheme:
Gr๐ต,๐ : ๐ รร
#
๐ฅ : ๐ ร ๐, ๐ : ๐ ห ๐ ร ๐บzp๐บ๐q๐ ,๐ผ a factorization of ๐|p๐ห๐qzฮ๐ฅ
through the
canonical map Specp๐q ร ๐บzp๐บ๐q๐ .
+
.
Here ฮ๐ฅ denotes the graph of the map ๐ฅ.
2.15. In the above notation, we have a canonical isomorphism:
Zยปรร Gr๐ยด,๐ ห
Gr๐บ,๐
Gr๐ต,๐ .
Indeed, this is immediate from the definitions.
Note that Gr๐ต,๐ has a canonical map to Gr๐,๐ โลก
Pฮ Gr๐,๐ . Letting Gr๐ต,๐ be the fiber overthe corresponding connected component of Gr๐,๐ , we obtain:
Zยปรร Gr๐ยด,๐ ห
Gr๐บ,๐
Gr๐ต,๐ .
2.16. By S2.7, we have an action of Divฮ๐๐๐
eff on ๐ต so that the morphism ๐ is Divฮ๐๐๐
eff -equivariant.
We let act๐ต denote the action map Divฮ๐๐๐
eff ห๐ต ร ๐ต. We abuse notation in denoting the induced
map Divฮ๐๐๐
eff ห๐๐ต ร ๐ต by act ๐
๐ต(that does not define an action on
๐๐ต, i.e., this map does not factor
through๐๐ต).
For P ฮ acting on ๐ต defines the map:
act๐ต : Divฮ๐๐๐
eff ห๐ต ร ๐ต.
For ๐ P ฮ๐๐๐ we use the notation act๐,๐ต for the induced map:
act๐,๐ต : Div๐eff ห๐ต ร ๐ต `๐.
Similarly, we have the maps act๐๐ต
and act,๐๐๐ต
.
The following lemma is well-known (see e.g. [BFGM]).
Lemma 2.16.1. For every , ๐ P ฮ๐๐๐ , the act,๐๐ต is finite morphism and the map act,๐๐๐ต
is a locally
closed embedding. For fixed the set of locally closed subschemes of ๐ต :
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 23
tact๐,๐๐ตpDiv๐eff ห
๐๐ต qu `๐โ
,๐Pฮ๐๐๐
forms a stratification.
2.17. Intersection cohomology of Zastava. For P ฮ๐๐๐ we now review the description from[BFGM] of the fibers of the intersection cohomology ๐ท-module IC๐ต along the strata describedabove, i.e., the ๐ท-modules:
act๐,,!๐๐ต
pIC๐ตq P ๐ทpDiv๐eff ห๐๐ต q, ๐, P ฮ๐๐๐ , ` ๐ โ .
Theorem 2.17.1. (1) With notation as above, the regular holonomic ๐ท-module:
act๐,,!๐๐ต
pIC๐ตq P ๐ทpDiv๐eff ห๐๐ต q (2.17.1)
is concentrated in constructible cohomological degree ยดdim๐๐ต .
(2) For ๐ฅ P ๐ a point, the further ห-restriction of (2.17.1) to๐Z๐ฅ is a lisse sheaf in constructible
degree ยดdim๐๐ต isomorphic to:
๐pnqp๐q b ๐๐Z๐ฅ
rdim๐๐ต s
where ๐pnqp๐q indicates the ๐-weight space.
(3) The !-restriction of (2.17.1) to๐๐ต is a sum of sheaves:
โpartitions ๐โ
ล๐๐โ1
๐
๐ a positive coroot
๐๐Z๐ฅ
rยด๐ ` dim๐๐ต s (2.17.2)
Remark 2.17.2. Recall from the above that,๐๐ต is equidimensional with dim๐ต โ 2p๐, q.
Remark 2.17.3. For clarity, in (2.17.2) we sum over all partitions of ๐ as a sum of positive coroots(where two partitions are the same if the multiplicity of each coroot is the same). We emphasizethat the ๐ are not assumed to be simple coroots, so the total number of summands is given bythe Kostant partition function.
Remark 2.17.4. This theorem is a combination of Theorem 4.5 and Lemma 4.3 of [BFGM] usingthe inductive procedure of loc. cit.
2.18. Locality. For ๐ a smooth (possibly affine) curve with choice of ฮฉ12๐ , we obtain an iden-
tical geometric picture. One can either realize this by restriction from a compactification, or byreinterpreting e.g. the map ๐ต ร G๐ through residues instead of through global cohomology.
3. Limiting case of the Casselman-Shalika formula
3.1. The goal for this section is to prove Theorem 3.4.1, on the vanishing of the IC-Whittakercohomology groups of Zastava spaces. This vanishing will play a central role in the remainder ofthe paper.
Remark 3.1.1. The method of proof is essentially by a reduction to the geometric Casselman-Shalikaformula of [FGV].
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24 SAM RASKIN
Remark 3.1.2. We are grateful to Dennis Gaitsgory for suggesting this result to us.
3.2. Artin-Schreier sheaves. We define the !-Artin-Schreier ๐ท-module ๐ P ๐ทpG๐q to be theexponential local system normalized cohomologically so that ๐rยด1s P ๐ทpG๐q
. Note that ๐ ismultiplicative with respect to !-pullback.
3.3. For P ฮ๐๐๐ , let ๐๐ต P ๐ทp๐ต q denote the !-pullback of the Artin-Schreier ๐ท-module ๐ alongthe composition:
๐ต ร Bun๐ยดcanรรร G๐.
Note that ๐๐ต
!b IC๐ต P ๐ทp๐ต q.
We then also define:
๐ ๐๐ตโ ๐ฅ,!p๐๐ตq.
3.4. The main result of this section is the following:
Theorem 3.4.1. If โฐ 0, then:
๐ห,๐๐ pIC๐ต
!b๐๐ตq โ 0.
The proof will be given in S3.6 below.This theorem is etale local on ๐, and therefore we may assume that we have ๐ โ A1. In
particular, we have a fixed trivialization of ฮฉ12๐ .
3.5. Central fibers via affine Schubert varieties. In the proof of Theorem 3.4.1 we will useProposition 3.5.1 below. We note that it is well-known, though we do not know a published reference.
Throughout S3.5, we work only with reduced schemes and indschemes, so all symbols refer tothe reduced indscheme underlying the corresponding indscheme. Note that this restriction does notaffect ๐ท-modules on the corresponding spaces.
Let ๐ p๐พq๐ denote the group indscheme over ๐ of meromorphic jets into ๐ (so the fiber of๐ p๐พq๐ at ๐ฅ P ๐ is the loop group ๐ p๐พ๐ฅq). Because we have chosen an identification ๐ ยป A1, wehave a canonical homomorphism:
Gr๐,๐ ยป A1 ห ฮ ร ๐ p๐พq๐ ยป A1 ห ๐ p๐พq
p๐ฅ, q รร p๐ฅ, p๐กqq
where ๐ก is the uniformizer of A1 (of course, the formula Gr๐,๐ ยป A1 ห ฮ is only valid at thereduced level). This induces an action of the ๐-group indscheme Gr๐,๐ on Gr๐ต,๐ , Gr๐บ,๐ and
Gr๐ยด,๐ โ Gr0๐ตยด,๐ .
Using this action, we obtain a canonical isomorphism:
Z โ Gr0๐ตยด,๐ หGr๐บ,๐
Gr๐ต,๐
ยปรร Gr๐
๐ตยด,๐ห
Gr๐บ,๐
Gr`๐๐ต,๐
of ๐-schemes for every ๐ P ฮ.
Proposition 3.5.1. For ๐ deep enough24 in the dominant chamber we have:
24This should be understood in a way depending on .
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 25
Gr๐๐ตยด,๐
หGr๐บ,๐
Gr`๐๐ต,๐ โ Gr๐
๐ตยด,๐ห
Gr๐บ,๐
Gr`๐๐บ,๐ .
This equality also identifies:
Gr๐๐ตยด,๐
หGr๐บ,๐
Gr`๐๐ต,๐ โ Gr๐๐ตยด,๐
หGr๐บ,๐
Gr`๐๐บ,๐ .
Proof. It suffices to verify the result fiberwise and therefore we fix ๐ฅ โ 0 P ๐ โ A1 (this is really
just a notational convenience here). We let Z๐ฅ (resp.๐Z๐ฅ) denote the fiber of Z (resp.
๐Z) at ๐ฅ. Let
๐ก P ๐พ๐ฅ be a coordinate at ๐ฅ.
Because there are only finitely many 0 ฤ ฤ and because each๐Z๐ฅ is finite type, for ๐ deep
enough in the dominant chamber we have:
๐Z๐ฅ โ Gr๐ยด,๐ฅXAdยด๐p๐กqp๐p๐๐ฅqq ยจ p๐กq
(p๐กq being regarded as a point in Gr๐บ,๐ฅ here and the intersection symbol is short-hand for fiber
product over Gr๐บ,๐ฅ) for all 0 ฤ ฤ . Choosing ๐ possibly larger, we can also assume that ๐ `
is dominant for all 0 ฤ ฤ . Then we claim that such a choice ๐ suffices for the purposes of theproposition.
Observe that for each 0 ฤ ฤ we have:
Gr๐๐ตยด,๐ฅ
XGr`๐๐ต,๐ฅ โ ๐p๐กq ยจ๐Z๐ฅ ฤ Gr๐
๐ตยด,๐ฅX
ห
๐p๐๐ฅq ยจ p` ๐qp๐กq
ห
ฤ Gr๐๐ตยด,๐ฅ
XGr`๐๐บ,๐ฅ .
Recall (c.f. [MV]) that Gr`๐๐ต,๐ฅ is a union of strata:
Gr`๐๐ต,๐ฅ , ฤ
while for :
Gr๐๐ตยด,๐ฅ
XGr`๐๐ต,๐ฅ โ H
unless ฤ 0. Therefore, Gr๐๐ตยด,๐ฅ
intersects Gr๐ต,๐ฅ only in the strata Gr`๐๐ต,๐ฅ for 0 ฤ ฤ .
The above analysis therefore shows that:
Gr๐๐ตยด,๐ฅ
XGr`๐๐ต,๐ฅ ฤ Gr๐
๐ตยด,๐ฅXGr`๐๐บ,๐ฅ .
Now observe that ๐ตp๐๐ฅq ยจ p` ๐qp๐กq is open in Gr. Therefore, we have:
Gr`๐๐บ,๐ฅ ฤ Gr`๐๐ต
giving the opposite inclusion above.
It remains to show that the equality identifies๐Z๐ฅ in the desired way. We have already shown
that:
Gr๐๐ตยด,๐ฅ
XGr`๐๐ต,๐ฅ ฤ Gr๐๐ตยด,๐ฅ
XGr`๐๐บ,๐ฅ .
so it remains to prove the opposite inclusion. Suppose that ๐ฆ is a geometric point of the right handside. Then, by the Iwasawa decomposition, ๐ฆ P Gr`๐๐ต,๐ฅ for some (unique) P ฮ and we wish to show
that โ .
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26 SAM RASKIN
Because:
๐ฆ P Gr`๐๐ต,๐ฅ XGr`๐๐บ,๐ฅ โฐ H
we have ฤ . We also have:
๐ฆ P Gr`๐๐ต,๐ฅ XGr๐๐ตยด,๐ฅ
โฐ H
which implies ฤ 0. Therefore, by construction of ๐ we have:
๐ฆ P Gr๐๐ตยด,๐ฅ
XGr`๐๐ต,๐ฅ ฤ Gr๐๐ตยด,๐ฅ
XGr`๐๐บ,๐ฅ ฤ Gr`๐๐บ,๐ฅ
but Gr`๐๐บ,๐ฅ XGr`๐๐บ,๐ฅ โ H if โฐ (because ` ๐ and ` ๐ are assumed dominant) and therefore
we must have โ as desired.
We continue to use the notation introduced in the proof of Proposition 3.5.1.
Recall that ๐ฝ (resp. 7) denotes the closed embedding Z รฃร ๐ต (resp.๐Z รฃร
๐๐ต ). For ๐ฅ P ๐,
let ๐ฝ๐ฅ (resp. 7๐ฅ) denote the closed embedding Z๐ฅ รฃร ๐ต (resp.๐Z๐ฅ รฃร
๐๐ต ).
Corollary 3.5.2. For every ๐ฅ P ๐, the cohomology:
๐ปห๐๐
ยด
Z๐ฅ, ๐ฝ,!๐ฅ pIC๐ต
!b๐๐ตq
ยฏ
(3.5.1)
is concentrated in non-negative cohomological degrees, for for 0 โฐ , it is concentrated in strictlypositive cohomological degrees.
Remark 3.5.3. It follows a posteriori from Theorem 3.4.1 that the whole cohomology vanishes for0 โฐ .
Proof. First, we claim that when either:
โ ๐ ฤ 0, or:โ ๐ โ 0 and โฐ 0
we have:
๐ป ๐๐๐
ยด ๐Z๐ฅ, 7
,!๐ฅ pIC ๐
๐ต
!b๐ ๐
๐ตq
ยฏ
โ 0 (3.5.2)
Indeed, from the smoothness of๐๐ต , we see that IC ๐
๐ต
!b7,!๐ฅ p๐ ๐
๐ตq is a rank one local system
concentrated in perverse cohomological degree:
dimp๐๐ต q ยด dimp
๐Z๐ฅq โ dimp
๐Z๐ฅq.
This gives the desired vanishing in negative degrees.Moreover, from Proposition 3.5.1 and the Casselman-Shalika formula ([FGV] Theorem 1), we
deduce that, for โฐ 0, the restriction of our rank one local system to every irreducible component
of๐Z๐ฅ is moreover non-constant. This gives (3.5.2).To complete the argument, note that by Theorem 2.17.1 (3), for 0 ฤ ฤ , the !-restriction of
IC๐ต to๐Z๐ฅ lies in perverse cohomological degrees ฤ p๐, q, with strict inequality for โฐ .
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 27
By lisseness of ๐๐ต , we deduce that for 0 ฤ ฤ , ๐ฝ,!๐ฅ pIC๐ต
!b๐๐ตq has !-restriction to
๐Z๐ฅ in
perverse cohomological degrees strictly greater than p๐, q โ dimp๐ต q. Therefore, the non-positivecohomologies of these restrictions vanish.
We find that the cohomology is the open stratum can contribute to the non-positive cohomology,but this vanishes by (3.5.2).
Corollary 3.5.4. If 0 โฐ P ฮ๐๐๐ then we have the vanishing Euler characteristic:
๐
ห
๐ปห๐๐
ยด
Z๐ฅ, ๐ฝ,!๐ฅ pIC๐ต
!b๐๐ตq
ยฏ
ห
โ 0.
Proof. The key point is to establish the following equality:
r๐ฝ,!๐ฅ pIC๐ตqs โ r๐!pIC
Gr`๐๐บ,๐ฅ
qs P ๐พ0p๐ท๐โ๐๐pZ
๐ฅqq (3.5.3)
in the Grothendieck group of complexes of (coherent and) holonomic ๐ท-modules on Z๐ฅ. Here themap ๐ is defined as:
Z๐ฅยปรร Gr๐
๐ตยด,๐ฅXGr`๐๐บ,๐ฅ ร Gr`๐๐บ,๐ฅ .
It suffices to show that for each 0 ฤ ฤ , the !-restrictions of these classes coincide in theGrothendieck group of:
Gr๐๐ตยด,๐ฅ
XGr`๐๐บ,๐ฅ .
Indeed, these locally closed subvarieties form a stratification.
First, note that the !-restriction of ICGr`๐
๐บ,๐ฅ
to Gr`๐บ,๐ฅ has constant cohomologies (by ๐บp๐q-
equivariance). Moreover, by [Lus] the corresponding class in the Grothendieck group is the dimen-sion of the weight component:
dim`
๐ ยด๐ค0p`๐qpยดยด ๐qห
ยจ rICGr`
๐บ,๐ฅ
s.
Further !-restricting to Gr๐๐ตยด,๐ฅ
XGr`๐๐บ,๐ฅ , we obtain that the right hand side of our equation is given
by:
dim๐ ยด๐ค0p`๐qpยดยด ๐q ยจ rICGr๐๐ตยด,๐ฅ
XGr`๐๐บ,๐ฅs.
By having ๐pnq act on a lowest weight vector of ๐ `๐, we observe that for ๐ large enough, we have:
๐ ยด๐ค0p`๐qpยดยด ๐q ยป ๐pnqpยด q.
The similar identification for the left hand side follows from the choice of ๐ (so that Gr๐๐ตยด,๐ฅ
XGr`๐๐บ,๐ฅ
identifies with๐Z๐ฅ) and Theorem 2.17.1 (3).
Appealing to (3.5.3), we see that in order to deduce the corollary, it suffices to prove that:
๐
ห
๐ปห๐๐
ยด
Z๐ฅ, ๐!pIC
Gr`๐๐บ,๐ฅ
q!b ๐ฝ,!๐ฅ p๐๐ตq
ยฏ
ห
โ 0.
Even better: by the geometric Casselman-Shalika formula [FGV], this cohomology itself vanishes.
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28 SAM RASKIN
3.6. Now we give the proof of Theorem 3.4.1.
Proof of Theorem 3.4.1. We proceed by induction on p๐, q, so we assume the result holds for all
0 ฤ ฤ . By factorization and induction, we see that F :โ ๐ห,๐๐ pIC๐ต
!b๐๐ตq is concentrated on
the main diagonal ๐ ฤ Diveff .
The (ห โ!-)restriction of F to ๐ is the ห-pushforward along Z ร ๐ of ๐ฝ,!pIC๐ต
!b๐๐ตq. More-
over, since Z ร ๐ is a Zariski-locally trivial fibration, the cohomologies of F on ๐ are lisse andthe fiber at ๐ฅ P ๐ is:
๐ปห๐๐
ยด
Z๐ฅ, ๐ฝ,!๐ฅ pIC๐ต
!b๐๐ตq
ยฏ
.
Because ๐ is affine and IC๐ต
!b๐๐ต is a perverse sheaf, F lies in perverse degrees ฤ 0. Moreover,
by Corollary 3.5.2, its !-fibers are concentrated in strictly positive degrees. Since F is lisse along ๐,this implies that F is actually perverse. Now Corollary 3.5.4 provides the vanishing of the Eulercharacteristics of the fibers of F, giving the result.
4. Identification of the Chevalley complex
4.1. The goal for this section is to identify the Chevalley complex in the cohomology of Zastavaspace with coefficients in the Whittaker sheaf: this is the content of Theorem 4.6.1.
The argument combines Theorem 3.4.1 with results from [BG2].
Remark 4.1.1. Theorem 4.6.1 is one of the central results of this text: as explained in the introduc-tion, it provides a connection between Whittaker sheaves on the semi-infinite flag variety and thefactorization algebra ฮฅn, and therefore relates to the main conjecture of the introduction.
4.2. We will use the language of graded factorization algebras.The definition should encode the following: a Zฤ0-graded factorization algebra is a system A๐ P
๐ทpSym๐๐q such that we have, for every pair ๐,๐ we have isomorphisms:
ยด
A๐ bA๐
ยฏ
|rSym๐๐หSym๐๐s๐๐๐ ๐ยปรร
ยด
A๐`๐
ยฏ
|rSym๐๐หSym๐๐s๐๐๐ ๐
satisfying (higher) associativity and commutativity. Note that the addition map Sym๐๐หSym๐๐ ร
Sym๐`๐๐ is etale when restricted to the disjoint locus, and therefore the restriction notation aboveis unambiguous.
Formally, the scheme Sym๐ โลก
๐ Sym๐๐ is naturally a commutative algebra under correspon-dences, where the multiplication is induced by the maps:
rSym๐๐ ห Sym๐๐s๐๐๐ ๐
tt ))Sym๐๐ ห Sym๐๐ Sym๐`๐๐.
Therefore, as in [Ras1] S6, we can apply the formalism of loc. cit. S5 to obtain the desired theory.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 29
Remark 4.2.1. We will only be working with graded factorization algebras in the heart of the ๐ก-structure, and therefore the language may be worked out โby handโ as in [BD], i.e., without needingto appeal to [Ras1].
Similarly, we have the notion of ฮ๐๐๐ -graded factorization algebra: it is a collection of ๐ท-modules
on the schemes Diveff with similar identifications as above.
4.3. Recall that [BG2] has introduced a certain ฮ๐๐๐ -graded commutative factorization algebra,
i.e., a commutative factorization ๐ท-module on Divฮ๐๐๐
eff . This algebra incarnates the homologicalChevalley complex of n. In loc. cit., this algebra is denoted by ฮฅpn๐q: we use the notation ฮฅn
instead. We denote the component of ฮฅn on Diveff by ฮฅn . Recall from loc. cit. that each ฮฅ
n lies in
๐ทpDiveffq.25
Remark 4.3.1. To remind the reader of the relation between ฮฅn and the homological Chevalleycomplex ๐ถโpnq of n, we recall that the ห-fiber of ฮฅn at a ฮ๐๐๐ -colored divisor
ล๐๐โ1 ๐ ยจ ๐ฅ๐ (here
๐ P ฮ๐๐๐ and the ๐ฅ๐ P ๐ are distinct closed points) is canonically identified with:
๐b๐โ1๐ถโpnq
๐
where ๐ถโpnq๐ denotes the ๐-graded piece of the complex.
Remark 4.3.2. The ฮ๐๐๐ -graded vector space:
n โ โ a positive coroot
n
gives rise to the ๐ท-module:
n๐ :โ โ a positive coroot
โห,๐๐ pn
b ๐๐q P ๐ทpDivฮeffq
where for P ฮ, โ : ๐ ร Diveff is the diagonal embedding. The Lie algebra structure on n givesa Lie-ห structure on n๐ .
Then ฮฅn is tautologically given as the factorization algebra associated to the chiral envelopingalgebra of this Lie-ห algebra.
Remark 4.3.3. We emphasize the miracle mentioned above and crucially exploited in [BG2] (andbelow): although ๐ถโpnq is a cocommutative (DG) coalgebra that is very much non-classical, its๐ท-module avatar does lie in the heart of the ๐ก-structure. Of course, this is no contradiction, sincethe ห-fibers of a perverse sheaf need only live in degrees ฤ 0.
4.4. Observe that ๐ฅห,๐๐ pIC ๐๐ตq naturally factorizes on ๐ต. Therefore, sห,๐๐ ๐ฅห,๐๐ pIC ๐
๐ตq is naturally a
factorization ๐ท-module in ๐ทpDivฮ๐๐๐
eff q.The following key identification is essentially proved in [BG2], but we include a proof with
detailed references to loc. cit. for completeness.
Theorem 4.4.1. There is a canonical identification:
๐ป0psห,๐๐ ๐ฅห,๐๐ pIC ๐๐ตqq
ยปรร ฮฅn
of ฮ๐๐๐ -graded factorization algebras.
25We explicitly note that in this section we exclusively use the usual (perverse) ๐ก-structure.
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30 SAM RASKIN
Remark 4.4.2. To orient the reader on cohomological shifts, we note that for P ฮ๐๐๐ fixed, IC ๐๐ต
is concentrated in degree 0 and therefore the above ๐ป0 is the maximal cohomology group of thecomplex sห,๐๐ ๐ฅห,๐๐ pIC ๐
๐ตq.
Proof of Theorem 4.4.1. Let ๐ : Divฮ๐๐๐ ,simpleeff รฃร Divฮ๐๐๐
eff denote the open consisting of simpledivisors, i.e., its geometric points are divisors of the form
ล๐๐โ1 ๐ ยจ๐ฅ๐ for ๐ a positive simple coroot
and the points t๐ฅ๐u pairwise distinct. For each P ฮ๐๐๐ , we let ๐ : Div,simpleeff ร Diveff denote the
corresponding open embedding. Note that ๐ and each embedding ๐ is affine.
Observe that Divฮ๐๐๐ ,simpleeff has a factorization structure induced by that of Diveff . The restric-
tion of ฮฅn to Divฮ๐๐๐ ,simpleeff identifies canonically with the exterior product over ๐ P โ๐บ of the
corresponding โsignโ (rank 1) local systems under the identification:
Div,simpleeff ยป
ลบ
๐Pโ๐บ
Sym๐๐,simple๐
where โล
๐Pโ๐บ ๐๐๐ and on the right the subscript simple means โsimple effective divisorโ in thesame sense as above. Moreover, these identifications are compatible with the factorization structurein the natural sense.
Let๐๐ตsimple and
๐๐ต ,simple denote the corresponding opens in
๐๐ต and
๐๐ต obtained by fiber product.
Let ssimple and s,simple denote the corresponding restrictions of s and s.
Then๐๐ต ,simple ยป
รร Div,simpleeff หGp๐,q๐ as a Div,simple
eff -scheme by (2.11.1), and these identificationsare compatible with factorization.
Therefore, we deduce an isomorphism:
๐ป0pssimple,ห,๐๐ ๐ฅห,๐๐ pIC ๐๐ตsimple
qqยปรร ๐!pฮฅnq
of factorization ๐ท-modules on Divฮ๐๐๐ ,simpleeff (note that the sign local system appears on the left by
the Koszul rule of signs).Therefore, we obtain a diagram:
๐!๐ป0pssimple,ห,๐๐ ๐ฅหpIC ๐
๐ตsimpleqq
ยป //
๐!๐!pฮฅnq
๐ป0psห,๐๐ ๐ฅห,๐๐ pIC ๐
๐ตqq ฮฅn
(4.4.1)
Note that the top horizontal arrow is a map of factorization algebras on Divฮ๐๐๐
eff .By (the Verdier duals to) [BG2] Lemma 4.8 and Proposition 4.9, the vertical maps in (4.4.1)
are epimorphisms in the abelian category ๐ทpDivฮ๐๐๐
eff q. Moreover, by the analysis in loc. cit. S4.10,there is a (necessarily unique) isomorphism:
๐ป0psห,๐๐ ๐ฅห,๐๐ pIC ๐๐ตqq
ยปรร ฮฅn
completing the square (4.4.1). By uniqueness, this isomorphism is necessarily an isomorphism offactorizable ๐ท-modules.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 31
4.5. Observe that the ๐ท-module ๐ ๐๐ต
canonically factorizes on๐๐ต. Therefore, ๐ฅห,๐๐ p๐ ๐
๐ตq factorizes
in ๐ทp๐ตq.By Theorem 4.4.1, we have for each P ฮ๐๐๐ we have a map:
๐ฅห,๐๐ pIC ๐๐ตq ร sห,๐๐ ๐ป
0ยด
s,ห,๐๐ ๐ฅห,๐๐ pIC ๐๐ตq
ยฏ
โ sห,๐๐ pฮฅnq. (4.5.1)
These maps are compatible with factorization as we vary .
Lemma 4.5.1. The map (4.5.1) is an epimorphism in the abelian category ๐ทp๐ต q.
Proof. Let F P ๐ทp๐ต qฤ0. Then the canonical map:
F ร sห,๐๐ s,ห,๐๐ pFq (4.5.2)
has kernel given by restricting to and then !-extending from the complement to the image of s.Since this is an open embedding, the kernel of (4.5.2) is concentrated in cohomological degrees ฤ 0.
Taking the long exact sequence on cohomology, we see that the map:
F ร ๐ป0psห,๐๐ s,ห,๐๐ pFqq โ sห,๐๐ ๐ป
0ps,ห,๐๐ pFqq P ๐ทp๐ต q
is an epimorphism.
Applying this to F โ ๐ฅห,๐๐ pIC ๐๐ตq gives the claim.
4.6. Applying ๐๐ต!bยด to (4.5.1) and using the canonical identifications s,!,๐๐ p๐๐ตq
ยปรร ๐
Diveff, we
obtain maps:
๐ : ๐ฅห,๐๐ p๐ ๐๐ต
!b IC ๐
๐ตq ร sห,๐๐ pฮฅ
nq.
Because everything above is compatible with factorization as we vary , the maps ๐ are as well.
We let ๐ : ๐ฅห,๐๐ p๐ ๐๐ต
!bIC ๐
๐ตq ร sห,๐๐ pฮฅnq denote the induced map of factorizable ๐ท-modules on ๐ต.
Theorem 4.6.1. The map:
๐๐ห,๐๐ p๐ ๐
๐ต
!b IC ๐
๐ตq โ ๐ห,๐๐ ๐ฅห,๐๐ p๐ ๐
๐ต
!b IC ๐
๐ตq๐ห,๐๐ p๐qรรรรรร ๐ห,๐๐ sห,๐๐ pฮฅnq โ ฮฅn (4.6.1)
is an equivalence of factorizable ๐ท-modules on Divฮ๐๐๐
eff .
Remark 4.6.2. In particular, the theorem asserts that๐๐ห,๐๐ p๐ ๐
๐ต
!b IC ๐
๐ตq is concentrated in cohomo-
logical degree 0.
Proof of Theorem 4.6.1. It suffices to show for fixed P ฮ๐๐๐ that ๐ห,๐๐ p๐q is an equivalence.
Recall from [BG2] Corollary 4.5 that we have an equality:
r๐ฅห,๐๐ pIC ๐๐ตqs โ
รฟ
,๐Pฮ๐๐๐
`๐โ
ract๐,๐ต,ห,๐๐ pฮฅ๐n b IC๐ตqs P ๐พ0p๐ท
๐โ๐๐p๐ต qq. (4.6.2)
in the Grothendieck group of (coherent and) holonomic ๐ท-modules. Therefore, because ๐๐ต is lisse,we obtain a similar equality:
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32 SAM RASKIN
r๐ฅห,๐๐ p๐ ๐๐ต
!b IC ๐
๐ตqs โ
รฟ
,๐Pฮ๐๐๐
`๐โ
ract๐,๐ต,ห,๐๐
ยด
ฮฅ๐n b p๐ ๐
๐ต
!b IC๐ตq
ยฏ
s (4.6.3)
by the projection formula.For every decomposition ` ๐ โ , we have:
๐ห,๐๐ act๐,๐ต,ห,๐๐
ยด
ฮฅ๐n b p๐ ๐
๐ต
!b IC๐ตq
ยฏ
โ add๐,ห,๐๐
ยด
ฮฅ๐n b ๐
ห,๐๐ p๐ ๐๐ต
!b IC๐ตq
ยฏ
.
By Theorem 3.4.1, this term vanishes for โฐ 0.Therefore, we see that the left hand side of (4.6.1) is concentrated in degree 0, and that it agrees
in the Grothendieck group with the right hand side.
Moreover, by affineness of ๐, the functor ๐ห,๐๐ is right exact. Therefore, by Lemma 4.5.1, the
map ๐ห,๐๐ p๐q is an epimorphism in the heart of the ๐ก-structure; since the source and target agree
in the Grothendieck group, we obtain that our map is an isomorphism.
5. Hecke functors: Zastava calculation over a point
5.1. Next, we compare Theorem 4.6.1 with the geometric Satake equivalence.More precisely, given a representation ๐ of the dual group , there are two ways to associate a
factorization ฮฅn-module: one is through its Chevalley complex ๐ถโpn, ๐ q, and the other is througha geometric procedure explained below, relying on geometric Satake and Theorem 4.6.1. In whatfollows, we refer to these two operations as the spectral and geometric Chevalley functors respec-tively.
The main result of this section, Theorem 5.14.1, identifies the two functors.
Notation 5.1.1. We fix a ๐-point ๐ฅ P ๐ in what follows.
5.2. Polar Drinfeld structures. Suppose ๐ is proper for the moment.Recall the ind-algebraic stack Bun
8ยจ๐ฅ๐ยด from [FGV]: it parametrizes ๐ซ๐บ a ๐บ-bundle on ๐ and
non-zero maps:26
ฮฉbp๐,๐q๐ ร ๐ ๐
๐ซ๐บp8 ยจ ๐ฅq
defined for each dominant weight ๐ and satisfying the Plucker relations.
Example 5.2.1. Let ๐บ โ ๐๐ฟ2. Then Bun8ยจ๐ฅ๐ยด classifies the datum of an ๐๐ฟ2-bundle E and a non-zero
map ฮฉ12๐ ร Ep8 ยจ ๐ฅq.27
Example 5.2.2. For ๐บ โ G๐, Bun8ยจ๐ฅ๐ยด is the affine Grassmannian for ๐ at ๐ฅ.
26Here if p๐, ๐q is half integral, we appeal to our choice of ฮฉ12๐ .
27Here we are slightly abusing notation in letting E denote the rank two vector bundle underlying our ๐๐ฟ2-bundle.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 33
5.3. Hecke action. The key feature of Bun8ยจ๐ฅ๐ยด is that Hecke functors at ๐ฅ act on ๐ทpBun
8ยจ๐ฅ๐ยด q.
More precisely, the action of the Hecke groupoid on Bun๐บ lifts in the obvious way to an action onBun
8ยจ๐ฅ๐ยด .
For definiteness, we introduce the following notation. Let H๐ฅ๐บ denote the Hecke stack at ๐ฅ,
parametrizing pairs of ๐บ-bundles on ๐ identified away from ๐ฅ. Let โ1 and โ2 denote the twoprojections H๐ฅ
๐บ ร Bun๐บ.Define the Drinfeld-Hecke stack H๐ฅ
๐บ,Drin as the fiber product:
H๐ฅ๐บ ห
Bun๐บBun
8ยจ๐ฅ๐ยด
where we use the map โ1 : H๐ฅ๐บ ร Bun๐บ in order to form this fiber product. We abuse notation in
using the same notation for the two projections H๐ฅ๐บ,Drin ร Bun
8ยจ๐ฅ๐ยด .
Example 5.3.1. Let ๐บ โ ๐๐ฟ2. Then H๐ฅ๐บ,Drin parametrizes a pair of ๐๐ฟ2-bundles E1 and E2 identified
away from ๐ฅ and a non-zero map ฮฉ12๐ ร E1p8 ยจ ๐ฅq. The two projections โ1 and โ2 correspond to
the maps to Bun8ยจ๐ฅ๐ยด sending a datum as above to:
`
E1,ฮฉ12๐ ร E1p8 ยจ ๐ฅq
ห
`
E2,ฮฉ12๐ ร E1p8 ยจ ๐ฅq
ยปรร E2p8 ยจ ๐ฅq
ห
respectively.
We have the usual procedure for producing objects of H๐ฅ๐บ from objects of Sph๐บ,๐ฅ :โ ๐ทpGr๐บ,๐ฅq
๐บp๐q๐ฅ .These give Hecke functors acting on ๐ทpBun๐บq using the correspondence H๐ฅ
๐บ from ๐ทpBun๐บq to it-self and the kernel induced by this object of Sph๐บ,๐ฅ. We normalize our Hecke functors so that we
!-pullback along โ1 and ห-pushforward along โ2. The same discussion applies for ๐ทpBun8ยจ๐ฅ๐ยด q.
We use ห to denote the action by convolution of Sph๐บ,๐ฅ on these categories.
5.4. Polar Zastava space. We let๐๐ต8ยจ๐ฅ denote the indscheme defined by the ind-open embedding:
๐ต8ยจ๐ฅ ฤ Bun๐ต หBun๐บ
Bun8ยจ๐ฅ๐ยด
given by the usual generic transversality condition.
Note that๐๐ต ฤ
๐๐ต8ยจ๐ฅ is the fiber of
๐๐ต8ยจ๐ฅ along Bun๐ยด ฤ Bun
8ยจ๐ฅ๐ยด .
Remark 5.4.1. As in the case of usual Zastava, note that๐๐ต8ยจ๐ฅ is of local nature with respect to ๐:
i.e., the definition makes sense for any smooth curve, and is etale local on the curve. Therefore, wetypically remove our requirement that ๐ is proper in what follows.
5.5. Let Divฮ๐๐๐ ,8ยจ๐ฅeff be the indscheme parametrizing ฮ-valued divisors on ๐ that are ฮ๐๐๐ -valued
away from ๐ฅ.As for usual Zastava space, we have the map:
๐๐ต8ยจ๐ฅ
๐๐8ยจ๐ฅรรรร Divฮ๐๐๐ ,8ยจ๐ฅ
eff .
Remark 5.5.1. There is a canonical map deg : Divฮ๐๐๐ ,8ยจ๐ฅeff ร ฮ (considering the target as a discrete
๐-scheme) of taking the total degree of a divisor.
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34 SAM RASKIN
5.6. Factorization patterns. Note that Divฮ๐๐๐ ,8ยจ๐ฅeff is a unital factorization module space for
Divฮ๐๐๐
eff . This means that e.g. we have a correspondence:
โ
ww $$
Divฮ๐๐๐
eff หDivฮ๐๐๐ ,8ยจ๐ฅeff Divฮ๐๐๐ ,8ยจ๐ฅ
eff .
For this action, the left leg of the correspondence is the open embedding encoding disjointness ofpairs of divisors, while the right leg is given by addition. (For the sake of clarity, let us note that
the only reasonable notion of the support of a divisor in Divฮ๐๐๐ ,8ยจ๐ฅeff requires that ๐ฅ always lie in
the support).
Therefore, as in S4.2, we can talk about unital factorization modules in Divฮ๐๐๐ ,8ยจ๐ฅeff for a unital
graded factorization algebra A P ๐ทpDivฮ๐๐๐
eff q. We denote this category by Aโmodfact๐ข๐,๐ฅ.
Remark 5.6.1. The factorization action of Divฮ๐๐๐
eff on Divฮ๐๐๐ ,8ยจ๐ฅeff is commutative in the sense of
[Ras1] S7. Indeed, it comes from the obvious action of the monoid Divฮ๐๐๐
eff on Divฮ๐๐๐ ,8ยจ๐ฅeff .
Remark 5.6.2. We emphasize that there is no Ran space appearing here: all the geometry occurson finite-dimensional spaces of divisors.
5.7. There is a similar picture to the above for Zastava. More precisely,๐๐ต8ยจ๐ฅ is a unital factor-
ization module space for๐๐ต in a way compatible with the structure maps to and from the spaces of
divisors.
Therefore, for a unital factorization algebra B on๐๐ต, we can form the category Bโmodfact๐ข๐ p
๐๐ต8ยจ๐ฅq.
Moreover, for โณ P Bโmodfact๐ข๐ p๐๐ต8ยจ๐ฅq, ๐
๐8ยจ๐ฅห,๐๐ pโณq is tautologically an object of๐๐ห,๐๐ pBqโmodfact๐ข๐,๐ฅ.
We denote the corresponding functor by:
๐๐8ยจ๐ฅห,๐๐ : Bโmodfact๐ข๐ p
๐๐ต8ยจ๐ฅq ร ๐
๐ห,๐๐ pBqโmodfact๐ข๐,๐ฅ.
5.8. Construction of the geometric Chevalley functor. We now define a functor:
Chevgeomn,๐ฅ : Reppq ร ฮฅnโmodfact๐ข๐,๐ฅ
using the factorization pattern for Zastava space.
Remark 5.8.1. Following our conventions, Reppq denotes the DG category of representations of.
Remark 5.8.2. We will give a global interpretation of the induced functor to ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅeff q in
S5.12; this phrasing may be easier to understand at first pass.
5.9. First, observe that there is a natural โcompactificationโ ๐ต8ยจ๐ฅ of๐๐ต8ยจ๐ฅ: for ๐ proper, it is the
appropriate ind-open locus in:
๐ต8ยจ๐ฅ ฤ Bun8ยจ๐ฅ๐ต ห
Bun๐บBun
8ยจ๐ฅ๐ยด .
Here Bun8ยจ๐ฅ๐ต is defined analogously to Bun
8ยจ๐ฅ๐ยด ; we remark that it has a structure map to Bun๐
with fibers the variants of Bun8ยจ๐ฅ๐ยด for other bundles. Again, ๐ต8ยจ๐ฅ is of local nature on the curve ๐.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 35
The advantage of ๐ต8ยจ๐ฅ is that there is a Hecke action here, so Sph๐บ,๐ฅ acts on ๐ทp๐ต8ยจ๐ฅq. Note
that !-pullback from Bun8ยจ๐ฅ๐ยด commutes with Hecke functors.
There is again a canonical map to Divฮ๐๐๐ ,8ยจ๐ฅeff , and the factorization pattern of S5.7 carries over
in this setting as well, that is, ๐ต8ยจ๐ฅ is a unital factorization module space for ๐ต. Moreover, thisfactorization schema is compatible with the Hecke action.
5.10. Define Y ฤ ๐ต8ยจ๐ฅ as the preimage of Bun๐ยด ฤ Bun8ยจ๐ฅ๐ยด in ๐ต8ยจ๐ฅ: again, Y is of local nature
on ๐.
Remark 5.10.1. The notation๐๐ต8ยจ๐ฅ would be just as appropriate for Y as for the space we have
denoted in this way: both are polar versions of๐๐ต, but for
๐๐ต8ยจ๐ฅ we allow poles for the ๐ยด-bundle,
while for Y we allow poles for the ๐ต-bundle.
There is a canonical map Y ร G๐ which e.g. for ๐ proper comes from the canonical mapBun๐ยด ร G๐. We can !-pullback the exponential ๐ท-module ๐ on G๐ (normalized as always to bein perverse degree -1): we denote the resulting ๐ท-module by ๐Y P ๐ทpYq.
We then cohomologically renormalize: define ๐ICY by:
๐ICY :โ ๐Yrยดp2๐, degqs.
Here we recall that we have a degree map Y ฤ ๐ต8ยจ๐ฅ ร ฮ, so pairing with 2๐, we obtain an integervalued function on Y: we are shifting accordingly.
Remark 5.10.2. The reason for this shift is the normalization of Theorem 4.6.1: this shift is implicit
there in the notation!b IC ๐
๐ต. This is also the reason for our notation ๐IC
Y .
5.11. Recall that ๐ฅ denotes the embedding๐๐ต รฃร ๐ต. We let ๐ฅ8ยจ๐ฅ denote the map
๐๐ต8ยจ๐ฅ รฃร ๐ต8ยจ๐ฅ.
Let:
Sat๐ฅ : Reppqยปรร Sph๐บ,๐ฅ
denote the geometric Satake equivalence. Then let:
Sat๐๐๐๐ฃ๐๐ฅ : Reppq ร Sph๐บ,๐ฅ
denote the induced functor.We then define Chevgeom
n,๐ฅ as the following composition:
ReppqSat๐๐๐๐ฃ๐
๐ฅรรรรรร Sph๐บ,๐ฅ
ยดห๐ICY
รรรรร
๐ฅห,๐๐ p๐ ๐๐ต
!b IC ๐
๐ตqโmodfact๐ข๐ p๐ต8ยจ๐ฅq
๐ฅ8ยจ๐ฅ,!รรรร p๐ ๐
๐ต
!b IC ๐
๐ตqโmodfact๐ข๐ p
๐๐ต8ยจ๐ฅq
๐๐8ยจ๐ฅห,๐๐ รรรร ฮฅnโmodfact๐ข๐,๐ฅ.
(5.11.1)
Here in the last step, we have appealed to the identification:
๐๐ห,๐๐ p๐ ๐
๐ต
!b IC ๐
๐ตqq โ ฮฅn
of Theorem 4.6.1. We also abuse notation in not distinguishing between ๐ICY and its ห-pushforward
to ๐ต8ยจ๐ฅ.
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36 SAM RASKIN
5.12. Global interpretation. As promised in Remark 5.8.2, we will now give a description of thefunctor Chevgeom
n,๐ฅ in the case ๐ is proper.
Since ๐ is proper, we can speak about Bun๐ยด and its relatives. Let Wโ๐๐ก P ๐ทpBun๐ยดq denotethe canonical Whittaker sheaf, i.e., the !-pullback of the exponential sheaf on G๐ (normalized asalways to be in perverse degree ยด1). We then have the functor:
Reppq ร ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅeff q
given by applying geometric Satake, convolving with the pห โ!q-pushforward of Wโ๐๐ก to ๐ทpBun8ยจ๐ฅ๐ยด q,
and then !-pulling back to๐๐ต8ยจ๐ฅ and ห-pushing forward along
๐๐8ยจ๐ฅ.
Since !-pullback from Bun8ยจ๐ฅ๐ยด to ๐ต8ยจ๐ฅ commutes with Hecke functors, up to the cohomological
shifts by degrees, this functor computes the object of ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅeff q underlying the factorization
ฮฅn-module coming from Chevgeomn,๐ฅ .
5.13. Spectral Chevalley functor. We need some remarks on factorization modules for ฮฅn:Recall from Remark 4.3.2 that ฮฅn is defined as the chiral enveloping algebra of the graded Lie-ห
algebra n๐ P ๐ทpDivฮ๐๐๐
eff q. By Remark 5.6.1, we may speak of Lie-ห modules for n๐ on Divฮ๐๐๐ ,8ยจ๐ฅeff :
the definition follows [Ras1] S7.19. Let n๐โmod๐ฅ denote the DG category of Lie-ห modules for
n๐ supported on Gr๐,๐ฅ ฤ Divฮ๐๐๐ ,8ยจ๐ฅeff (this embedding is as divisors supported at ๐ฅ). We have a
tautological equivalence:
n๐โmod๐ฅ ยป nโmodpRepp๐ qq (5.13.1)
coming from identifying Repp๐ q with the DG category of ฮ-graded vector spaces. Note that theright hand side of this equation is just the category of ฮ-graded n-representations.
Moreover, by [Ras1] S7.19, we have an induction functor Ind๐โ : n๐โmod๐ฅ ร ฮฅnโmodfact๐ข๐,๐ฅ.
We then define Chevspecn,๐ฅ : Reppq ร ฮฅnโmodfact๐ข๐,๐ฅ as the composition:
ReppqOblvรรรร Reppq
Oblvรรรร nโmodpRepp๐ qq
(5.13.1)ยป n๐โmod๐ฅ
Ind๐โรรรร ฮฅnโmodfact๐ข๐,๐ฅ. (5.13.2)
5.14. Formulation of the main result. We can now give the main result of this section.
Theorem 5.14.1. There exists a canonical isomorphism between the functors Chevspecn,๐ฅ and Chevgeom
n,๐ฅ .
The proof will be given in S5.16 below after some preliminary remarks.
Remark 5.14.2. As stated, the result is a bit flimsy: we only claim that there is an identificationof functors. The purpose of S7 is essentially to strengthen this identification so that it preservesstructure encoding something about the symmetric monoidal structure of Reppq.
5.15. Equalizing the Hecke action. Suppose temporarily that ๐ is a smooth proper curve. Onethen has the following relationship between Hecke functors acting on Bun
8ยจ๐ฅ๐ต and Hecke functors
acting on Bun8ยจ๐ฅ๐ยด .
Let ๐ผ (resp. ๐ฝ) denote the projection ๐ต8ยจ๐ฅ ร Bun8ยจ๐ฅ๐ยด (resp. ๐ต8ยจ๐ฅ ร Bun
8ยจ๐ฅ๐ต ). Recall that ๐ผ!
and ๐ฝ! commute with the actions of Sph๐บ,๐ฅ.
Let ๐8ยจ๐ฅ denote the canonical map ๐ต8ยจ๐ฅ ร Divฮ๐๐๐ ,8ยจ๐ฅeff .
Lemma 5.15.1. For F P ๐ทpBun8ยจ๐ฅ๐ยด q, G P ๐ทpBun
8ยจ๐ฅ๐ต q, and S P Sph๐บ,๐ฅ, there is a canonical identi-
fication:
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 37
๐8ยจ๐ฅห,๐๐
ยด
๐ผ!pSห Fq!b ๐ฝ!pGq
ยฏ
q ยป ๐8ยจ๐ฅห,๐๐
ยด
๐ผ!pFq!b ๐ฝ!pSห Gq
ยฏ
.
Proof. By base-change, each of these functors is constructed using a kernel on some correspondencebetween Bun
8ยจ๐ฅ๐ยด ห๐บp๐๐ฅqzGr๐บ,๐ฅหBun
8ยจ๐ฅ๐ต and Div8ยจ๐ฅeff .
In both cases, one finds that this correspondence is just the Hecke groupoid (at ๐ฅ) for Zastava,
mapping via โ1 to Bun8ยจ๐ฅ๐ยด and via โ2 to Bun
8ยจ๐ฅ๐ต , with the kernel being defined by S.
5.16. We now give the proof of Theorem 5.14.1.
Proof of Theorem 5.14.1. As Reppq is semi-simple, we reduce to showing this for ๐ โ ๐ anirreducible highest weight representation with highest weight P ฮ`.
Our technique follows that of Theorem 4.4.1.
Step 1. Let ๐ : ๐ รฃร Divฮ๐๐๐ ,8ยจ๐ฅeff be the locally closed subscheme parametrizing divisors of the form:
๐ค0pq ยจ ๐ฅ`รฟ
๐ ยจ ๐ฅ๐
where ๐ฅ๐ P ๐ are pairwise disjoint and distinct from ๐ฅ (this is the analogue of the open Divฮ๐๐๐ ,simpleeff ฤ
Divฮ๐๐๐
eff which appeared in the proof of Theorem 4.4.1).We have an easy commutative diagram:
๐!๐! Chevgeom
n,๐ฅ p๐ q
ยป // ๐!๐! Chevspec
n,๐ฅ p๐q โ ๐ก
๐ค0pqห,๐๐ pฮฅnq
Chevgeomn,๐ฅ p๐ q Chevspec
n,๐ฅ p๐q.
(5.16.1)
One easily sees that the right vertical map is an epimorphism (this is [BG2] Lemma 9.2).It suffices to show that the left vertical map in (5.16.1) is an epimorphism, and that there exists
a (necessarily unique) isomorphism in the bottom row of the diagram (5.16.1).This statement is local on ๐, and therefore we can (and do) assume that ๐ is proper in what
follows.
Step 2. We claim that Chevgeomn,๐ฅ p๐ q lies in the heart of the ๐ก-structure, and that rChevgeom
n,๐ฅ p๐ qs โ
rChevspecn,๐ฅ p๐
qs in the Grothendieck group.
By Lemma 5.15.1, for every representation ๐ of we have:
๐8ยจ๐ฅห,๐๐
`
๐ผ!pSat๐ฅp๐ qหWโ๐๐กq!b ๐ฝ!pICBun๐ต q
ห
ยป
๐8ยจ๐ฅห,๐๐
`
๐ผ!pWโ๐๐กq!b ๐ฝ!pSat๐ฅp๐ qห ICBun๐ต q
ห
.
(5.16.2)
Here ICBun๐ต indicates the ห-extension of this ๐ท-module to Bun8ยจ๐ฅ๐ต .
By definition, Chevgeomn,๐ฅ p๐ q is the left hand side of (5.16.2). Therefore, Theorem 3.4.1 and the
discussion of [BG2] S8.7 gives the claim.
Step 3. We will use (a slight variant of) the following construction.28
28As Dennis Gaitsgory pointed out to us, one can argue somewhat more directly, by combining Lemma 5.15.1 withTheorem 8.11 from [BG2] (and the limiting case of the Casselman-Shalika formula, Theorem 3.4.1).
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38 SAM RASKIN
Suppose that ๐ is a variety and F P ๐ทp๐ ห A1qG๐ is G๐-equivariant for the action of G๐ byhomotheties on the second factor, and that F is concentrated in negative (perverse) cohomologicaldegrees.
For ๐ P ๐, let ๐๐ denote the embedding ๐ ห t๐u รฃร ๐ ห A1.Then, for each ๐ P Z, the theory of vanishing cycles furnishes specialization maps:
๐ป๐p๐!1pFqq ร ๐ป๐p๐!0pFqq P ๐ทp๐ q (5.16.3)
that are functorial in F, and which is an epimorphism for ๐ โ 0. Indeed, these maps arise from theboundary map in the triangle:29
๐!0pFq ร ฮฆ๐ข๐pFqvarรรร ฮจ๐ข๐pFq
`1รรร
when we use G๐-equivariance to identify ฮจ๐ข๐pFq with F1r1s. The ๐ก-exactness of ฮฆ๐ข๐ and theassumption that F is in degrees ฤ 0 shows that (5.16.3) is an epimorphism for ๐ โ 0:
. . .ร ๐ปยด1pฮจ๐ข๐pFqq โ ๐ป0p๐!1pFqq ร ๐ป0p๐!0pF0qq ร ๐ป0pฮฆ๐ข๐pFqq โ 0
Step 4. We now apply the previous discussion to see that:
Chevgeomn,๐ฅ p๐ q ยป Chevspec
n,๐ฅ p๐ q
as objects of ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅeff q for ๐ P Reppq finite-dimensional.
Forget for the moment that we chose Chevalley generators t๐๐u and let ๐ denote the vector
space pnยดrnยด, nยดsqห. Note that ๐ acts on ๐ through its adjoint action on nยด. Let๐๐ ฤ๐ denote
the open subscheme corresponding to non-degenerate characters.Then we have a canonical map:
Yห๐ ร G๐
by imitating the construction of the map can : ๐ต ร G๐ of (2.8.2). Note that this map is ๐ -equivariant for the diagonal action on the source and the trivial action on the target.30
Let W P ๐ทp๐ต8ยจ๐ฅ ห๐ q๐ denote the result of !-pulling back of the exponential ๐ท-module on G๐
to Yห๐ and then ห-extending. We then define:
rW :โ p๐๐8ยจ๐ฅ ห id๐ qห,๐๐ p๐ฅ
8ยจ๐ฅ ห id๐ q!ยด
Sat๐๐๐๐ฃ๐๐ฅ p๐ qหWrยดp2๐,degqsยฏ
P ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅeff ห๐ q๐ .
Here the ๐ -equivariance now refers to the ๐ -action coming from the trivial action on Divฮ๐๐๐ ,8ยจ๐ฅeff .
The notation for the cohomological shift is as in S5.10.
By ๐ -equivariance, the cohomologies of our rW are constant along the open stratum Divฮ๐๐๐ ,8ยจ๐ฅeff ห
๐๐ .
Moreover, note that rW is concentrated in cohomological degrees ฤ ยด rankp๐บq โ ยดdimp๐ q: this
again follows from Lemma 5.15.1, S8.7 of [BG2], and ind-affineness of๐๐8ยจ๐ฅ.
Therefore, !-restricting to the line through our given non-degenerate character, Step 3 gives usthe specialization map:
๐ป0` ๐๐8ยจ๐ฅห,๐๐ ๐ฅ
8ยจ๐ฅ,!pSat๐๐๐๐ฃ๐๐ฅ p๐ qห ๐Yrยดp2๐, degqsqqห
ร ๐ป0pChevgeomn,๐ฅ p๐ qq P ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅ
eff q.
29Our nearby and vanishing cycles functors are normalized to preserve perversity.30We use the canonical ๐ -action on ๐ต, coming from the action of ๐ on Bun๐ยด induced by its adjoint action on ๐ยด.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 39
By Step 3, this specialization map is an epimorphism.However, the Zastava space version of Theorem 8.8 from [BG2] (which is implicit in loc. cit.
and easy to deduce from there) implies that the left hand term coincides with Chevspecn,๐ฅ p๐ q, and
therefore this map is an isomorphism by the computation in the Grothendieck group.Moreover, one immediately sees that this picture is compatible with the diagram (5.16.1), and
therefore we actually do obtain an isomorphism of factorization modules, as desired.
6. Around factorizable Satake
6.1. Our goal in S7 is to prove a generalization of Theorem 5.14.1 in which we treat severalpoints t๐ฅ1, . . . , ๐ฅ๐u ฤ ๐, allowing these points to move and collide (in the sense of the Ran spaceformalism). This section plays a supplementary and technical role for this purpose.
6.2. Generalizing the geometric side of Theorem 5.14.1 is an old idea: one should use the Beilinson-Drinfeld affine Grassmannian Gr๐บ,๐๐ผ and the corresponding factorizable version of the Satakecategory.
Therefore, we need a geometric Satake theorem over powers of the curve. This has been treatedin [Gai1], but the treatment of loc. cit. is inconvenient for us, relying too much on specific aspectsof perverse sheaves that do not generalize to non-holonomic ๐ท-modules.
6.3. The goal for this section is to give a treatment of factorizable geometric Satake for ๐ท-modules.However, most of the work here actually goes into treating formal properties of the spectral side
of this equivalence. Here we have DG categories Reppq๐๐ผ which provide factorizable versions ofthe category Reppq appearing in the Satake theory.
These categories arise from a general construction, taking C a symmetric monoidal object ofDGCat๐๐๐๐ก (so we assume the tensor product commutes with colimits in each variable), and pro-ducing C๐๐ผ P ๐ทp๐๐ผqโmod. As we will see, this construction is especially well-behaved for C rigidmonoidal (as for C โ Reppq).
6.4. Structure of this section. We treat the construction and general properties of the categoriesC๐๐ผ in S6.5-6.18, especially treating the case where C is rigid. We specialize to the case where C isrepresentations of an affine algebraic group in S6.19.
We then discuss the (naive) factorizable Satake theorem from S6.28 until the end of this section.
6.5. Let C P ComAlgpDGCat๐๐๐๐กq be a symmetric monoidal DG category. We denote the monoidaloperation in C by b.
6.6. Factorization. Recall from [Ras1] S7 that we have an operation attaching to each finite set๐ผ a ๐ทp๐๐ผq-module category C๐๐ผ .31
We will give an essentially self-contained treatment of this construction below, but first giveexamples to give the reader a feeling for the construction.
Example 6.6.1. For ๐ผ โ ห, we have C๐ โ Cb๐ทp๐q.
Example 6.6.2. Let ๐ผ โ t1, 2u. Let ๐ denote the open embedding ๐ โ ๐ ห๐z๐ รฃร ๐ ห๐.Then we have a fiber square:
31In [Ras1], we use the notation ฮp๐๐ผ๐๐ , Loc๐๐ผ
๐๐ pCqq in place of C๐๐ผ .
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40 SAM RASKIN
C๐2 //
Cb๐ทp๐2q
idCb๐!
`
Cb Cห
b๐ทp๐qpยดbยดqbid๐ทp๐2q // Cb๐ทp๐q.
We emphasize that pยด b ยดq indicates the tensor product morphism Cb Cร C.
Example 6.6.3. If ๐ค is an affine algebraic group and we take C โ Repp๐ค q, then the above says thatRepp๐ค q๐2 parametrizes a representation of ๐ค over ๐2
๐๐ with the structure of a ๐ค ห๐ค -representationon the complement to the diagonal, compatible under the diagonal embedding ๐ค รฃร ๐ค ห ๐ค .
6.7. For the general construction of C๐๐ผ , we need the following combinatorics.First, for any surjection ๐ : ๐ผ ๐ฝ of finite sets, let ๐p๐q denote the open subscheme of points
p๐ฅ๐q๐P๐ผ with ๐ฅ๐ โฐ ๐ฅ๐1 whenever ๐p๐q โฐ ๐p๐1q.
Example 6.7.1. For ๐ : ๐ผ ร ห, we have ๐p๐q โ ๐๐ผ . For ๐ : ๐ผidรร ๐ผ, ๐p๐q is the locus ๐๐ผ
๐๐๐ ๐ of
pairwise disjoint points in ๐๐ผ .
We let S๐ผ denote the p1, 1q-category indexing data ๐ผ๐ ๐ฝ
๐ ๐พ, where we allow morphisms of
diagrams that are contravariant in ๐ฝ and covariant in ๐พ, and surjective termwise.
6.8. For every ฮฃ โ p๐ผ๐ ๐ฝ
๐ ๐พq in S๐ผ , define Cฮฃ P ๐ทp๐
๐ผqโmod as:
Cฮฃ โ ๐ทp๐p๐qq b Cb๐พ .
For ฮฃ1 ร ฮฃ2 P S๐ผ , we have a canonical map Cฮฃ1 ร Cฮฃ2 P ๐ทp๐๐ผqโmod constructed as follows. If
the morphism ฮฃ1 ร ฮฃ2 is induced by the diagram:
๐ผ๐1 // // ๐ฝ1
๐1 // // ๐พ1
๐ผ
๐ผ๐2 // // ๐ฝ2
OOOO
๐2 // // ๐พ2
then our functor is given as the tensor product of:
Cb๐พ1 ร Cb๐พ2
b๐P๐พ1
F๐ รร b๐1P๐พ2
p b๐2P๐ผยด1p๐1q
F๐2q
and the ๐ท-module restriction along the map ๐p๐2q ร ๐p๐1q.It is easy to upgrade this description to the homotopical level to define a functor:
S๐ผ ร ๐ทp๐๐ผqโmod.
We define C๐๐ผ as the limit of this functor.
Example 6.8.1. It is immediate to see that this description recovers our earlier formulae for ๐ผ โ หand ๐ผ โ t1, 2u.
Remark 6.8.2. This construction unwinds to say the following: we have an object F P C b๐ทp๐๐ผq
such that for every ๐ : ๐ผ ๐ฝ , its restriction to C b ๐ทp๐p๐qq has been lifted to an object ofCb๐ฝ b๐ทp๐p๐qq.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 41
Example 6.8.3. For C โ Repp๐ค q with ๐ค an affine algebraic group, this construction is a derivedversion of the construction of [Gai1] S2.5.
Remark 6.8.4. Obviously each C๐๐ผ is a commutative algebra in ๐ทp๐๐ผqโmod. Indeed, each Cฮฃ โ
๐ทp๐p๐qq b Cb๐พ , is and the structure functors are symmetric monoidal. We have an obvious sym-metric monoidal functor:
Loc โ Loc๐๐ผ : Cb๐ผ ร C๐๐ผ
for each ๐ผ, with these functors being compatible under diagonal maps.
6.9. Factorization. It follows from [Ras1] S7 that the assignment ๐ผ รร C๐๐ผ defines a commutativeunital chiral category on ๐๐๐ . For the sake of completeness, the salient pieces of structure here aretwofold:
(1) For every pair of finite sets ๐ผ1 and ๐ผ2, we have a symmetric monoidal map:
C๐๐ผ1 b C๐๐ผ2 ร C๐๐ผ1ลก
๐ผ2
of๐ทp๐๐ผ1ลก
๐ผ2q-module categories that is an equivalence after tensoring with๐ทpr๐๐ผ1 ห๐๐ผ2s๐๐๐ ๐q.(2) For every ๐ผ1 ๐ผ2, an identification:
C๐๐ผ1 b๐ทp๐๐ผ1 q
๐ทp๐๐ผ2q ยป C๐๐ผ2 .
These should satisfy the obvious compatibilities, which we do not spell out here because in thehomotopical setting they are a bit difficult to say: we refer to [Ras1] S7 for a precise formulation.
We will construct these maps in S6.10 and 6.11.
6.10. First, suppose ๐ผ โ ๐ผ1ลก
๐ผ2.
Define a functor S๐ผ ร S๐ผ1 as follows. We send ๐ผ๐ ๐ฝ
๐ ๐พ to ๐ผ1 Imagep๐|๐ผ1q Imagep๐ห๐|๐ผ1q.
It is easy to see that this actually defines a functor. We have a similar functor S๐ผ ร S๐ผ2 , so weobtain S๐ผ ร S๐ผ1 ห S๐ผ2 .
Given ๐ผ๐ ๐ฝ
๐ ๐พ as above, let e.g. ๐ผ1
๐1 ๐ฝ1
๐1 ๐พ1 denote the corresponding object of S๐ผ1 .
We have a canonical map:
๐p๐q รฃร ๐p๐1q ห ๐p๐1q ฤ ๐๐ผ1 ห๐๐ผ2 โ ๐๐ผ .
We also have a canonical map Cb๐พ1 b Cb๐พ2 ร Cb๐พ induced by tensor product and the obviousmap ๐พ1
ลก
๐พ2 ร ๐พ. Together, we obtain maps:
p๐ทp๐p๐1qq b Cb๐พ1q b p๐ทp๐p๐2qq b Cb๐พ2q ร ๐ทp๐p๐qq b C๐พ
that in passage to the limit define
C๐๐ผ1 b C๐๐ผ2 ร C๐๐ผ .
That this map is an equivalence over the disjoint locus follows from a cofinality argument.
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42 SAM RASKIN
6.11. Next, suppose for ๐ : ๐ผ1 ๐ผ2 is given. We obtain S๐ผ2 ร S๐ผ1 by restriction.
Moreover, for any given ๐ผ2๐ ๐ฝ
๐ ๐พ P S๐ผ2 , we have the functorial identifications:
๐ทp๐p๐qq b Cb๐พ ยป p๐ทp๐p๐ ห ๐qq b๐ทp๐๐ผ1 q
๐ทp๐๐ผ2qq b Cb๐พq
that give a map:
C๐๐ผ1 b๐ทp๐๐ผ1 q
๐ทp๐๐ผ2q ร C๐๐ผ2 .
An easy cofinality argument shows that this map is an equivalence.
6.12. A variant. We now discuss a variant of the preceding material a categorical level down.
6.13. First, if ๐ด is a commutative algebra in Vect, then there is an assignment ๐ผ รร ๐ด๐๐ผ P
๐ทp๐๐ผq defining a commutative factorization algebra. Indeed, it is given by the same procedure asbeforeโwe have:
๐ด๐๐ผ :โ limp๐ผ
๐๐ฝ
๐๐พqPS๐ผ
๐๐,ห,๐๐ p๐ดb๐พ b ๐๐p๐qq P ๐ทp๐
๐ผq. (6.13.1)
The structure maps are as before.
6.14. More generally, when C is as before and ๐ด P C is a commutative algebra, we can attach a(commutative) factorization algebra ๐ผ รร ๐ด๐๐ผ P C๐๐ผ .
We will need this construction in this generality below. However, the above formula does notmake sense, since there is no way to make sense of ๐๐,ห,๐๐ p๐๐p๐qqb๐ด
b๐พ as an object of C๐๐ผ . So weneed the following additional remarks:
We do have ๐ด๐๐ผ defined as an object of ๐ทp๐๐ผqbC by the above formula. Moreover, as in S6.10,for every ๐ : ๐ผ ๐ฝ we have canonical โmultiplicationโ maps:
b๐P๐ฝ๐ด๐๐ผ๐ ร ๐ด๐๐ผ P ๐ทp๐๐ผq b C
where ๐ผ๐ is the fiber of ๐ผ at ๐ P ๐ฝ , and where our exterior product should be understood as a mix ofthe tensor product for C and the exterior product of ๐ท-modules. This map is an equivalence over๐p๐q.
This says that for every ๐ as above, the restriction of ๐ด๐๐ผ to ๐p๐q has a canonical structure asan object of ๐ทp๐p๐qq b Cb๐ฝ , lifting its structure of an object of ๐ทp๐p๐qq b C. Moreover, this iscompatible with further restrictions in the natural sense. This is exactly the data needed to upgrade๐ด๐๐ผ to an object of C๐๐ผ (which we denote by the same name).
6.15. ULA objects. For the remainder of the section, assume that C is compactly generated andrigid : recall that rigidity means that this means that the unit 1C is compact and every ๐ P C
compact admits a dual.Under this rigidity assumption, we discuss ULA aspects of the categories C๐๐ผ : we refer the reader
to Appendix B for the terminology here, which we assume for the remainder of this section.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 43
6.16. Recall that QCohp๐๐ผ ,C๐๐ผ q denotes the object of QCohp๐๐ผqโmod obtained from C๐๐ผ P
๐ทp๐๐ผqโmod by induction along the (symmetric monoidal) forgetful functor ๐ทp๐๐ผq ร QCohp๐๐ผq.
Proposition 6.16.1. For F P Cb๐ผ compact, Loc๐๐ผ pFq P C๐๐ผ is ULA.
We will deduce this from the following lemma.Let 1C๐๐ โ Loc๐๐ผ p1Cq denote the unit for the (๐ทp๐๐ผq-linear) symmetric monoidal structure on
C๐๐ .
Lemma 6.16.2. 1C๐๐ is ULA.
Proof. By 1-affineness (see [Gai4]) of ๐๐๐ and ๐, the induction functor:
๐ทp๐qโmodร QCohp๐qโmod
commutes with limits.It follows that QCohp๐๐ผ ,C๐๐ผ q is computed by a similar limit as defines C๐๐ผ , but with QCohp๐p๐qq
replacing ๐ทp๐p๐qq everywhere.Since this limit is finite and since each of the terms corresponding to Oblvp1C๐๐ q P QCohp๐
๐ผ ,C๐๐ผ q
is compact, we obtain the claim.
Proof of Proposition 6.16.1. Since the functor Cb๐ผ ร C๐๐ผ is symmetric monoidal and since eachcompact object in Cb๐ผ admits a dual by assumption, we immediately obtain the result from Lemma6.16.2.
Remark 6.16.3. Proposition 6.16.1 fails for more general C: the tensor product CbCร C typicallyfails to preserve compact objects, which implies that Loc๐2 does not preserve compacts.
6.17. We now deduce the following result about the categories C๐๐ผ (for the terminology, seeDefinition B.6.1).
Theorem 6.17.1. C๐๐ผ is ULA over ๐๐ผ .
We will use the following lemma, which is implicit but not quite stated in [Gai4].
Lemma 6.17.2. Let ๐ be a (possibly DG) scheme (almost) of finite type, and let ๐ : ๐ รฃร ๐ be aclosed subscheme with complement ๐ : ๐ รฃร ๐. For D P QCohp๐qโmod, the composite functor:
Kerp๐ห : Dร D๐ q รฃร Dร D bQCohp๐q
QCohp๐^๐ q (6.17.1)
is an equivalence, where ๐^๐ is the formal completion of ๐ along ๐ .
Proof. By [Gai4] Proposition 4.1.5, the restriction functor:
QCohp๐^๐ qโmodร QCohp๐qโmod
is fully-faithful with essential image being those module categories on which objects of QCohp๐q ฤQCohp๐q act by zero. But the endofunctor Kerp๐หq of QCohp๐qโmod is a localization functor forthe same subcategory, giving the claim.
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44 SAM RASKIN
Proof of Theorem 6.17.1. Suppose G P QCohp๐๐ผ ,C๐๐ผ q is some object with:
HomQCohp๐๐ผ ,C๐๐ผ qpPbOblv Loc๐๐ผ pFq,Gq โ 0
for all P P QCohp๐๐ผq perfect and all F P Cb๐ผ compact. Then by Proposition 6.16.1, it suffices toshow that G โ 0.
Fix ๐ : ๐ผ ๐ฝ . We will show by decreasing induction on |๐ฝ | that the restriction of G to ๐p๐q iszero.
We have the closed embedding ๐๐ฝ๐๐๐ ๐ รฃร ๐p๐q with complement being the union:
๐p๐qzp๐๐ฝ๐๐๐ ๐q โ
ฤ
๐ผ๐๐ฝ 1
๐1
โฐ๐ฝ,๐1๐โ๐
๐p๐q.
In particular, the inductive hypothesis implies that the restriction of G to this complement is zero.Let X denote the formal completion of ๐๐ฝ
๐๐๐ ๐ in ๐p๐q and let ๐๐ : X รฃร ๐p๐q denote the embedding.By Lemma 6.17.2, it suffices to show that:
๐ห๐pGq โ 0 P QCohpX,C๐๐ผ q :โ QCohp๐๐ผ ,C๐๐ผ q bQCohp๐๐ผq
QCohpXq.
The map Xร ๐๐ผ๐๐ factors through ๐๐ฝ
๐๐๐ ๐,๐๐ (embedded via ๐), so by factorization we have:
QCohp๐๐ผ ,C๐๐ผ q bQCohp๐๐ผq
QCohpXq โ C๐๐ผ b๐ทp๐๐ผq
QCohpXq ยป Cb๐ฝ b QCohpXq.
This identification is compatible with the functors Loc in the following way. Let๐b : Cb๐ผ ร Cb๐ฝ
denote the map induced by the tensor structure on C. We then have a commutative diagram:
Cb๐ผLoc
๐๐ผ //
๐b
C๐๐ผ// QCohp๐๐ผ ,C๐๐ผ q
๐ห๐
Cb๐ฝidbOX // Cb๐ฝ b QCohpXq.
by construction.Since QCohpXq is compactly generated by objects of the form ๐ห๐pPq with P P QCohp๐p๐qq perfect
(and with set-theoretic support in ๐๐ฝ๐๐๐ ๐), we reduce to the following:
Each F P Cb๐ฝ compact then defines a continuous functor ๐นF : Cb๐ฝ b QCohpXq ร QCohpXq, andour claim amounts to showing that an object in Cb๐ฝ bQCohpXq is zero if and only if each functor๐นF annihilates it, but this is obvious e.g. from the theory of dualizable categories.
6.18. Dualizability. Next, we record the following technical result.
Lemma 6.18.1. For every D P ๐ทp๐๐ผqโmod, the canonical map:
C๐๐ผ b๐ทp๐๐ผq
D โ limp๐ผ
๐๐ฝ
๐๐พqPS๐ผ
ยด
Cb๐พ b๐ทp๐p๐qqยฏ
b๐ทp๐๐ผq
Dร limp๐ผ
๐๐ฝ
๐๐พqPS๐ผ
ยด
Cb๐พ b๐ทp๐p๐qq b๐ทp๐๐ผq
Dยฏ
is an equivalence.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 45
This proof is digressive, so we postpone the proof to Appendix A, assuming it for the remainderof this section.
We obtain the following consequence.32
Corollary 6.18.2. C๐๐ผ is dualizable and self-dual as a ๐ทp๐๐ผq-module category.
Remark 6.18.3. In fact, one can avoid the full strength of Lemma 6.18.1 for our purposes: we includeit because it gives an aesthetically nicer treatment, and because it appears to be an importanttechnical result that should be included for the sake of completeness.
With that said, we apply it below only for D โ Sph๐บ,๐๐ผ , and here it is easier: it follows from the
dualizability of Sph๐บ,๐๐ผ as a ๐ทp๐๐ผq-module category, which is much more straightforward.
6.19. Let ๐ค be an affine algebraic group. We now specialize the above to the case C โ Repp๐ค q.
6.20. Induction. Our main tool in treating Repp๐ค q๐๐ผ is the good behavior of the induction functorAv๐ค๐๐ผ ,ห : ๐ทp๐๐ผq ร Repp๐ค q๐๐ผ introduced below.
6.21. The symmetric monoidal forgetful functor Oblv : Repp๐ค q ร Vect induces a conservativefunctor Oblv๐๐ผ : Repp๐ค q๐๐ผ ร ๐ทp๐๐ผq compatible with ๐ทp๐๐ผq-linear symmetric monoidal struc-tures.
We abuse notation in also letting Oblv๐๐ผ denote the QCohp๐๐ผq-linear functor:
Oblv๐๐ผ : QCohp๐๐ผ ,Repp๐ค q๐๐ผ q ร QCohp๐๐ผq
promising the reader to always take caution to make clear which functor we mean in the sequel.
6.22. Applying the discussion of S6.14, we obtain O๐ค,๐๐ผ P Repp๐ค q๐๐ผ factorizable corresponding tothe regular representation O๐ค P Repp๐ค q of ๐ค (so we are not distinguishing between the sheaf O๐คand its global sections in this notation).33
Proposition 6.22.1. (1) The functor Oblv๐๐ผ : Repp๐ค q๐๐ผ ร ๐ทp๐๐ผq admits a ๐ทp๐๐ผq-linearright adjoint34 Av๐ค๐๐ผ ,ห : ๐ทp๐๐ผq ร Repp๐ค q๐๐ผ compatible with factorization.35
(2) The functor Av๐ค๐๐ผ ,ห maps ๐๐๐ผ to the factorization algebra O๐ค,๐๐ผ introduced above.
Proof. By Proposition B.7.1 and Theorem 6.17.1, it suffices to show that Oblv๐๐ผ maps the ULAgenerators Loc๐๐ผ p๐ q of Repp๐ค q๐๐ผ to ULA objects of ๐ทp๐๐ผq, which is obvious.
For the second part, note that the counit map O๐ค ร ๐ P ComAlgpVectq induces a map Oblv๐๐ผ O๐ค,๐๐ผ ร
๐๐๐ผ P ๐ทp๐๐ผq factorizably, and therefore induces factorizable maps:
O๐ค,๐๐ผ ร Av๐ค๐๐ผ ,หp๐๐๐ผ q.
By factorization, it is enough to show that this map is an equivalence for ๐ผ โ ห, where it is clear.
32We remark that this result is strictly weaker than the above, and more direct to prove.33The ๐ท-module Oblv๐๐ผ pO๐ค,๐๐ผ q P ๐ทp๐๐ผ
q (or its shift cohomologically up by |๐ผ|, depending on oneโs conventions)
appears in [BD] as factorization algebra associated with the the constant ๐ท๐ -scheme ๐ค ห๐๐ผร ๐๐ผ .
34The superscript ๐ค stands for โweak,โ and is included for compatibility with [FG2] S20.35More generally, the proof below shows that the analogous statement holds more generally for any symmetricmonoidal functor ๐น : C ร D P DGCat๐๐๐๐ก with C rigid, where this is generalizing the forgetful functor Oblv :Repp๐ค q ร Vect.
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46 SAM RASKIN
6.23. Coalgebras. We now realize the categories Repp๐ค q๐๐ผ in more explicit terms.
Lemma 6.23.1. The functor Oblv๐๐ผ is comonadic, i.e., satisfies the conditions of the comonadicBarr-Beck theorem.
In fact, we will prove the following strengthening:
Lemma 6.23.2. For any D P ๐ทp๐๐ผqโmod, the forgetful functor:
Oblv๐๐ผ b idD : Repp๐ค q๐๐ผ b๐ทp๐๐ผq
Dร D
is comonadic.
Proof. Using Lemma 6.18.1, we deduce that Oblv๐๐ผ b idD arises by passage to the limit over S๐ผfrom the compatible system of functors:
Repp๐ค qb๐พ b๐ทp๐p๐qq b๐ทp๐๐ผq
Dร ๐ทp๐p๐qq b๐ทp๐๐ผq
D.
Therefore, it suffices to show that each of these functors is conservative and commutes with Oblv-split totalizations.
But by [Gai4] Theorem 2.2.2 and Lemma 5.5.4, the functor Repp๐ค๐q b E ร E is comonadic forany E P DGCat๐๐๐๐ก. This obviously gives the claim.
6.24. ๐ก-structures. It turns out that the categories Repp๐ค q๐๐ผ admit particularly favorable ๐ก-structures.
Proposition 6.24.1. There is a unique ๐ก-structure on Repp๐ค q๐๐ผ (resp. QCohp๐๐ผ ,Repp๐ค q๐๐ผ q) suchthat Oblv๐๐ผ is ๐ก-exact. This ๐ก-structure is left and right complete.
Proof. We first treat the quasi-coherent case.
For every p๐ผ๐ ๐ฝ
๐ ๐พq P S๐ผ , the category:
Repp๐ค qb๐ฝ b QCohp๐p๐qq โ QCohpB๐ค ๐ฝ ห ๐p๐qqadmits a canonical ๐ก-structure, since it is quasi-coherent sheaves on an algebraic stack. This ๐ก-structure is left and right exact, and the forgetful functor to QCohp๐p๐qq is obviously ๐ก-exact.Moreover, the structure functors corresponding to maps in S๐ผ are ๐ก-exact, and therefore we obtaina ๐ก-structure with the desired properties on the limit, which is QCohp๐๐ผ ,Repp๐ค q๐๐ผ q.
We now deduce the ๐ท-module version. We have the adjoint functors:36
QCohp๐๐ผ ,Repp๐ค q๐๐ผ qInd // Repp๐ค q๐๐ผ .Oblvoo
Since the monad Oblv Ind is ๐ก-exact on QCohp๐๐ผ ,Repp๐ค q๐๐ผ q and since Oblv is conservative, itfollows that Repp๐ค q๐๐ผ admits a unique ๐ก-structure such that the functor37 Oblvrdimp๐๐ผqs โ
Oblvr|๐ผ|s : Repp๐ค q๐๐ผ ร QCohp๐๐ผ ,Repp๐ค q๐๐ผ q is ๐ก-exact. Since this functor is continuous and com-mutes with limits (being a right adjoint), this ๐ก-structure on Repp๐ค q๐๐ผ is left and right complete.
36Apologies are due to the reader for using the different functors Oblv and Oblv๐๐ผ in almost the same breath.37We use a cohomological shift here since for ๐ smooth, Oblv : ๐ทp๐q ร QCohp๐q only ๐ก-exact up to shift by thedimension, since Oblvp๐๐q โ O๐ . This is because we are working with the so-called left forgetful functor, not theright one.
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It remains to see that Oblv๐๐ผ : Repp๐ค q๐๐ผ ร ๐ทp๐๐ผq is ๐ก-exact. This is immediate: we see that
the ๐ก-structure we have constructed is the unique one for which the composition Repp๐ค q๐๐ผOblvr|๐ผ|sรรรรรร
QCohpRepp๐ค q๐๐ผ qOblv
๐๐ผรรรรรร QCohp๐๐ผq is ๐ก-exact, and this composition coincides with Repp๐ค q๐๐ผ
Oblv๐๐ผ
รรรรรร
๐ทp๐๐ผqOblvr|๐ผ|sรรรรรร QCohp๐๐ผq. We obtain the claim, since the standard ๐ก-structure on ๐ทp๐๐ผq is the
unique one for which Oblv๐๐ผ r|๐ผ|s : ๐ทp๐๐ผq ร QCohp๐๐ผq is ๐ก-exact.
Proposition 6.24.2. The functor Av๐ค๐๐ผ ,ห : ๐ทp๐๐ผq ร Repp๐ค q๐๐ผ is ๐ก-exact for the ๐ก-structure
of Proposition 6.24.1, and similarly for the corresponding quasi-coherent functor QCohp๐๐ผq ร
QCohp๐๐ผ ,Repp๐ค q๐๐ผ q.
We will use the following result of [BD]. We include a proof for completeness.
Lemma 6.24.3. Let ๐ด P Vect be a classical (unital) commutative algebra and let ๐ผ รร ๐ด๐๐ผ P
๐ทp๐๐ผq be the corresponding factorization algebra. Then ๐ด๐๐ผ rยด|๐ผ|s P ๐ทp๐๐ผq.
Proof. We can assume |๐ผ| ฤ 1, since otherwise the result is clear.Choose ๐, ๐ P ๐ผ distinct. Let ๐ผ ๐ผ be the set obtained by contracting ๐ and ๐ onto a single
element (so |๐ผ| โ |๐ผ| ยด 1).
The map ๐ผ ๐ผ defines a diagonal closed embedding โ : ๐๐ผ ร ๐๐ผ . Let ๐ : ๐ รฃร ๐๐ผ denote thecomplement, which here is affine.
Since โ!p๐ด๐๐ผ q โ ๐ด๐๐ผ , the result follows inductively if we show that the map ๐ห,๐๐ ๐
!p๐ด๐๐ผ q ร
โห,๐๐ โ!p๐ด๐๐ผ qr1s is surjective after taking cohomology in degree ยด|๐ผ|.
Writing ๐ผ โ t๐uลก
๐ผ using the evident splitting, we obtain the following commutative diagramfrom unitality of ๐ด and from the commutative factorization structure:
๐๐ b๐ด๐๐ผ
// ๐ห,๐๐ ๐!p๐๐ b๐ด
๐๐ผ q //
โห,๐๐ p๐ด๐๐ผ qr1s
ยป
๐ด๐ b๐ด๐๐ผ
๐ห,๐๐ ๐!p๐ด๐ b๐ด
๐๐ผ q
๐ด๐๐ผ
// ๐ห,๐๐ ๐!p๐ด๐๐ผ q // โห,๐๐ โ!p๐ด๐๐ผ qr1s
The top line is obviously (by induction) a short exact sequence in the |๐ผ|-shifted heart of the๐ก-structure. Since the right vertical map is an isomorphism, this implies the claim.
Proof of Proposition 6.24.2. E.g., in the quasi-coherent setting: it suffices to show that Av๐ค๐๐ผ ,ห หOblv๐๐ผ
is ๐ก-exact. This composition is given by tensoring with O๐ค,๐๐ผ P ๐ทp๐๐ผq by construction, which we
have just seen is in the heart of the ๐ก-structure (since Oblv : ๐ทp๐๐ผq ร QCohp๐๐ผq is ๐ก-exact onlyafter a shift by |๐ผ|).
It follows that this functor is right ๐ก-exact, since it is given by tensoring with something in theheart. But it is also left ๐ก-exact, since it is right adjoint to the ๐ก-exact functor Oblv๐๐ผ .
Corollary 6.24.4. Repp๐ค q๐๐ผ is the derived category of the heart of this ๐ก-structure.
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48 SAM RASKIN
Proof. At the level of bounded below derived categories, this is a formal consequence of the corre-sponding fact for ๐ทp๐๐ผq and the fact that Oblv๐๐ผ and Av๐ค๐๐ผ ,ห are ๐ก-exact.
To treat unbounded derived categories, it suffices to show that the derived category of Repp๐ค q๐๐ผ
is left complete, but this is clear: the category has finite homological dimension.
6.25. Constructibility. We now show how to recover Repp๐ค q๐๐ผ from a holonomic version.This material is not necessary for our purposes, but we include it for completeness. The reader
may safely skip straight to S6.28.
6.26. Let ๐ทโ๐๐p๐๐ผq ฤ ๐ทp๐๐ผq denote the ind-completion of the subcategory of ๐ทp๐๐ผq formed by
compact objects (i.e., coherent ๐ท-modules) that are holonomic in the usual sense. We emphasizethat we allow infinite direct sums of holonomic objects to be counted as such.
Definition 6.26.1. Define the holonomic subcategory Repp๐ค q๐๐ผ ,โ๐๐ of Repp๐ค q๐๐ผ to consist of those
objects that map into ๐ทโ๐๐p๐๐ผq under the forgetful functor.
Remark 6.26.2. We have:
Repp๐ค q๐๐ผ ,โ๐๐ ยป limp๐ผ
๐๐ฝ
๐๐พq
Repp๐ค qb๐พ b๐ทโ๐๐p๐p๐qq ฤ
limp๐ผ
๐๐ฝ
๐๐พq
Repp๐ค qb๐พ b๐ทp๐p๐qq โ: Repp๐ค q๐๐ผ .(6.26.1)
Indeed, the key point is that Repp๐ค qb๐พ b ๐ทโ๐๐p๐p๐qq ร Repp๐ค qb๐พ b ๐ทp๐p๐qq is actually fully-faithful, and this follows from the general fact that tensoring a fully-faithful functor (here๐ทโ๐๐p๐p๐qq รฃร๐ทp๐p๐qq with a dualizable category (here Repp๐ค qb๐พ) gives a fully-faithful functor.
Since e.g. for each ๐ : ๐ผ ๐ฝ , ๐ทโ๐๐p๐๐ฝq is dualizable as a ๐ทโ๐๐p๐
๐ผq-module category (forthe same reason as for the non-holonomic categories), we deduce that Repp๐ค q๐๐ผ ,โ๐๐ satisfies thesame factorization patterns at Repp๐ค q๐๐ผ , but with holonomic ๐ท-module categories being usedeverywhere. Indeed, the arguments we gave were basically formal cofinality arguments, and thereforeapply verbatim.
6.27. We have the following technical result.
Proposition 6.27.1. The functor:
Repp๐ค q๐๐ผ ,โ๐๐ b๐ทโ๐๐p๐๐ผq
๐ทp๐๐ผq ร Repp๐ค q๐๐ผ
is an equivalence.
Remark 6.27.2. In light of (6.26.1), this amounts to commuting a limit with a tensor product.However, we are not sure how to use this perspective to give a direct argument, since ๐ทp๐๐ผq is(almost surely) not dualizable as a ๐ทโ๐๐p๐
๐ผq-module category.
Proof of Proposition 6.27.1. The idea is to appeal to use Proposition B.8.1.
Step 1. Let ๐ P Repp๐ค qb๐ผ be given. We claim that Loc๐๐ผ p๐ q lies in Repp๐ค q๐๐ผ ,โ๐๐ and that induced
object of Repp๐ค q๐๐ผ ,โ๐๐ b๐ทโ๐๐p๐๐ผq
๐ทp๐๐ผq is ULA in this category (considered as a ๐ทp๐๐ผq-module
category in the obvious way) if ๐ is compact.Indeed, that Loc๐๐ผ p๐ q is holonomic follows since Oblv๐๐ผ p๐ q is lisse. The ULA condition then
follows from Proposition B.5.1 and Remark B.5.2.
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Step 2. Next, we claim that Repp๐ค q๐๐ผ ,โ๐๐ is generated as a ๐ทโ๐๐p๐๐ผq-module category by the
objects Loc๐๐ผ p๐ q, ๐ P Repp๐ค qb๐ผ , i.e., the minimal ๐ทโ๐๐p๐๐ผq-module subcategory of Repp๐ค q๐๐ผ ,โ๐๐
containing the Loc๐๐ผ p๐ q is the whole category.Indeed, this follows as in the proof of Theorem 6.17.1.
Step 3. We now claim that Repp๐ค q๐๐ผ ,โ๐๐ b๐ทโ๐๐p๐๐ผq
๐ทp๐๐ผq is ULA as a ๐ทp๐๐ผq-module category.
We have to show that Repp๐ค q๐๐ผ b๐ทโ๐๐p๐๐ผq
QCohp๐๐ผq is generated as a QCohp๐๐ผq-module category
by objects coming from Loc๐๐ผ p๐ q. But this is clear from Step 2.
Step 4. Finally, we apply Proposition B.8.1 to obtain the result:Our functor sends a set of ULA generators to ULA objects. And moreover, by Remark 6.26.2,
this functor is an equivalence after tensoring with ๐ทp๐๐ฝ๐๐๐ ๐q for each ๐ : ๐ผ ๐ฝ , giving the result.
Remark 6.27.3. Taking (6.26.1) as a definition of C๐๐ผ ,โ๐๐ for general rigid C, the above argumentshows that the analogue of Proposition 6.27.1 is true in this generality.
6.28. The naive Satake functor. We now specialize the above to ๐ค โ .
6.29. Digression: more on twists. We will work with Grassmannians and loop groups twistedby ๐ซ๐๐๐
๐ as in S2.14.To define Gr๐ค,๐๐ผ for ๐ค P t๐,๐ต,๐ยด, ๐บu, one exactly follows S2.14.
Similarly, we have a group scheme (resp. group indscheme) ๐ค p๐q๐๐ผ (resp. ๐ค p๐พq๐๐ผ ) over ๐๐ผ for๐ค as above, where ๐ค p๐พq๐๐ผ acts on Gr๐บ,๐๐ผ . Trivializing ๐ซ๐๐๐
๐ locally on ๐๐ผ , the picture becomes theusual picture for factorizable versions of the arc and loop groups: c.f. [BD] and [KV] for example.
6.30. Let Sph๐บ,๐๐ผ denote the spherical Hecke category ๐ทpGr๐บ,๐๐ผ q๐บp๐q
๐๐ผ . The assignment ๐ผ รรSph๐บ,๐๐ผ defines a factorization monoidal category.
Our goal for the remainder of this section is to construct and study certain monoidal functors:
Sat๐๐๐๐ฃ๐๐๐ผ : Reppq๐๐ผ ร Sph๐บ,๐๐ผ
compatible with factorization.
Remark 6.30.1. We follow Gaitsgory in calling this functor naive because it is an equivalence onlyon the hearts of the ๐ก-structures (indeed, it is not an equivalence on Exts between unit objects,since equivariant cohomology appears in the right hand side but not the left).
6.31. The following results provide toy models for constructing the functors Sat๐๐๐๐ฃ๐๐๐ผ .
Lemma 6.31.1. For D P DGCat๐๐๐๐ก, the map:
t๐น : Repp๐ค q ร D P DGCat๐๐๐๐กu ร O๐ค โcomodpDq
๐น รร ๐น pO๐ค q
is an equivalence.
Proof. Since Repp๐ค q is self-dual and since Repp๐ค q bDOblvรรรร Vect bD โ D is comonadic (c.f. the
proof of Lemma 6.23.1), we obtain the claim.
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50 SAM RASKIN
Lemma 6.31.2. For D P AlgpDGCat๐๐๐๐กq a monoidal (in the cocomplete sense) DG category, themap:
t๐น : Repp๐ค q ร D continuous and lax monoidalu ร AlgpO๐ค โcomodpDqq
๐น รร ๐น pO๐ค q
is an equivalence. Here O๐ค โcomodpDq is equipped with the obvious monoidal structure, induced fromthat of D.
Remark 6.31.3. Here is a heuristic for Lemma 6.31.2:Given ๐ด P O๐ค โcomodpDq, the corresponding functor Repp๐ค q ร D is given by the formula
๐ รร p๐ b๐ดq๐ค (where the invariants here are of course derived). If ๐ด is moreover equipped with a๐ค -equivariant algebra structure, we obtain the canonical maps:
p๐ b๐ดq๐ค b p๐ b๐ดq๐ค ร p๐ b๐ดb๐ b๐ดq๐ค โ p๐ b๐ b๐ดb๐ดq๐ค ร p๐ b๐ b๐ดq๐ค
as desired, where the last map comes from the multiplication on ๐ด.
Proof of Lemma 6.31.2. This follows e.g. from the identification of the monoidal structure of Repp๐ค qbD with the Day convolution structure on the functor category HomDGCat๐๐๐๐กpRepp๐ค q,Dq, identifyingthe two via self-duality of Repp๐ค q.
6.32. We will use the following more sophisticated version of the above lemmas.
Lemma 6.32.1. For D P ๐ทp๐๐ผqโmod, the functor:
t๐น : Repp๐ค q๐๐ผ ร D P ๐ทp๐๐ผqโmodu ร Repp๐ค q๐๐ผ b๐ทp๐๐ผq
DLem. 6.23.2
โ O๐ค,๐๐ผโcomodpDq
๐น รร ๐น pO๐ค,๐๐ผ q
is an equivalence. Giving a lax monoidal structure in the left hand side amounts to giving an algebrastructure on the right hand side.
Proof. By Lemma 6.18.1, ๐ทp๐๐ผq-linear functors Repp๐ค q๐๐ผ ร D are equivalent to objects ofRepp๐ค q๐๐ผ b๐ทp๐๐ผq D.
The result then follows from Lemma 6.23.2 and Lemma 6.31.2.
6.33. Construction of the functor. By Lemma 6.32.1, to construct Sat๐๐๐๐ฃ๐๐๐ผ as a lax monoidal
functor, we need to specify an object of Reppq๐๐ผ with an algebra structure.Such objects โ๐โ
๐๐ผ P Reppq๐๐ผ b๐ทp๐๐ผq Sph๐บ,๐๐ผ are defined in a factorizable way in Appendix
B of [Gai1] (they go by the name chiral Hecke algebra and were probably first constructed byBeilinson).38 For each ๐ผ, โ๐โ
๐๐ผ is concentrated in cohomological degree ยด|๐ผ|.
Example 6.33.1. For ๐ผ โ ห, โ๐โ๐ comes from the regular representation of under geometric Satake.
Remark 6.33.2. We emphasize that the general construction (and the data required to define theoutput) is purely abelian categorical, and comes from the usual construction of the geometric Satakeequivalence.
Lemma 6.33.3. The lax monoidal functors Sat๐๐๐๐ฃ๐๐๐ผ are actually monoidal.
38In the notation of [Gai1], we have โ๐โ๐๐ โ R๐
๐ r๐s โ๐R๐
๐ r๐s.
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Proof. We need to check that some maps between some objects of Sph๐บ,๐๐ผ are isomorphisms. It
suffices to do this after restriction to strata on ๐๐ผ , and by factorization, we reduce to the case ๐ผ โ หwhere it follows from usual geometric Satake and the construction of the chiral Hecke algebra.
6.34. We have the following important fact:
Proposition 6.34.1. Sat๐๐๐๐ฃ๐๐๐ผ is ๐ก-exact.
We begin with the following.
Lemma 6.34.2. The functor Sph๐บ,๐๐ผ ร Sph๐บ,๐๐ผ defined by convolution with โ๐โ๐๐ผ is ๐ก-exact.
Proof. Recall that for each ๐ผ and ๐ฝ , there is the exterior convolution functor:
Sph๐บ,๐๐ผ b Sph๐บ,๐๐ฝ ร Sph๐บ,๐๐ผลก
๐ฝ
which is a morphism of ๐ทp๐๐ผลก
๐ฝq-module categories.39 The relation to usual convolution is thatfor ๐ฝ โ ๐ผ, convolution is obtained by applying exterior convolution and then !-restricting to thediagonal.
The usual semi-smallness argument shows that exterior convolution is ๐ก-exact. Therefore, sinceโ๐โ๐๐ผ lies in degree ยด|๐ผ|, we deduce from the above that convolution with โ๐โ
๐๐ผ has cohomologicalamplitude rยด|๐ผ|, 0s: in particular, it is right ๐ก-exact.
It remains to see that this convolution functor is left ๐ก-exact. For a given partition ๐ : ๐ผ ๐ฝ ,let ๐๐ : ๐๐ฝ
๐๐๐ ๐ ร ๐๐ผ denote the embedding of the corresponding stratum of ๐๐ผ . The !-restriction
of โ๐โ๐๐ผ to ๐๐ฝ
๐๐๐ ๐ is concentrated in cohomological degree ยด|๐ฝ |, and is the object corresponding tothe regular representation under geometric Satake. It follows that the functor of convolution with๐๐,ห,๐๐ ๐
!๐pโ๐โ
๐๐ผ q is left ๐ก-exact from the exactness of convolution in the Satake category for a point.We now obtain the claim by devissage.
Proof of Proposition 6.34.1. First, we claim that our functor is left ๐ก-exact.We can write Sat๐๐๐๐ฃ๐๐๐ผ as a composition of tensoring F with the delta ๐ท-module on the unit of
Gr๐บ,๐๐ผ , convolving with โ๐โ๐๐ผ , and then taking invariants with respect to the โdiagonalโ actions
for the ๐ฝ . The first step is obviously ๐ก-exact, and the second step is ๐ก-exact by Lemma 6.34.2; thethird step is obviously left ๐ก-exact.
It remains to show that it is right ๐ก-exact.First, let ๐ P Repp๐ผq โ Reppqb๐ผ,. We claim that convolution with Sat๐๐๐๐ฃ๐๐๐ผ pLoc๐๐ผ p๐ qq is
๐ก-exact (as an endofunctor of Sph๐บ,๐๐ผ ).
It suffices to show this for ๐ finite-dimensional, and then duality of ๐ and monoidality of Sat๐๐๐๐ฃ๐๐๐ผ
reduces us to showing exactness in either direction: we show that this convolution functor is left๐ก-exact. This then follows by the same stratification argument as in the proof of Lemma 6.34.2.
In particular, convolving with the unit, we see that Sat๐๐๐๐ฃ๐๐๐ผ pLoc๐๐ผ p๐ qq is concentrated in coho-
mological degree ยด|๐ผ|, and more generally, Sat๐๐๐๐ฃ๐๐๐ผ หLoc๐๐ผ is ๐ก-exact up to this same cohomologicalshift.
For simplicity, we localize on ๐ to assume ๐ is affine. Then by Theorem 6.17.1, Reppqฤ0๐๐ผ is
generated under colimits by objects of the form Ind OblvpLoc๐๐ผ p๐ qq for ๐ P Repp๐ค ๐ผqฤ|๐ผ|: indeed,this follows from the observation that Ind Oblv is ๐ก-exact, which is true since after applying Oblv
39We emphasize that ๐ผ and ๐ฝ play an asymmetric role in the definition, i.e., the definition depends on an orderedpair of finite sets, not just a pair of finite sets.
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52 SAM RASKIN
again, it is given by tensoring with the ind-vector bundle of differential operators on ๐๐ผ . The samereasoning shows that Ind Oblv is ๐ก-exact on Sph๐บ,๐๐ผ , giving the result.
6.35. The naive Satake theorem. We will not need the following result, but include a proof forcompleteness. Since we are not going to use it, we permit ourselves to provide substandard detail.
Theorem 6.35.1. The functor Sat๐๐๐๐ฃ๐๐๐ผ induces an equivalence between the hearts of the ๐ก-structures:
Sat๐๐ผ : Reppq
๐๐ผ
ยปรร Sph
๐บ,๐๐ผ .
We will give an argument in S6.37.
Remark 6.35.2. In the setting of perverse sheaves, Theorem 6.35.1 is proved in [Gai1] AppendixB. We provide a different argument from loc. cit. that more easily deals with the problem of non-holonomic ๐ท-modules.
6.36. Spherical Whittaker sheaves. Our argument for Satake will appeal to the following. Let
Whit๐ ๐โ๐๐ผ denote the category of Whittaker๐ท-modules on Gr๐บ,๐๐ผ , i.e., ๐ท-modules equivariant against
๐ยดp๐พq๐๐ผ equipped with its standard character (we use the ๐p๐๐q-twist here).
We have a canonical functor Sph๐บ,๐๐ผ ร Whit๐ ๐โ๐๐ผ given by convolution with the unit object
unitWhit๐ ๐โ
๐๐ผP Whit๐ ๐โ
๐๐ผ , i.e., the canonical object cleanly extended from Gr๐ยด,๐๐ผ (i.e., the ห and
!-extensions coincide here).
Theorem 6.36.1 (Frenkel-Gaitsgory-Vilonen, Gaitsgory, Beraldo). The composite functor:
Reppq๐๐ผ
Sat๐๐๐๐ฃ๐๐๐ผ
รรรรรร Sph๐บ,๐๐ผ รWhit๐ ๐โ๐๐ผ
is an equivalence.
Proof. We will appeal to Proposition B.8.1.
It is easy to see that the unit object of Whit๐ ๐โ๐๐ผ is ULA: this follows from the usual cleanness
argument. We then formally deduce from dualizability of ULA objects in Reppq๐๐ผ and monoidalityof Sat๐๐๐๐ฃ๐๐๐ผ that the above functor sends ULA objects to ULA objects.
Then since these sheaves of categories are locally constant along strata (by factorizability), weobtain the claim by noting that this functor is an equivalence over a point, as follows from [FGV]and the comparison of local and global40 definitions of spherical Whittaker categories, as has beendone e.g. in the unpublished work [Gai2].
We also use the following fact about Whittaker categories.
Lemma 6.36.2. The object Av๐บp๐q๐๐ผ ,หpunit
Whit๐ ๐โ๐๐ผq P Sph๐บ,๐๐ผ lies in cohomological degrees ฤ ยด|๐ผ|.
The adjunction map:
unitSph๐บ,๐๐ผ
ร Av๐บp๐q๐๐ผ ,หpunit
Whit๐ ๐โ๐๐ผq
is an equivalence on cohomology in degree ยด|๐ผ|. (Here unitSph๐บ,๐๐ผ
is the delta ๐ท-module on ๐๐ผ
ห-pushed forward to Gr๐บ,๐๐ผ using the tautological section).
40I.e., using Drinfeldโs compactifications as in [FGV].
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Proof. The corresponding fact over a point is obvious: the fact that Sph๐บ,๐ฅ ร Whit๐ ๐โ๐ฅ is ๐ก-exacton hearts of ๐ก-structures implies that its right adjoint left ๐ก-exact, so applying the above averagingto the unit, one obtains an object in degrees ฤ 0. The adjunction map is an equivalence on 0thcohomology because Sph๐บ,๐ฅ รWhit๐ ๐โ๐ฅ is an equivalence on hearts of ๐ก-structures.
We then deduce that from factorization that for each ๐ : ๐ผ ๐ฝ , the !-restriction of:
CokerpunitSph๐บ,๐๐ผ
ร Av๐บp๐q๐๐ผ ,หpunit
Whit๐ ๐โ๐๐ผqq
to the corresponding stratum ๐๐ฝ๐๐๐ ๐ defined by ๐ is concentrated in cohomological degrees ฤ ยด|๐ฝ |,
which immediately gives the claim.
6.37. We now deduce factorizable Satake.
Proof of Theorem 6.35.1. We have an adjunction Sph๐บ,๐๐ผ// Whit๐ ๐โ
๐๐ผoo where the left adjoint is
convolution with the unit and the right adjoint is ห-averaging with respect to ๐บp๐q๐๐ผ .From Theorem 6.36.1, we obtain the adjunction:
Sph๐บ,๐๐ผ// Reppq๐๐ผ .
Sat๐๐๐๐ฃ๐๐๐ผ
oo
Since Sat๐๐๐๐ฃ๐๐๐ผ is ๐ก-exact, we obtain a corresponding adjunction between the hearts of the ๐ก-structure.Lemma 6.36.2 implies that the left adjoint is fully-faithful at the abelian categorical level, and the
right adjoint Sat๐๐๐๐ฃ๐,๐๐ผ is conservative by Theorem 6.36.1, so we obtain the claim.
7. Hecke functors: Zastava with moving points
7.1. As in S5, the main result of this section, Theorem 7.9.1, will compare geometrically andspectrally defined Chevalley functors. However, in this section, we work over powers of the curve:we are giving a compatibility now between Theorem 4.6.1 and the factorizable Satake theorem ofS6.
7.2. Structure of this section. In S7.3-7.9, we give โmoving pointsโ analogues of the construc-tions of S5 and formulate our main theorem.
The remainder of the section is dedicated to deducing this theorem from Theorem 7.9.1.There are two main difficulties in proving the main theorem: working over powers of the curve
presents difficulties, and the fact that we are giving a combinatorial (i.e., involving Langlandsduality) comparison functors in the derived setting.
The former we treat by exploiting ULA objects: c.f. Appendix B and S6. These at once exhibitgood functoriality properties and provide a method for passing from information the disjoint locus๐๐ผ๐๐๐ ๐ to the whole of ๐๐ผ .We treat the homotopical difficulties by exploiting a useful ๐ก-structure on factorization ฮฅn-
modules, c.f. Proposition 7.11.1.
7.3. Define the indscheme Divฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ over ๐๐ผ as parametrizing an ๐ผ-tuple ๐ฅ โ p๐ฅ๐q of points of
๐ and a ฮ-valued divisor on ๐ that is ฮ๐๐๐ -valued on ๐zt๐ฅ๐u.
Warning 7.3.1. The notation 8 ยจ ๐ฅ in the superscript belies that ๐ฅ is a dynamic variable: it is usedto denote our ๐ผ-tuple of points in ๐. We maintain this convention in what follows, keeping thesubscript ๐๐ผ to indicate that we work over powers of the curve now.
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Remark 7.3.2. We again have a degree map Divฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ ร ฮ.
Let ฮฅnโmodfact๐ข๐,๐๐ผ denote the DG category of unital factorization modules for ฮฅn on Divฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ .
The two functors we will compare will go from Reppq๐๐ผ to ฮฅnโmodfact๐ข๐,๐๐ผ .
7.4. Geometric Chevalley functor. To construct the geometric Chevalley functor, we imitatemuch of the geometry that appeared in S5.2-5.14.
7.5. For starters, define Bun8ยจ๐ฅ๐ยด,๐๐ผ ร ๐๐ผ as parametrizing ๐ฅ โ p๐ฅ๐q๐P๐ผ P ๐
๐ผ , a ๐บ-bundle ๐ซ๐บ on๐, and non-zero maps:
ฮฉbp๐,๐q๐ ร ๐ ๐
๐ซ๐บp8 ยจ ๐ฅq
defined for each dominant weight ๐ and satisfying the Plucker relations, in the notation of S5.2.Here the notation of twisting by O๐p8 ยจ ๐ฅq makes sense in ๐-points: for ๐ฅ โ p๐ฅ๐q๐P๐ผ : ๐ ร ๐๐ผ , wetake the sum of the Cartier divisors on ๐ ห ๐ associated with the graphs of the maps ๐ฅ๐ to defineO๐ห๐p๐ฅq.
7.6. We can imitate the other constructions in the same fashion, giving the indscheme๐๐ต8ยจ๐ฅ๐๐ผ (resp.
๐ต8ยจ๐ฅ) over ๐๐ผ and the map๐๐8ยจ๐ฅ๐๐ผ :
๐๐ต8ยจ๐ฅ๐๐ผ ร Divฮ๐๐๐ ,8ยจ๐ฅ
eff,๐๐ผ (resp. ๐8ยจ๐ฅ๐๐ผ : ๐ต8ยจ๐ฅ
๐๐ผ ร Divฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ ).
Let ๐ด๐๐ผ be the inverse image of ๐๐ผ ห Bun๐ยด ฤ Bun8ยจ๐ฅ๐ยด,๐๐ผ . We have a distinguished object
๐๐ด๐๐ผP ๐ทp๐ด๐๐ผ q, obtained by !-pullback from ๐๐๐ผ bWโ๐๐ก P ๐ทp๐๐ผ ห Bun๐ยดq.
We also have a ๐ทp๐๐ผq-linear action of Sph๐บ,๐๐ผ on ๐ทp๐ต8ยจ๐ฅ๐๐ผ q.
We obtain a ๐ทp๐๐ผq-linear functor:
Chevgeomn,๐๐ผ : Reppq๐๐ผ ร ฮฅnโmodfact๐ข๐,๐๐ผ
imitating our earlier functor Chevgeomn,๐ฅ . Indeed, we use the naive Satake functor, convolution with
(the ห-pushforward of) ๐Y๐๐ผ
, !-restriction to๐๐ต8ยจ๐ฅ๐๐ผ , and then ห-pushforward to Divฮ๐๐๐ ,8ยจ๐ฅ
eff,๐๐ผ , exactly
as in S5.
7.7. Spectral Chevalley functor. To construct Chevspecn,๐๐ผ , we will use the following.
Lemma 7.7.1. The category n๐โmod๐๐ผ of Lie-ห modules on ๐๐ผ for n๐ P Repp๐ q๐ is canonicallyidentified with the category nโmodpRepp๐ qq๐๐ผ , i.e., the ๐ทp๐๐ผq-module category associated with thesymmetric monoidal DG category nโmodpRepp๐ qq by the procedure of S6.6.
Proof. Let ฮ ฤ ๐ ห ๐๐ผ be the union of the graphs of the projections ๐๐ผ ร ๐. Let ๐ผ (resp. ๐ฝ)denote the projection from ฮ to ๐ (resp. ๐๐ผ).
Since ๐ฝ is proper, one finds that:
๐ฝห,๐๐ ๐ผ! : ๐ทp๐q ร ๐ทp๐๐ผq
is colax symmetric monoidal, and in particular maps Lie coalgebras for p๐ทp๐q,!bq to Lie coalgebras
for p๐ทp๐๐ผq,!bq.
Moreover, if ๐ฟ P ๐ทp๐q is a Lie-ห algebra and compact as a ๐ท-module, then its Verdier dual
D๐ ๐๐๐๐๐๐p๐ฟq is a Lie coalgebra in p๐ทp๐q,!bq, and ๐ฟ-modules on๐๐ผ are equivalent to ๐ฝห,๐๐ ๐ผ
!pD๐ ๐๐๐๐๐๐p๐ฟqq-comodules. We have an obvious translation of this for the โgradedโ case, where e.g. ๐ทpGr๐,๐๐ผ q
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 55
replaces ๐ทp๐๐ผq. (See [Roz] Proposition 4.5.2 for a non-derived version of this; essentially the sameargument works in general).
One then easily finds that for ๐ P Vect, one has:
๐ฝห,๐๐ ๐ผ!p๐ b ๐๐q lim
p๐ผ๐ฝ๐พqPS๐ผ๐๐,ห,๐๐ p๐
โ๐พ b ๐๐p๐qq
where the notation is as in S6. We remark that this limit is a โlogarithmโ of the one appearing in(6.13.1): we use the addition maps ๐ โ๐พ ร ๐ โ๐พ
1
for ๐พ ๐พ 1 to give the structure maps in thelimit, i.e., the canonical structure of commutative algebra on ๐ in pVect,โq.
Moreover, this identification is compatible with Lie cobrackets, so that the Lie coalgebra pn_qb๐๐maps to the Lie coalgebra:
๐ฝห,๐๐ ๐ผ!pn_ b ๐๐q lim
p๐ผ๐ฝ๐พqPS๐ผ๐๐,ห,๐๐ ppn
_qโ๐พ b ๐๐p๐qq.
This immediately gives the claim.
Remark 7.7.2. We identify n๐โmod๐๐ผ and nโmodpRepp๐ qq๐๐ผ in what follows. We emphasize thatalthough the ฮ-grading does not appear explicitly in the notation, it is implicit in the fact that n๐is always considered as ฮ-graded.
We obtain the restriction functor:
Reppq๐๐ผ ร n๐โmod๐๐ผ .
Using the chiral induction functor Ind๐โ : n๐โmod๐๐ผ ร ฮฅnโmodfact๐ข๐,๐๐ผ and the restriction functor
from to , we obtain:
Chevspecn,๐๐ผ : Reppq๐๐ผ ร ฮฅnโmodfact๐ข๐,๐๐ผ
as desired.
7.8. For convenience, we record the following consequence of Lemma 7.7.1. The reader may skipthis section.
Recall from [Ras1] S6.12 and S8.14 that the external fusion construction defines a lax unitalfactorization category structure on the assignment:
๐ผ รร ฮฅnโmodfact๐ข๐,๐๐ผ .
Corollary 7.8.1. The lax factorization structure is a true factorization structure. I.e., for every๐ผ, ๐ฝ P Setฤ8, the external fusion functor:
rฮฅnโmodfact๐ข๐,๐๐ผbฮฅnโmodfact๐ข๐,๐๐ฝ s b๐ทp๐๐ผ
ลก
๐ฝ q
๐ทpr๐๐ผห๐๐ฝ s๐๐๐ ๐q ร rฮฅnโmodfact๐ข๐,๐๐ผ
ลก
๐ฝ b๐ทp๐๐ผ
ลก
๐ฝ q
๐ทpr๐๐ผห๐๐ฝ s๐๐๐ ๐q
is an equivalence.
Proof. The corresponding result for Lie-ห modules over n๐ follows from Lemma 7.7.1. Using theadjoint functors pInd๐โ,Oblv๐โq, we see that factorization modules for ฮฅn are modules for a monadon Lie-ห modules, and the two monads obviously match up e.g. by the chiral PBW theorem.
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56 SAM RASKIN
7.9. Formulation of the main theorem. Observe that formation of each of the functors Chevspecn,๐๐ผ
and Chevgeomn,๐๐ผ are compatible with factorization as we vary the finite set ๐ผ (here we use the external
fusion construction ฮฅnโmodfact๐ข๐,๐๐ผ ).
Theorem 7.9.1. The factorization functors ๐ผ รร Chevspecn,๐๐ผ and ๐ผ รร Chevgeom
n,๐๐ผ are canonically
isomorphic as factorization functors.
The proof of Theorem 7.9.1 will occupy the remainder of this section.
Remark 7.9.2. Here is the idea of the argument: since both functors factorize, we know the resultover strata of ๐๐ผ by Theorem 5.14.1. We glue these isomorphisms over all of ๐๐ผ by analyzing ULAobjects.
Remark 7.9.3. This theorem is somewhat loose as stated, as it does not specify how they areisomorphic. This is because the construction of the isomorphism is somewhat difficult, due in partto the difficulty of constructing anything at all in the higher categorical setting.
However, we remark that for ๐บ simply-connected, we will see that such an isomorphism offactorization functors is uniquely characterized as such. Similarly, for ๐บ a torus, it is easy to writedown such an isomorphism by hand (just as it is easy to write down the (naive) geometric Satakeby hand in this case). This should be taken to indicate the existence of a canonical isomorphism ingeneral. We refer to Remark 7.10.2 and S7.22 for further discussion of this point.
7.10. First, we observe the following.
Lemma 7.10.1. Chevspecn,๐๐ผ and Chevgeom
n,๐๐ผ are canonically isomorphic for ๐ผ โ ห.
Proof. We are comparing two ๐ทp๐q-linear functors:
Reppq๐ โ Reppq b๐ทp๐q ร ฮฅnโmodfact๐ข๐,๐
or equivalently, two continuous functors:
Reppq ร ฮฅnโmodfact๐ข๐,๐ .
By lisseness along ๐, we obtain the result from Theorem 5.14.1 (alternatively: the methods ofTheorem 5.14.1 work when the point ๐ฅ is allowed to vary, giving the result).
Remark 7.10.2. In what follows, we will see that the isomorphism of Theorem 7.9.1 is uniquelypinned down by a choice of isomorphism over ๐, i.e., an isomorphism as in Lemma 7.10.1. In-deed, this will follow from Proposition 7.18.2. Note that we have constructed such an isomorphismexplicitly in the proof of Theorem 5.14.1, and therefore this completely pins down Theorem 7.9.1.
7.11. Digression: a ๐ก-structure on factorization modules. We now digress to discussion thefollowing result.
Proposition 7.11.1. (1) There is a (necessarily unique) ๐ก-structure on ฮฅnโmodfact๐ข๐,๐๐ผ such that
the forgetful functor:
Oblvฯn : ฮฅnโmodfact๐ข๐,๐๐ผ ร ๐ทpDiv8ยจ๐ฅeff,๐๐ผ q (7.11.1)
is ๐ก-exact.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 57
(2) With respect to this ๐ก-structure, the chiral induction functor:
Ind๐โ : n๐โmod๐๐ผ ร ฮฅnโmodfact๐ข๐,๐๐ผ
is ๐ก-exact with respect to the ๐ก-structure on the left hand side coming from Proposition 7.7.1.(3) This ๐ก-structure is left and right complete.
Proof. Note that we have a commutative diagram:
ฮฅnโmodfact๐ข๐,๐๐ผ//
Oblvฯn
n๐โmod๐๐ผ
๐ทpDivฮ๐๐๐ ,8ยจ๐ฅ
eff,๐๐ผ q๐! // ๐ทpGr๐,๐๐ผ q
(7.11.2)
where we use ๐ to denote the map Gr๐,๐๐ผ ร Divฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ .
Define pฮฅnโmodfact๐ข๐,๐๐ผ qฤ0 as the subcategory generated under colimits by n๐โmodฤ0
๐๐ผ by Ind๐โ.
This defines a ๐ก-structure in the usual way. Note that an object lies in pฮฅnโmodfact๐ข๐,๐๐ผ qฤ 0 if and only
if its image under Oblv๐โ lies in n๐โmodฤ 0๐๐ผ .
The main observation is that the composition Oblvฯn ห Ind๐โ is ๐ก-exact:
The PBW theorem for factorization modules [Ras1] S7.19 says that for๐ P n๐โmod๐๐ผ , Ind๐โp๐q
has a filtration as an object of ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ q with subquotients given by the ห-pushforward of:
n๐r1sb . . .b n๐r1slooooooooooomooooooooooon
๐ times
b๐ P ๐ท`
pDivฮ๐๐๐
eff q๐ หGr๐,๐๐ผ
ห
along the addition map to Divฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ . Formation of this exterior product is obviously ๐ก-exact, and
the ห-pushforward operation is as well by finiteness, giving our claim.Then from the commutative diagram (7.11.2), we see that Oblv๐โ ห Ind๐โ is left ๐ก-exact. This
immediately implies the ๐ก-exactness of Ind๐โ.It remains to show that Oblvฯn is ๐ก-exact. By the above computation of Oblvฯn Ind๐โ, it is right
๐ก-exact.Suppose ๐ P ฮฅnโmodfact๐ข๐,๐๐ผ with ๐! Oblvฯnp๐q P ๐ทpGr๐,๐๐ผ qฤ 0. By factorization and since ฮฅn P
๐ทpDivฮ๐๐๐
eff q, we deduce that Oblvฯnp๐q is in degree ฤ 0. By the commutative diagram (7.11.2),
this hypothesis is equivalent to assuming that ๐ P pฮฅnโmodfact๐ข๐,๐๐ผ qฤ 0, so we deduce our left ๐ก-
exactness.Finally, that this ๐ก-structure is left and right complete follows immediately from (1).
Corollary 7.11.2. The functor Chevspecn,๐๐ผ : Reppq๐๐ผ ร ฮฅnโmodfact๐ข๐,๐๐ผ is ๐ก-exact.
7.12. ULA objects. Next, we discuss the behavior of ULA objects under the Chevalley functors.In the discussion that follows, we use the term โULAโ as an abbreviation for โULA over ๐๐ผ .โ
7.13. We begin with a technical remark on the spectral side.
Proposition 7.13.1. (1) The functor Chevspecn,๐๐ผ maps ULA objects in Reppq๐๐ผ to ULA objects
in ฮฅnโmodfact๐ข๐,๐๐ผ .
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58 SAM RASKIN
(2) For every ๐ P Reppqb๐ผ , the object Oblvฯn Chevspecn,๐๐ผ p๐ q P ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅ
eff,๐๐ผ q underlying
Chevspecn,๐๐ผ p๐ q is ind-ULA.
More precisely, if ๐ is compact, then for every P ฮ, the restriction of this ๐ท-module tothe locus of divisors of total degree is ULA.41
Proof. The functor Reppq๐๐ผ ร n๐โmod๐๐ผ preserves ULA objects by the same argument as inProposition 6.16.1, and then the first part follows from ๐ทp๐๐ผq-linearity of the adjoint functors
n๐โmod๐๐ผ
Ind๐โ// ฮฅnโmodfact๐ข๐,๐๐ผoo .
For the second part, we claim more generally that Oblvฯn Ind๐โ maps ULA objects in Reppq๐๐ผ
to objects in ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ q whose restriction to each degree is ULA.
To this end, we immediately reduce to the case of one-dimensional representations of ๐ผ , sinceevery compact object of Reppqb๐ผ admits a finite filtration with such objects as the subquotients.
In the case of the trivial representation of , the corresponding object is the vacuum repre-sentation, which in this setting is obtained by ห-pushforward from ๐๐๐ผ b ฮฅn along the obviousmap:
๐๐ผ หDivฮ๐๐๐
eff ร Divฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ .
Since this map is a closed embedding, we obtain the claim since ๐๐๐ผ b ฮฅn obviously has thecorresponding property.
The general case of a 1-dimensional representation differs from this situation by a translation on
Divฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ , giving the claim here as well.
7.14. Next, we make the following observation on the geometric side.
Proposition 7.14.1. (1) For every ๐ P Reppqb๐ผ , Oblvฯn Chevgeomn,๐๐ผ pLoc๐๐ผ p๐ qq P ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅ
eff,๐๐ผ q
is ind-ULA.More precisely, for ๐ compact and P ฮ, the restriction of Oblvฯn Chevgeom
n,๐๐ผ pLoc๐๐ผ p๐ qq
to the locus of divisors of total degree is ULA.(2) For ๐ P Reppqb๐ผ,, Chevgeom
n,๐๐ผ pLoc๐๐ผ p๐ qq P ฮฅnโmodfact๐ข๐,๐๐ผ lies in cohomological degree
ยด|๐ผ|.
Proof. As in Proposition 7.13.1 suffices to show that for ๐ P Reppqb๐ผ, compact, then that
Chevgeomn,๐๐ผ pLoc๐๐ผ p๐ qq admits a filtration by ฮ๐ผ with -subquotient Ind๐โpโqb๐ pq, where ๐ pq is
the -weight space of ๐ and โ P Reppqb๐ผ, is the corresponding one dimensional representation.This follows exactly as in Step 2 of the proof of Theorem 5.14.1: the weight space of ๐ P Reppqb๐ผ
appears as a semi-infinite integral a la Mirkovic-Vilonen by the appropriate moving points versionof Lemma 5.15.1.
7.15. We now deduce the following key result, comparing Chevgeomn,๐๐ผ and Chevspec
n,๐๐ผ on ULA objects.
Proposition 7.15.1. The two functors:
41Note that this claim is wrong if we do not restrict to components, since ULA objects are compact.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 59
Chevgeomn,๐๐ผ หLoc๐๐ผ : Reppqb๐ผ ร ฮฅnโmodfact๐ข๐,๐๐ผ
Chevspecn,๐๐ผ หLoc๐๐ผ : Reppqb๐ผ ร ฮฅnโmodfact๐ข๐,๐๐ผ
are isomorphic.More precisely, there exists a unique such isomorphism extending the isomorphism between these
functors over ๐๐ผ๐๐๐ ๐ coming from Lemma 7.10.1 and factorization.
Proof. It suffices to produce an isomorphism between the restrictions of Chevgeomn,๐๐ผ and Chevspec
n,๐๐ผ to
the category of compact objects in the heart of Reppqb๐ผ,.Suppose ๐ P Reppqb๐ผ, is compact. By [Rei] IV.2.8,42 ULAness of Chevgeom
n,๐๐ผ pLoc๐๐ผ p๐ qq and
perversity (up to shift) imply that as a ๐ท-module, Chevgeomn,๐๐ผ pLoc๐๐ผ p๐ qq is concentrated in one
degree, and as such, it is middle extended from this disjoint locus. The same conclusion holds forChevspec
n,๐๐ผ pLoc๐๐ผ p๐ qq for the same reason.
Since the isomorphism above over Divฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ ห๐๐ผ๐๐ผ
๐๐๐ ๐ is compatible with factorization mod-
ule structures, we deduce that the factorization module structures on Chevgeomn,๐๐ผ pLoc๐๐ผ p๐ qq and
Chevspecn,๐๐ผ pLoc๐๐ผ p๐ qq are compatible with the middle extension construction, and we obtain that
these two are isomorphic as factorization modules for ฮฅn.
Corollary 7.15.2. The functor Chevgeomn,๐๐ผ is ๐ก-exact.
Proof. For simplicity, we localize to assume that ๐ is affine.First, we claim that Chevgeom
n,๐๐ผ is right ๐ก-exact.
Indeed, as in the proof of Proposition 6.34.1, Reppqฤ0๐๐ผ is generated under colimits by objects
of the form Ind OblvpLoc๐๐ผ p๐ qq โ ๐ท๐๐ผ
!b Loc๐๐ผ p๐ q for ๐ P Repp๐ผqฤ|๐ผ| โ Reppqb๐ผ,ฤ|๐ผ|.
The functor ๐ท๐๐ผ
!b ยด is ๐ก-exact on ๐ทpDivฮ๐๐๐ ,8ยจ๐ฅ
eff,๐๐ผ q (since after applying forgetful functors, it is
given by tensoring with the ind-vector bundle that is the pullback of differential operators on ๐๐ผ),and since:
Chevgeomn,๐๐ผ p๐ท๐๐ผ
!b Loc๐๐ผ p๐ qq โ ๐ท๐๐ผ
!bChevgeom
n,๐๐ผ pLoc๐๐ผ p๐ qqProp. 7.15.1
โ ๐ท๐๐ผ
!bChevspec
n,๐๐ผ pLoc๐๐ผ p๐ qq
we obtain the result from Corollary 7.11.2.For left ๐ก-exactness: let ๐ : ๐ผ ๐ฝ be given, and let ๐๐ denote the corresponding locally closed em-
bedding ๐๐ฝ๐๐๐ ๐ ร ๐๐ผ . Note that the functors ๐!๐ Chevgeom
n,๐๐ผ are left ๐ก-exact by factorization. Therefore,
since Chevgeomn,๐๐ผ is filtered by the functors ๐๐,ห,๐๐ ๐
!๐ Chevgeom
n,๐๐ผ , we obtain the claim.
Warning 7.15.3. It is not clear at this point that the isomorphisms of Proposition 7.15.1 arecompatible with restrictions to diagonals. Here we note that, as in the proof of loc. cit., this questionreduces to the abelian category, and here it becomes a concrete, yes-or-no question. The problemis that the isomorphism of Proposition 7.15.1 was based on middle extending from ๐๐ผ
๐๐๐ ๐ ฤ ๐๐ผ ,
42Note that loc. cit. only formulates its claim for complements to smooth Cartier divisors, since this reference onlydefines the ULA condition in this case. However, the claim from loc. cit. is still true in this generality, as one sees bycombining Beilinsonโs theory [Bei] and Corollary B.5.3.
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60 SAM RASKIN
and for ๐๐ฝ รฃร ๐๐ผ , ๐๐ผ๐๐๐ ๐ and ๐๐ฝ
๐๐๐ ๐ do not speak to one another. We will deal with this problem inS7.22.
7.16. Factoring through Reppq๐๐ผ . Next, we construct a functor:
1Chevgeom
n,๐๐ผ : Reppq๐๐ผ ร Reppq๐๐ผ
so that the composition:
Reppq๐๐ผ
1Chevgeom
n,๐๐ผ
รรรรรรร Reppq๐๐ผ ร n๐โmod๐๐ผInd๐โรรรร ฮฅnโmodfact๐ข๐,๐๐ผ
identifies with Chevgeomn,๐๐ผ .
Lemma 7.16.1. The ๐ก-exact functor:
Reppq๐๐ผ ร n๐โmod๐๐ผInd๐โรรรร ฮฅnโmodfact๐ข๐,๐๐ผ
is fully-faithful on the hearts of the ๐ก-structures.
Proof. The functor Reppq๐๐ผ ร n๐โmod๐๐ผ is obviously fully-faithful (even at the derived level),as is clear by writing both categories as limits and using the fully-faithfulness of the functorsReppqb๐ฝ ร nโmodpRepp๐ qb๐ฝ .
So it remains to show that Ind๐โ : n๐โmod๐๐ผ ร ฮฅnโmodfact๐ข๐,๐๐ผ is fully-faithful at the abelian
categorical level.This follows from the chiral PBW theorem, as in the proof of Proposition 7.11.1:Indeed, let Oblv๐โ denote the right adjoint to Ind๐โ. Then for ๐ P n๐โmod
๐๐ผ , Oblv๐โ Ind๐โp๐qis filtered as a ๐ท-module with associated graded terms:
๐! add๐,ห,๐๐
ยด
n๐r1sb . . .b n๐r1slooooooooooomooooooooooon
๐ times
b ๐ห,๐๐ p๐qยฏ
P ๐ทpGr๐,๐๐ผ q (7.16.1)
where add๐ is the addition map:
pDivฮ๐๐๐
eff q๐ หDivฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ ร Divฮ๐๐๐ ,8ยจ๐ฅ
eff,๐๐ผ
and ๐ is the embedding Gr๐,๐๐ผ รฃร Divฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ . It suffices to show that ๐ป0 of this term vanishes for
๐ โฐ 0.Observe that we have a fiber square:
Greff๐,๐๐ผ ห๐๐ผ. . . ห
๐๐ผGreff๐,๐๐ผ
loooooooooooooomoooooooooooooon
๐ times
ห๐๐ผ
Gr๐,๐๐ผ//
pDivฮ๐๐๐
eff q๐ หDivฮ๐๐๐ ,8ยจ๐ฅeff,๐๐ผ
add๐
Gr๐,๐๐ผ๐ // Divฮ๐๐๐ ,8ยจ๐ฅ
eff,๐๐ผ
where Greff๐,๐๐ผ is the locus of points in ๐๐ผ ห Diveff of pairs pp๐ฅ๐q๐P๐ผ , ๐ทq so that ๐ท is zero when
restricted to ๐zt๐ฅ๐u (so the reduced fiber of Greff๐,๐๐ผ over a point ๐ฅ P ๐ฮรร ๐๐ผ is the discrete
scheme ฮ๐๐๐ ).
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 61
Let ฮ ฤ ๐ ห๐๐ผ be the incidence divisor, as in the proof of Lemma 7.7.1. For given, we have a
canonical map ๐ฝ : ฮ ร Greff๐,๐๐ผ over ๐๐ผ , sending p๐ฅ, p๐ฅ๐q๐๐โ1q P ฮ to the divisor ยจ๐ฅ. More generally,
for every datum p๐q๐๐โ1 with ๐ P ฮ๐๐๐ , we obtain a map:
๐ฝp๐q๐๐โ1 : ฮ ห
๐๐ผ. . . ห
๐๐ผฮ ร Greff๐,๐๐ผ ห
๐๐ผ. . . ห
๐๐ผGreff๐,๐๐ผ .
By base-change, the !-restriction of n๐r1sb. . .bn๐r1sb๐ห,๐๐ p๐q to Greff๐,๐๐ผ ห๐๐ผ. . .ห
๐๐ผGreff๐,๐๐ผ ห
๐๐ผGr๐,๐๐ผ
is the direct sum of terms:
๐ฝp๐q
๐๐โ1
ห,๐๐
ยด
๐!1๐!pn1
!b ๐๐r1sq
!b . . .
!b ๐!๐๐
!pn๐!b ๐๐r1sq
!b ๐!๐`1p๐q
ยฏ
.
where the ๐๐ are the projections and ๐ is the map ฮ ร ๐, and where the sum runs over all ๐-tuplesp๐q
๐๐โ1 of positive coroots. Since ๐๐r1s โ ๐๐rยด1s, these terms are concentrated in cohomological
degree ฤ ๐, which gives the claim.
Proposition 7.16.2. The functor Chevgeomn,๐๐ผ |Reppq
๐๐ผfactors through Reppq
๐๐ผ ฤ ฮฅnโmodfact๐ข๐,๐๐ผ .
Proof. Since Reppqฤ0๐๐ผ is generated under colimits by objects of the form Ind OblvpLoc๐๐ผ p๐ qq โ
๐ท๐๐ผ
!bLoc๐๐ผ p๐ q for ๐ P Repp๐ผqฤ|๐ผ| โ Reppqb๐ผ,ฤ|๐ผ|, Reppq
๐๐ผ is generated under (for emphasis:possibly non-filtered) colimits by the top cohomologies of such objects, i.e., by objects of the form
๐ท๐๐ผ
!b Loc๐๐ผ p๐ q for ๐ P Reppqb๐ผ concentrated in degree |๐ผ|.
But we have seen that such objects map into Reppq๐๐ผ , giving the claim.
We now obtain the desired functor1Chevgeom
n,๐๐ผ from Corollary 6.24.4, i.e., from the fact that
Reppq๐๐ผ is the derived category of its heart. These functors factorize as one varies ๐ผ.
7.17. Kernels. By Lemma 6.32.1, the functor1Chevgeom
n,๐๐ผ is defined by a kernel:
Kgeom๐๐ผ P Reppห q๐๐ผ .
Recall that the object of Reppq๐๐ผ underlying Kgeom๐๐ผ is
1Chevgeom
n,๐๐ผ pO,๐๐ผ q. Moreover, we recall that
one recovers the functor1Chevgeom
n,๐๐ผ by noting that for F P Reppq๐๐ผ , F!bK๐๐ผ P Reppห ห q๐๐ผ ,
and then we take invariants with respect to ๐ฝ on each ๐p๐q (๐ : ๐ผ ๐ฝ), where ๐ฝ acts diagonallythrough the embedding รฃร ห ).
Let Kspec๐๐ผ P Reppห q๐๐ผ denote the kernel defining the tautological functor Reppq ร Reppq,
i.e., for each ๐ : ๐ผ ๐ฝ , Kspec๐๐ผ |๐p๐q is given by the regular representation O๐ฝ considered as a
p๐ฝ , ๐ฝq-bimodule by restriction from its p๐ฝ , ๐ฝq-bimodule structure (i.e., forgetting Kspec๐๐ผ down
to Reppq๐๐ผ , we recover O,๐๐ผ from S6).
7.18. We have the following preliminary observations about these kernels.
Lemma 7.18.1. Kgeom๐๐ผ and K
spec๐๐ผ are concentrated in cohomological degree ยด|๐ผ| in Reppห q๐2.
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62 SAM RASKIN
Proof. For Kspec๐๐ผ , this follows from Lemma 6.24.3.
By construction, we recover Kgeom๐๐ผ as an object of Reppq๐๐ผ by evaluating
1Chevgeom
n,๐๐ผ on O,๐๐ผ .
Since this object is concentrated in degree ยด|๐ผ| by Lemma 6.24.3, we obtain the claim from ๐ก-exactness of Chevgeom
n,๐๐ผ .
Proposition 7.18.2. The group43 of automorphisms of Kspec๐๐ผ restricting to the identity automor-
phism on ๐๐ผ๐๐๐ ๐ is trivial.
Proof. Note that the underlying object of Gpd underlying this group is a set by Lemma 7.18.1.Then automorphisms of Kspec
๐๐ผ inject into automorphisms of O,๐๐ผ P Reppq๐๐ผ , so it suffices toverify the claim here.
By adjunction, we have: 44
HomReppq๐๐ผpO,๐๐ผ ,O,๐๐ผ q โ Hom๐ทp๐๐ผqpO,๐๐ผ , ๐๐๐ผ q.
Therefore, it suffices to show that:
Hom๐ทp๐๐ผqpO,๐๐ผ , ๐๐๐ผ q ร Hom๐ทp๐๐ผ๐๐๐ ๐q
p๐!pO,๐๐ผ q, ๐๐๐ผ๐๐๐ ๐q (7.18.1)
is an injection, where ๐ denotes the open embedding ๐๐ผ๐๐๐ ๐ รฃร ๐๐ผ .
Note that ๐!pO,๐๐ผ q ยป ๐!pLoc๐๐ผ pOqq is obviously ind-lisse, so ๐! is defined on it. Let ๐ denote
the closed embedding of the union of all diagonal divisors into ๐๐ผ , so ๐ is the complementary openembedding. We then have the long exact sequence:
0 ร Homp๐ห,๐๐ ๐ห,๐๐ pO,๐๐ผ q, ๐๐๐ผ q ร HompO,๐๐ผ , ๐๐๐ผ q ร Homp๐!๐
!pO,๐๐ผ q, ๐๐๐ผ q โ
Homp๐!pO,๐๐ผ q, ๐๐๐ผ๐๐๐ ๐q ร . . . .
We can compute the first term as:
Homp๐ห,๐๐ ๐ห,๐๐ pO,๐๐ผ q, ๐๐๐ผ q โ Homp๐ห,๐๐ pO,๐๐ผ q, ๐
!p๐๐๐ผ qq
which we then see vanishes, since ๐ห,๐๐ pO,๐๐ผ q is obviously concentrated in cohomological degrees
ฤ ยด|๐ผ| (since O,๐๐ผ is in degree ยด|๐ผ|), while ๐!p๐๐๐ผ q is the dualizing sheaf of a variety of dimension
|๐ผ| ยด 1, and therefore is concentrated in cohomological degrees ฤ ยด|๐ผ| ` 1.
Remark 7.18.3. Note that by factorization and by the |๐ผ| โ 1 case, we have an isomorphismbetween K
geom๐๐ผ and K
spec๐๐ผ over the disjoint locus. We deduce from Proposition 7.18.2 that there is
at most one isomorphism extending this given isomorphism, or equivalently, there is at most oneisomorphism between
1Chevgeom
n,๐๐ผ and the functor of restriction of representations that extends the
known isomorphism over the disjoint locus.
43Here by group, we mean a group object of Gpd.44We emphasize here that Hom means the the groupoid of maps, not the whole chain complex of maps. In particular,these Homs are actually sets, not more general groupoids.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 63
7.19. Commutative structure. The following discussion will play an important role in the sequel.By factorization:
๐ผ รร Kgeom๐๐ผ :โ
1Chevgeom
n,๐๐ผ pO,๐๐ผ q P Reppห q๐๐ผ
is a factorization algebra in a commutative factorization category.
Lemma 7.19.1. ๐ผ รร Kgeom๐๐ผ is a commutative factorization algebra.
Remark 7.19.2. Since each term Kgeom๐๐ผ is concentrated in cohomological degree ยด|๐ผ|, this factor-
ization algebra is classical, i.e., of the kind considered in [BD]. In particular, its commutativity isa property, not a structure.
Proof of Lemma 7.19.1. Let ฮ denote the functor:
ฮ : Reppห q๐ b Reppห q๐ ร Reppห q๐2 .
By [BD] S3.4, we only need to show that there is a map:
ฮpKgeom๐ bK
geom๐ q ร K
geom๐2 P Reppห Reppq๐2 (7.19.1)
extending the factorization isomorphism on ๐2z๐.Let ๐ denote diagonal embedding ๐ รฃร ๐2 and let ๐ denote the complementary open embedding
๐2๐๐๐ ๐ รฃร ๐2.
Since ๐!pKgeom๐2 q โ K
geom๐ is in cohomological degree ยด1, we have a short exact sequence:
0 ร Kgeom๐2 ร ๐ห,๐๐ ๐
!pKgeom๐2 q ร ๐ห,๐๐ pK
geom๐ qr1s ร 0
in the shifted heart of the ๐ก-structure.Therefore, the obstruction to a map (7.19.1) is the existence of a non-zero map:
Kgeom๐ bK
geom๐ ร ๐ห,๐๐ pK
geom๐ qr1s.
We know (from the ๐ผ โ ห case of S7.10) that Kgeom๐ โ Loc๐pOq, so K
geom๐ b K๐ is similarly
localized. It follows that ๐ห,๐๐ pKgeom๐ b K
geom๐ q is concentrated in cohomological degree ยด3, while
Kgeom๐ r1s is concentrated in cohomological degeree ยด2, giving the claim.
7.20. Lemma 7.19.1 endows Kgeom๐ with the structure of commutative algebra object of Reppห
q๐ . Moreover, since Kgeom๐ is isomorphic to O,๐ , this object lies in the full subcategory:
Reppห q๐ ฤ Reppห q๐ .
Moreover, the Beilinson-Drinfeld theory [BD] S3.4 then implies that Kgeom๐๐ผ can be recovered from
Kgeom๐ equipped with its commutative algebra structure. For example, this observation already buys
us that for every ๐ผ, K๐๐ผ P Reppห q๐๐ผ ฤ Reppห q๐๐ผ , and that ๐ผ รร K๐๐ผ has a factorizationcommutative algebra structure.
Using Lemma 6.32.1, it follows that the factorization functor1Chevgeom
n is induced from a sym-
metric monoidal functor equivalence ๐น : Reppqยปรร Reppq by composing ๐น with the restriction
functor to Reppq and applying the functoriality of the construction C รร p๐ผ รร C๐๐ผ q from S6.
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64 SAM RASKIN
7.21. We claim that ๐น is equivalent as a symmetric monoidal functor to the identity functor.Indeed, this follows from the next lemma.
Lemma 7.21.1. Let r๐น : Reppq ร Reppq be a symmetric monoidal equivalence such that for
every P ฮ`, r๐น p๐ q is equivalent to ๐ in Reppq. Then r๐น is equivalent (non-canonically) to theidentity functor as a symmetric monoidal functor.
Proof. By the Tannakian formalism, r๐น is given by restriction along an isomorphism ๐ : ยปรร .
We need to show that ๐ is an inner automorphism. We now obtain the result, since the outerautomorphism group of a reductive group is the automorphism group of its based root datumand since our assumption implies that the corresponding isomorphism is the identity on ฮ` andtherefore on all of ฮ.
7.22. Trivializing the central gerbe. The above shows that there exists an isomorphism of thefactorization functors Chevgeom
n and Chevspecn .
However, the above technique is not strong enough yet to produce a particular isomorphism.Indeed, the isomorphism of Lemma 7.21.1 is non-canonical: the problem is that the identity functorof Reppq admits generally admits automorphisms as a symmetric monoidal functor: this automor-phism group is the canonical the set of ๐-points of the center ๐pq.
Unwinding the above constructions, we see that the data of a factorizable isomorphism Chevgeomn
and Chevspecn form a trivial ๐pq-gerbe.
In order to trivialize this gerbe, it suffices (by Proposition 7.18.2, c.f. Remark 7.18.3) to showthe following.
Proposition 7.22.1. There exists a (necessarily unique) isomorphism of factorization functorsChevgeom
n ยป Chevspecn whose restriction to ๐ is the one given by Lemma 7.10.1.
Remark 7.22.2. Even when ๐pq โ ห, this assertion is not obvious: c.f. Warning 7.15.3. Essentially,the difficulty is that the identity functor of Reppq admits many automorphisms that are not tensorautomorphisms.
7.23. We will deduce the above proposition using the following setup.
Lemma 7.23.1. Suppose that we are given a symmetric monoidal functor ๐น : Reppq ร Reppqsuch that ๐น is (abstractly) isomorphic to the identity as a tensor functor, and such that we aregiven a fixed isomorphism:
๐ผ : Res๐ห๐น ยป Res
๐
of symmetric monoidal functors Reppq ร Repp๐ q (Res indicates the restriction functor here).Then there exists an isomorphism of symmetric monoidal functors between ๐น and the identity
functor on Reppq inducing ๐ผ if and only if, for every ๐ P Reppq irreducible, there exists an
isomorphism ๐ฝ๐ : ๐น p๐ qยปรร ๐ P Reppq inducing the map:
๐ผp๐ q : Res๐๐น p๐ q ยป Res
๐p๐ q P Repp๐ q
upon application of Res๐.
Moreover, a symmetric monoidal isomorphism between ๐น and the identity compatible with ๐ผ isunique if it exists. At the level of objects, it is given by the maps ๐ฝ๐ .
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 65
Remark 7.23.2. In words: an isomorphism ๐ผ as above may not be compatible with any tensorisomorphism between ๐น and the identity. Indeed, consider the case where is adjoint, so that atensor isomorphism between ๐น and the identity is unique if it exists, while there are many choicesfor ๐ผ as above. However, if this isomorphism exists, it is unique. Moreover, there is an object-wisecriterion to test whether or not such an isomorphism exists.
Proof. Choose some isomorphism ๐ฝ between ๐น and the identity functor (of symmetric monoidal
functors). From ๐ผ, we obtain a symmetric monoidal automorphism of Res๐
. By Tannakian theory,
this is given by the action of some ๐ก P ๐ p๐q.Since the symmetric monoidal automorphism group of the identity functor of Reppq is the center
of this group, it suffices to show that ๐ก lies in the center of . (Moreover, we immediately deducethe uniqueness from this observation).
To this end, it suffices to show that ๐ก acts by a scalar on every irreducible representation on .But by Schurโs lemma, this is follows from our hypothesis.
7.24. We now indicate how to apply Lemma 7.23.1 in our setup.
7.25. First, we give factorizable identifications of the composite functors:
Reppq๐๐ผ
1Chevgeom
n,๐๐ผ
รรรรรรร Reppq๐๐ผ ร Repp๐ q๐๐ผ
with the functors induced from Res๐
.Indeed, we have done this implicitly already in the proof of Proposition 7.14.1: one rewrites the
functors Chevgeomn,๐๐ผ using (the appropriate generalization of) Lemma 5.15.1, and then uses the (fac-
torizable45 of the) Mirkovic-Vilonen identification of restriction as cohomology along semi-infiniteorbits.
7.26. Now suppose that ๐ P Reppq is irreducible.Then for ๐ฅ P ๐, Theorem 5.14.1 produces a certain isomorphism between Chevgeom
n,๐ฅ p๐ q and
Chevspecn,๐ฅ p๐ q in Reppq ฤ ฮฅnโmodfact,๐ข๐,๐ฅ .
To check that the conditions of Lemma 7.23.1 are satisfied, it suffices to show that this isomor-phism induces the isomorphism of of S7.25 when we map to Repp๐ q.
Indeed, the isomorphism of Theorem 5.14.1 was constructed using a related isomorphism from[BG2] Theorem 8.8. The isomorphism of [BG2] has the property above, as is noted in loc. cit.Since the construction in Theorem 5.14.1 for reducing to the setting of [BG2] is compatible furtherrestriction to Repp๐ q, we obtain the claim.
Appendix A. Proof of Lemma 6.18.1
A.1. Suppose that we have a diagram ๐ รร C๐ P DGCat๐๐๐๐ก of categories with each C๐ dualizablewith dual C_๐ in the sense of [Gai3].
In this case, we can form the dual diagram ๐ รร C_๐ .We can ask: when is C :โ lim๐PI๐๐ C๐ dualizable with dual colim๐PI C
_๐ ? More precisely, there is a
canonical Vect valued pairing between the limit and colimit here, and we can ask when it realizesthe two categories as mutually dual.
As in [Gai3], we recall that this occurs if and only if colim๐PI C_๐ is dualizable, which occurs if
and only if, for every D P DGCat๐๐๐๐ก, the canonical map:
45This generalization is straightforward given the Mirkovic-Vilonen theory and the methods of this section and S6.
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66 SAM RASKIN
`
lim๐PI๐๐
C๐ห
bDร lim๐PI๐๐
`
C๐ bDห
is an equivalence.This section gives a criterion, Lemma A.2.1, in which this occurs, and which we will use to deduce
Lemma 6.18.1 in SA.3
A.2. A dualizability condition. Suppose we have a diagram:
C2
๐
C1๐น // C3
of dualizable categories. Let C denote the fiber product of this diagram.The main result of this section is the following.
Lemma A.2.1. Suppose that ๐ and ๐น have right adjoints ๐ and ๐บ respectively. Suppose in additionthat ๐บ is fully-faithful.
Then if each C๐ is dualizable, C is dualizable as well. Moreover, for each D P DGCat๐๐๐๐ก, thecanonical map:
CbDร C1 bD หC3bD
C2 bD (A.2.1)
is an equivalence.
The proof of this lemma is given in SA.7.
A.3. Proof of Lemma 6.18.1. We now explain how to deduce Lemma 6.18.1.
Proof that Lemma A.2.1 implies Lemma 6.18.1. Fix ๐ผ a finite set. We proceed by induction on |๐ผ|,the case |๐ผ| โ 1 being obvious.
Recall that we have C P DGCat๐๐๐๐ก rigid and symmetric monoidal, and ๐ a smooth curve.By 1-affineness of ๐๐ผ
๐๐ and ๐๐ผ (c.f. [Gai4]), we easily reduce to checking the corresponding
fact in the quasi-coherent setting. Note that by rigidity of QCohp๐๐ผq, dualizability questions inQCohp๐๐ผqโmod are equivalent to dualizability questions in DGCat๐๐๐๐ก.
Let ๐ ฤ ๐๐ผ be the complement of the diagonally embedded ๐ รฃร ๐๐ผ . We can then express C๐๐ผ
as a fiber product:
QCohp๐๐ผ ,C๐๐ผ q //
QCohp๐๐ผ ,C๐๐ผ q bQCohp๐๐ผq
QCohp๐q
QCohp๐๐ผq b C // QCohp๐q b C.
The two structure functors involved in defining this pullback admit continuous right adjoints,and the right adjoint to the bottom functor is fully-faithful. Moreover, the bottom two terms areobviously dualizable. Therefore, by Lemma A.2.1, it suffices to see that formation of the limitinvolved in defining the top right term commutes with tensor products over QCohp๐q.
Note that ๐ is covered by the open subsets ๐p๐q for ๐ : ๐ผ ๐ฝ with |๐ฝ | ฤ 1. By Zariski descentfor sheaves of categories, it suffices to check the commutation of tensor products and limits after
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 67
restriction to each ๐p๐q. But this follows from factorization and induction, using the same cofinalityresult as in S6.10.
A.4. The remainder of this section is devoted to the proof of Lemma A.2.1.
A.5. Gluing. Define the glued category Glue to consist of the triples pF,G, ๐q where F P C1, G P C2,and ๐ is a morphism ๐ : ๐pGq ร ๐น pFq P C3.
Note that the limit C :โ C1 หC3 C2 is a full subcategory of Glue.
Lemma A.5.1. The functor C รฃร Glue admits a continuous right adjoint.
Proof. We construct this right adjoint explicitly:
For pF,G, ๐q as above, define rF P C1 as the fiber product:
rF //
๐บ๐pGq
๐บp๐q
F // ๐บ๐น pFq.
Since ๐บ is fully-faithful, the map ๐ : ๐น prFq ร ๐น๐บ๐pGq ยป ๐pGq is an isomorphism, and therefore
prF,G, ๐q defines an object of C. It is easy to see that the resulting functor is the desired right adjoint.
A.6. Let D P DGCat๐๐๐๐ก be given.Define GlueD as with Glue, but instead use the diagram:
C2 bD
๐bidD
C1 bD๐นbidD// C3 bD
Lemma A.6.1. The canonical functor:
GluebDร GlueD
is an equivalence.
Proof. First, we give a description of functors Glueร E P DGCat๐๐๐๐ก for a test object E:We claim that such a functor is equivalent to the datum of a pair ๐0 : C1 ร E and ๐1 : C2 ร E
of continuous functors, plus a natural transformation:
๐1๐๐น ร ๐0
of functors C1 ร E.Indeed, given a functor ฮ : Glue ร E as above, we obtain such a datum as follows: for F P C1,
we let ๐0pFq :โ ฮpFrยด1s, 0, 0q, for G P C2 we let ๐1pGq :โ ฮp0,G, 0q (here we write objects of Glueas triples as above). The natural transformation comes from the boundary morphism for the exacttriangle Glue:
pF, 0, 0q ร pF, ๐๐น pFq, ๐Fq ร p0, ๐๐น pFq, 0q`1รรร
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68 SAM RASKIN
where ๐F is the adjunction map ๐๐๐น pFq ร ๐น pFq. It is straightforward to see that this constructionis an equivalence.
This universal property then makes the above property clear.
A.7. We now deduce the lemma.
Proof of Lemma A.2.1. We need to see that for every D P DGCat๐๐๐๐ก, the map (A.2.1) is an equiv-alence.
First, observe that each of these categories is a full subcategory of GlueD. Indeed, for the lefthand side of (A.2.1), this follows from Lemma A.5.1, and for the right hand side, this follows fromLemma A.6.1. Moreover, this is compatible with the above functor by construction.
Let ๐ฟ denote the right adjoint to ๐ : C รฃร Glue, and let ๐ฟD denote the right adjoint to theembedding:
๐D : C1 bD หC3bD
C2 bD รฃร GlueD.
We need to show that:
p๐ ห ๐ฟq b idD โ ๐D ห ๐ฟD
as endofunctors of GlueD, since the image of the left hand side is the left hand side of (A.2.1), andthe image of the right hand side is the right hand side of (A.2.1).
But writing GlueD as GluebD, this becomes clear.
Appendix B. Universal local acyclicity
B.1. Notation. Let ๐ be a scheme of finite type and let C be a ๐ทp๐q-module category in DGCat๐๐๐๐ก.Let QCohp๐,Cq denote the category Cb๐ทp๐q QCohp๐q.
Remark B.1.1. Everything in this section works with ๐ a general DG scheme almost of finite type.The reader comfortable with derived algbraic geometry may therefore happily understand โschemeโin the derived sense everywhere here.
B.2. The adjoint functors:46
QCohp๐qInd // ๐ทp๐qOblv
oo
induce adjoint functors:
QCohp๐,CqInd // C.Oblv
oo
Lemma B.2.1. The functor Oblv : Cร QCohp๐,Cq is conservative.
Proof. This is shown in [GR] in the case C โ ๐ทp๐q.In the general case, it suffices to show that Ind : QCohp๐,Cq ร C generates the target under
colimits. It suffices to show that the functor:
46Throughout this section, we use only the โleftโ forgetful and induction functors from [GR].
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 69
QCohp๐q b Cร ๐ทp๐q b Cร ๐ทp๐q b๐ทp๐q
C
generates, as it factors through Ind. But the first term generates by the [GR] result, and the secondterm obviously generates.
B.3. Universal local acyclicity. We have the following notion.
Definition B.3.1. F P C is universally locally acyclic (ULA) over ๐ if OblvpFq P QCohp๐,Cq iscompact.
Notation B.3.2. We let C๐๐ฟ๐ด ฤ C denote the full (non-cocomplete) subcategory of ULA objects.
B.4. We have the following basic consequences of the definition.
Proposition B.4.1. For every F P C๐๐ฟ๐ด and for every compact G P ๐ทp๐q, G!bF is compact in C.
Proof. Since Ind : QCohp๐q ร ๐ทp๐q generates the target, objects of the form IndpPq P ๐ทp๐q forP P QCohp๐q perfect generate the compact objects in the target under finite colimits and directsummands.
Therefore, it suffices to see that IndpPq!b F is compact for every perfect P P QCohp๐q.
To this end, it suffices to show:
IndpPbOblvpFqqยปรร IndpPq
!b F (B.4.1)
since the left hand side is obviously compact by the ULA condition on F. We have an obvious mapfrom the left hand side to the right hand side. To show it is an isomorphism, we localize to assume๐ is affine, and then by continuity this allows us to check the claim when P โ O๐ . Then the claimfollows because Ind and Oblv are ๐ทp๐q-linear functors.
Corollary B.4.2. Any F P C๐๐ฟ๐ด is compact in C.
Example B.4.3. Suppose that ๐ is smooth and C โ ๐ทp๐q. Then F is ULA if and only if F is compactwith lisse cohomologies. Indeed, if F is ULA, the cohomologies of OblvpFq P QCohp๐q are coherentsheaves and therefore the cohomologies of F are lisse.
Proposition B.4.4. Suppose that ๐น : C ร D is a morphism in ๐ทp๐qโmod with a ๐ทp๐q-linearright adjoint ๐บ. Then ๐น maps ULA objects to ULA objects.
Proof. We have the commutative diagram:
C๐น //
Oblv
D
Oblv
QCohp๐,Cq // QCohp๐,Dq
and the functor QCohp๐,Cq ร QCohp๐,Dq preserves compacts by assumption on ๐น .
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70 SAM RASKIN
B.5. Reformulations. For F P C, let HomCpF,ยดq : Cร ๐ทp๐q denote the (possibly non-continuous)functor right adjoint to ๐ทp๐q ร C given by tensoring with F.
Proposition B.5.1. For F P C, the following conditions are equivalent.
(1) F is ULA.(2) HomCpF,ยดq : Cร ๐ทp๐q is continuous and ๐ทp๐q-linear.(3) For every M P ๐ทp๐qโmod and every ๐ PM compact, the induced object:
F b๐ทp๐q
๐ P C b๐ทp๐q
M
is compact.
Proof. First, we show (1) implies (2).Proposition B.4.1, the functor ๐ทp๐q ร C of tensoring with F sends compacts to compacts, so its
right adjoint is continuous. We need to show that HomCpF,ยดq is ๐ทp๐q-linear.Observe first that Oblv HomCpF,ยดq computes47 HomQCohp๐,CqpOblvpFq,Oblvpยดqq : Cร QCohp๐q.
Indeed, both are right adjoints to pยด!b Fq ห Ind โ Ind หpยด b OblvpFqq, where we have identified
these functors by (B.4.1).Then observe that:
HomQCohp๐,CqpOblvpFq,ยดq : QCohp๐,Cq ร QCohp๐q
is a morphism of QCohp๐q-module categories: this follows from rigidity of QCohp๐q. This now easilygives the claim since Oblv is conservative.
Next, we show that (2) implies (3).Let M and ๐ PM be as given. The composite functor:
Vectยดb๐รรรรM โ ๐ทp๐q b
๐ทp๐qM
pยด!bFqbidM
รรรรรรรร C b๐ทp๐q
M
obviously sends ๐ P Vect to F b๐ทp๐q
M. But this composite functor also obviously admits a continuous
right adjoint: the first functor does because ๐ is compact, and the second functor does because๐ทp๐q ร C admits a ๐ทp๐q-linear right adjoint by assumption.
It remains to show that (3) implies (1), but this is tautological: take M โ QCohp๐q.
Remark B.5.2. Note that conditions (2) and (3) make sense for any algebra A P DGCat๐๐๐๐ก replacing๐ทp๐q and any F P C a right A-module category in DGCat๐๐๐๐ก. That (2) implies (3) holds in thisgenerality follows by the same argument.
Here is a sample application of this perspective.
Corollary B.5.3. For G P ๐ทp๐q holonomic and F P C๐๐ฟ๐ด, ๐!pG!b ๐!pFqq P C is defined, and the
natural map:
๐!pG!b ๐!pFqq ร ๐!pGq
!b F
is an isomorphism. In particular, ๐!pFq is defined.
47The notation indicates internal Hom for QCohp๐,Cq considered as a QCohp๐q-module category.
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CHIRAL PRINCIPAL SERIES CATEGORIES I: FINITE DIMENSIONAL CALCULATIONS 71
Proof. We begin by showing that there is an isomorphism:
๐!pHomCpF,ยดqq ยป HomC๐p๐!pFq, ๐!pยดqq
as functors Cร ๐ทp๐q. Indeed, we have:
๐ห,๐๐ ๐!pHomCpF,ยดqq โ ๐ห,๐๐ p๐๐ q
!bHomCpF,ยดq โ HomCpF, ๐ห,๐๐ p๐๐ q
!b pยดqq
and the right hand side obviously identifies with ๐ห,๐๐ HomC๐p๐!pFq, ๐!pยดqq.
Now for any rF P C, we see:
HomCp๐!pGq!b F, rFq โ Hom๐ทp๐qp๐!pGq,HomCpF,
rFqq โ Hom๐ทp๐qpG, ๐!HomCpF,
rFqq โ
Hom๐ทp๐qpG,HomCp๐!pFq, ๐!prFqqq โ HomC๐
pG!b ๐!pFq, ๐!prFqq
as desired.
B.6. We now discuss a ULA condition for ๐ทp๐q-module categories themselves.
Definition B.6.1. C as above is ULA over ๐ if QCohp๐,Cq is compactly generated by objects of theform PbOblvpFq with F P C๐๐ฟ๐ด and P P QCohp๐q perfect.
Example B.6.2. ๐ทp๐q is ULA. Indeed, ๐๐ is ULA with Oblvp๐๐q โ O๐ .
Lemma B.6.3. If C is ULA, then C is compactly generated.
Proof. Immediate from conservativity of Oblv.
B.7. In this setting, we have the following converse to Proposition B.4.4.
Proposition B.7.1. For C ULA, a ๐ทp๐q-linear functor ๐น : C ร D admits a ๐ทp๐q-linear rightadjoint if and only if ๐น preserves ULA objects.
Proof. We have already seen one direction in Proposition B.4.4. For the converse, suppose ๐น pre-serves ULA objects.
Since C is compactly generated and ๐น preserves compact objects, ๐น admits a continuous rightadjoint ๐บ.
We will check linearity using Proposition B.5.1:Suppose that F P ๐ทp๐q. We want to show that the natural transformation:
F!b๐บpยดq ร ๐บpF
!bยดq
of functors Dร C is an equivalence.It is easy to see that it is enough to show that for any G P C๐๐ฟ๐ด, the natural transformation of
functors Dร ๐ทp๐q induced by applying HomCpG,ยดq is an equivalence.But this follows from the simple identity HomDp๐น pGq,ยดq โ HomCpG, ๐บpยดqq. Indeed, we see:
HomCpG,F!b๐บpยดqq โ F
!bHomCpG, ๐บpยดqq โ F
!bHomDp๐น pGq, pยดqq โ
HomDp๐น pGq,F!b pยดqq โ HomD
`
G, ๐บpF!b pยดqq
ห
as desired.
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72 SAM RASKIN
B.8. Suppose that ๐ : ๐ รฃร ๐ is closed with complement ๐ : ๐ รฃร ๐.
Proposition B.8.1. Suppose C is ULA as a ๐ทp๐q-module category. Then ๐น : Cร D a morphismin ๐ทp๐qโmod is an equivalence if and only if if preserves ULA objects and the functors:
๐น๐ : C๐ :โ C b๐ทp๐q
๐ทp๐q ร D๐ :โ D b๐ทp๐q
๐ทp๐q
๐น๐ : C๐ :โ C b๐ทp๐q
๐ทp๐ q ร D๐ :โ D b๐ทp๐q
๐ทp๐ q
are equivalences.
Remark B.8.2. Note that a result of this form is not true without ULA hypotheses: the restrictionfunctor ๐ทp๐q ร ๐ทp๐q โ ๐ทp๐ q is ๐ทp๐q-linear and an equivalence over ๐ and over ๐ , but not anequivalence.
Proof of Proposition B.8.1. By Proposition B.7.1, the functor ๐น admits a ๐ทp๐q-linear right adjoint๐บ. We need to check that the unit and counit of this adjunction are equivalences.
By the usual Cousin devissage, we reduce to checking that the unit and counit are equivalencesfor objects pushed forward from ๐ and ๐ . But by ๐ทp๐q-linearity of our functors, this follows fromour assumption.
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Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139.E-mail address: [email protected]