Mirror Symmetry and the K Theory of a p-adic Group by Dmitry A. Vaintrob A. B., Harvard University (2011) Submitted to the Department of Mathematics MASSACHUSETTS INSTITUTE OF TECHNOLOG JUN 162016 LIBRARIES ARCHIVES in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2016 @ Massachusetts Institute of Technology 2016. All rights reserved. Signature redacted- Author .... ----------.................. Department of Mathematics May 13, 2016 Certified bySignature redacted / Roman Bezrukavnikov Professor of Mathematics Thesis Supervisor Accepted by Signature redacted.................. William Minicozzi raduate Chair, Department of Mathematics 2 Mirror Symmetry and the K Theory of a p-adic Group by Dmitry A. Vaintrob Submitted to the Department of Mathematics on May 13, 2016, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Let G be a split, semisimple p-adic group. We construct a derived localization func- tor Loc : Dbmfg D Sh from the compactified category of [BK2 associated to G to the category of equivariant sheaves on the Bruhat-Tits building whose stalks have finite-multiplicity isotypic components as representations of the stabilizer. Our construction is motivated by the "coherent-constructible correspondence" functor in toric mirror symmetry and a construction of [CCC]. We show that Loc has a number of useful properties, including the fact that the sections LIP, Loc(V) = V when V is an object of Smfg compactifying the finitely-generated representation V.We also con- struct a depth-< e "truncated" analogue Loc(e) which has finite-dimensional stalks, and satisfies the property RIP, Loc(e) (V) = V for any V of depth < e. We deduce that every finitely-generated representation of G has a bounded resolution by represen- tations induced from finite-dimensional representations of compact open subgroups, and use this to write down a set of generators for the K-theory of G in terms of K-theory of its parahoric subgroups. Thesis Supervisor: Roman Bezrukavnikov Title: Professor of Mathematics 3 4 Acknowledgments David Kazhdan once described the process of advising as "throw the student in the pool and let them learn to swim." I would like to thank Roman Bezrukavnikov for introducing me to a very beautiful and deep pool and explaining how to swim in it through many hours of guidance and discussions. Thanks are due also to Eric Larson, Isabel Vogt, David Nadler, David Treumann, Jacob Lurie, Ju-Lee Kim, Arnav Tripa- thy, Akhil Mathew, Alexander Polishschuk, Vadim Vologodsky, and Pavel Etingof for many helpful conversations. Thank you to Ren Lu for help with the editing process and her support throughout. Finally, thanks to the wonderful friends and relatives who put up with me through the many ins and outs of the writing process. 5 6 Contents 1 Introduction 11 1.1 K theory of the representation category . . . . . . . . . . . . . . . . . 11 1.2 Compactified category . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Localization on the building . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1 Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Toric mirror symmetry and corridors . . . . . . . . . . . . . . . . . . 17 1.5 The Yoneda philosophy and the Morita philosophy . . . . . . . . . . 19 2 Plan of paper 21 3 Reminders about the representation category and the Bruhat-Tits building 25 4 Geometry in the compactified category 27 4.1 Polarization of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Definition of Sm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3 Root toric variety and the geometric center of Sm . . . . . . . . . . . 30 4.3.1 Local projectivity and extension . . . . . . . . . . . . . . . . . 32 4.3.2 Internal and external tensor product . . . . . . . . . . . . . . 33 4.4 Formal charts and higher Hom . . . . . . . . . . . . . . . . . . . . . . 33 4.5 Noncommutative pushforwards . . . . . . . . . . . . . . . . . . . . . 35 4.6 Dg functors out of Sm . . . . . . . . . . . . . . . . . . . . . . . . . . 36 7 5 Algebra on the building 37 5.1 Models for sheaves and cosheaves . . . . . . . . . . . . . . . . . . . . 37 5.2 Projective and injective objects . . . . . . . . . . . . . . . . . . . . . 39 5.3 Constant sheaves on orbifold subsets . . . . . . . . . . . . . . . . . . 40 6 Definition of the localization functor 43 6.1 C orridors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2 The localization kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7 The truncated localization functor 51 7.1 Building combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.2 Consistent systems of idempotents . . . . . . . . . . . . . . . . . . . . 53 8 List of Figures 1-1 Example of a corridor for SL 2 (Q2 ) . .. .. . ...... . . . . . . 6-1 D1 for SL2 . . . . . . . .. . . . ................... . 6-2 Hyperbolic corridor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 18 44 44 To Ren, mom, and dad Viens avec moi par dessus les buildings Qa fait WHIN! quand on s'envole et puis KLING! Apres quoi SHEBAM! je fais TILT! POW! et ga fait BLOP! Serge Gainsbourg, Comic Strip 10 BOING! WIZZ! Chapter 1 Introduction 1.1 K theory of the representation category Let G be a split, semisimple p-adic group, and let Smfg(G) be the category of idem- potented finitely-generated representations of the Hecke algebra 7i(G) with values in C (equivalently, smooth finitely-generated representations of G, see e.g. [Ber]). The category Smfg(G) is extremely well-behaved: it is a direct sum of countably many Noetherian components, has enough projectives, and has finite homological dimension equal to the rank of the group. In particular, the category has a well-behaved K- theory, with K0 (Smfg(G)) the Grothendieck group of projectives in Smfg(G). Write K (G) := K0 (Smfg(G)). The group K0 (G) will be a central object of study in this paper. The rational coef- ficient version K0 (Smfg(G)) 0 Q was considered in the papers IBDK and [D], and is related to the character theory of admissible representations of G. Indeed, given any admissible representation A and finitely-generated representation V, the graded spaces Ext(A, V) are finite-dimensional and zero for i > n + 1 (for n the rank of G). The signed sum (A, [V]) := Z(-1)' dim Ext'(A, V) then defines, for each admissible A, an integral functional on K0 (G). Rationally, the group KO(G) was computed by Dat [D], who showed that the group K0 (G) 0 Q naturally pairs with the vector space 11 of central distributions on compact elements. Namely, we say that an element y E G is compact if it is contained in some compact subgroup K C G. Write G, C G for the (open and closed) subset of compact elements. Let 7, C R be the vector space of (compactly supported locally constant) functions supported on GC. The group G acts on R, by conjugation, and we write HHS for the space of coinvariants (7 c)G- Theorem (Dat). There is an isomorphism i : K 0 (G) 0 C -+ HHg. In terms of this isomorphism, the pairing (A, [V]) = (XA, t([V])), where XA is the Harish-Chandra character of A (here XA is viewed as a distribution on G, which acts on -c and by conjugation-invariance descends to conjugation coinvariants (Rc)G)- One way to get projectives in Sm(G) is by (compact) induction from compact subgroups. Suppose J C G is a compact open subgroup of G. Given a smooth representation V of J, write Ind5'(V) := V &H WG for the induced representation. Note that our functor Ind G is left adjoint to the forgetful functor, and is sometimes denoted Ind, lI (V) to distinguish it from the right adjoint, which is a non-isomorphic functor (since [J: G] is infinite). A representation V of a compact group J is finitely-generated and projective if and only if it is finite-dimensional (recall that smooth finite-dimensional representations of a compact group form a semisimple category). As both of these properties are obviously invariant with respect to induction, this gives us a collection of projectives Ind G(V) E Smfg(G) for pairs (J, V) with J C G compact and V finite-dimensional representations of J. Definition 1. We say that a representation is finitely induced if it is of the form Ind j(V) for some finite-dimensional representation V of an open compact J C G. Note that it is sufficient to consider maximal compact subgroups J, as if J is compact open and M c J is a maximal compact subgroup containing J, then Ind (V) IndG(Indm V), and Indm V is a finite-dimensional representation of M. Given a pair (J, V) as above, the class of the corresponding finitely induced represen- tation in Dat's K0 group is the normalized character 6 j -Xv, where Xv is the function (supported in J) such that Tr(h, V) = (h, Xv) for h E R supported in J and 6 1 is the 12 uniform distribution on J of norm 1. Note that 6 1 -Xv is supported on J C Ge, hence projects to the space of G-coinvariants HH. Because the characters Xv form a basis for the vector space of central functions, one sees that these projections of charac- ters span all of HHc over the complex numbers: in particular, the finitely induced representations rationally span the group K0 (G). In this paper we will consider the integral K group K0 (G). Our main result will be the following. Theorem 1. The classes [Indj V] of finitely induced representations integrally span the group K0 (G). An equivalent formulation of this result is as follows. Theorem 1'. For any projective object P of Smfg(G), there are finitely induced rep- resentations Indi1, Indy,1' such that P D Ind ,V' = Indi V. Since Smfg has enough projectives, this is equivalent to the following formulation. Theorem 1". Any object of Smfg(G) has a resolution by direct sums of finitely in- duced representations. The statement in this last form is a conjecture in Roman Bezrukavnikov's thesis, [BThes]. The functor Ind G : Smfd(J) -+ Smfg(G) takes direct sums to direct sums, hence induces a linear map on K0 groups [Ind G] : K0 Smfd(J) -+ K0 Smf9 (G). The maps [Ind5G] and [Ind G -] are intertwined by the isomorphism V H- -yV : K0 (J) - K0 (yJ-y 1 ) - hence, in particular, they have the same image in K0 (G). Further, [IndG] factors through Smfd(M) for M some maximal compact subgroup containing J. If we choose an Iwahori subgroup I C G, the collection Max, of maximal compact subgroups containing I is a set of representatives of maximal compact subgroups up to conjugation. With this in mind, we write down the following map. [Indmax] := [IndG] : K(M) -+ K(G). MEMaxi ME Maxi 13 Theorem 1 then implies that the map [Indmx.] is surjective. There are some classes obviously in the kernel of this map: namely, given a subgroup J c Mi n Mj, the two inductions [Indmi (V)] and [Indm/ (V)] have the same image under [Indm ] (viewed as elements of the corresponding direct summands). Note that it is enough to take J = Mi n Mj above. Write Kell for the quotient of ®MecM., K0 (M) by relations of the form [IndM'fnM. V] [Indd2 M, V]. The map [Indma] induces a map [Indcen] : K0el -÷ K0 (G). It can be shown from the formula of [D] and basic properties of parahoric subgroups that this map is an isomorphism rationally. Theorem 1 implies that it is a surjection integrally. Hence the map [Indcen] : K011 - K0 (G) is an isomorphism on torsion-free quotients. 1.2 Compactified category Our proof will proceed by constructing a resolution for an arbitrary object, in a way that is functorial up to a certain choice of a "normalization" of V. This choice of normalization is provided by the compactified category Sm defined in [BK2] and its subcategory Smfg of locally finitely-generated objects. This category is a powerful tool which in particular allows one to systematically normalize computations with finitely- generated representations of G. Namely, given two objects V, W of Smfg(G), the space Hom(V, W) is in general not finite-dimensional, but has action by the Bernstein cen- ter Z := HH(Wi), and is a finitely-generated representation of Z. Equivalently, this Hom space can be considered a sheaf Hom(V, W) over Specz which is coherent and supported over finitely many irreducible components. Similarly the derived Hom space can be written as a finite complex of coherent sheaves RHomsm(V, W) over Spec(Z). Now components of Spec(Z) are canonically scheme-theoretic quotients of tori (of dimension between 0 and n) by subgroups of the Weyl group W. Choosing W-equivariant toric compactifications of these tori (something that can be done in a consistent way), we get a canonical compactification Spec(Z)BK of the central spec- trum. The idea of [BK2] is to endow the objects V, W of Sm(G) with some additional data, giving objects V, W in some upgraded category Smfg, in order to be able to write 14 an inner Hom space Hom(V, W) as a coherent sheaf over Spec(Z)BK. One can then reconstruct Hom(V, W) as F (Spec(Z)BK, Hom(V, W)) , and Ext*(V, W) in Db'Eig as the hypercohomology of the double complex RF (Spec(Z)BK, RHom(V, W)). The wonderful advantage of this category and its derived category is that these cate- gories are proper (see e.g. 10]), and two objects (under suitable finite generation conditions) have finitely many finite-dimensional Ext spaces. This allows us to de- fine Yoneda functors from DbSmfg9 to Db Vectfd given by taking R Hom with any (finitely-generated) object. We will show that any representation V has a resolution by induced representations by choosing a compactification V e Smfg (something that is relatively easy to construct), and write down a resolution of the forgetful functor j:Sm - Sm(G) by a finite collection of functors given by direct sum of functors of the form y: V -+ R Hom(Xi, V) 9 Indji indexed by J running over corank-i parahoric subgroups of o containing some fixed Iwahori subgroup. 1.3 Localization on the building The resolution (ji, d) (as well as a version of this functor depending on depth) will be the focus of this paper, and is interesting independently of its application to K theory. Our construction will be topological in nature, and is motivated by a philosophy of p-adic localization introduced in the paper [BThes]. Namely, recall that (for G split and semisimple) the Bruhat-Tits building BG is a G-equivariant contractible cell complex with vertices parametrized by maximal compact subgroups and k-dimensional cells stabilized by parahoric subgroups of corank k. The space BG can be thought of as a p-adic analogue to the equivariant space G/K, either for G a real group and K a compact subgroup, or for G a complex group and K the 15 Borel. The combinatorially constructed topology on B then takes the place of the smooth or complex structure on the equivariant spaces. In particular, the appropriate analogue to the category of local systems on an equivariant space is the category of constructible sheaves on the building with finite-dimensional fibers, constructible with respect to the cellular stratification. This category (perhaps after imposing a suitable boundary condition) can be thought of as having action by something like the Lie algebra of the totally disconnected group G. To have action by all of G, we consider the category Sh G of G-equivariant constructible sheaves on the building with finite-dimensional stalks. One then is interested in the (compactly supported) global sections functor F: Sh G -+ Smfg(G) which is analogous to the inverse Beilinson-Bernstein localization functor arising in the theory of category- representations of semisimple Lie groups. Unlike the (inverse) localization functor in geometry, the functor r, is far from being an equivalence, and is not a faithful functor; nevertheless, Bezrukavnikov shows in [BThes that it becomes faithful after factoring out a certain Serre subcategory of ShGd of objects with trivial homology. In fact, the essential image of Bezrukavnikov's functor is precisely the full subcategory of representations in Smfg(G) consisting of representations which admit a resolution by direct sums of finitely induced representations. To motivate this, observe the functor F, comes endowed with a resolution (coming from the cellular structure) by functors IF : ShY -4 Smfg(G) with J running over the paraholics and FJ a Ind Stalk, canonically expressed as the induced representation from the stalk functor at a cell - stabilized by J. Thus in order to show that any finitely-generated representation has a resolution by finitely induced ones, it would be sufficient to construct a right inverse Locsm of the functor F : ShOd - Smfg(G): then the cell complex computing F(Locsm(V)) a V would give functorial such resolutions. Unfortunately, it is relatively easy to see that such a right inverse does not exist. Instead, what we do construct is (essentially) a de- rived, compactified version of the localization functor: a complex of sheaves Locg(V), which we will call Locv' : Dbgmfg -+ Sh Ggr fdl 16 with the property that the following diagram of functors commutes: Dm Loc DbShG Db Smfg(G). This commutative diagram, along with the existence for any object V E Sm of a (non-unique) compactified object V with jV 2 V, furnishes us with a resolution of every object by finitely induced representations. 1.3.1 Truncation The more canonical functor, and the one we will spend the most time studying, is a functor Loc : D'Smg, -+ Db CoShG into the category of cosheaves, not necessarily with finite-dimensional stalks. In order to get a functor Locv into sheaves we can use a standard Verdier-type equivalence between derived categories of sheaves and cosheaves (see [Cul). In order to further project to the category of sheaves with finite- dimensional stalks, we use a procedure of "truncation" and take stalkwise invariants with respect to a coefficient system of congruence subgroups of conductor depending on the depth of V. It is in fact somewhat surprising that "truncation" does not destroy commutativity of the global sections diagram above, and our proof of this (in section 7) uses extensively ideas of Meyer and Solleveld, [MS]. 1.4 Toric mirror symmetry and corridors The idea behind our construction of the localization functor comes from adapting to the context of buildings and noncommutative geometry a certain functor arising from mirror symmetry of toric varieties: namely, the coherent-constructible correspondence of [FLTZ] (especially in the interpretation of [T] and [CCC]). The basic idea underly- ing both the point of view of [CCC] and our construction of the localization functor is 17 one of descent: we express (countable) colimit-compatible dg functors DbSm -+ DbC (for arbitrary categories C) as collections of functors from noncommutative affine charts, with certain algebra actions and compatibilities between them. This converts the task of constructing the functor Loc : DbS m_ -+ DbShG to that of finding sev- eral compatible objects of ShG with appropriate algebra actions. These objects are constructed using (Verdier duals to) constant sheaves on a new class of contractible geometric subsets of the building which we call corridors (branched versions of dual toric cones which are used in the coherent-constructible correspondence of [TJ). Note Figure 1-1: Example of a corridor for SL 2(Q 2 ) that both the Beilinson-Bernstein localization functor and the coherent-constructible correspondence functor are in general fully faithful: not so for the localization functor here. Instead, we have a functor Col : Db CoShG -+ DbSm right adjoint to the local- ization functor with the property that Col o Loc: Sm -+ Sm is close to but not quite the identity functor (as would be the case if Loc were fully faithful). The question of whether it is possible to "fix" Loc to be fully faithful (and thus give an embedding of the compactified category Loc into the category of equivariant sheaves on B) is an interesting one, and one that the author is agnostic about at the moment. 18 1.5 The Yoneda philosophy and the Morita philoso- phy Before continuing, we point out a subtle point about the point of view we adopt in defining functors, which is in a sense dual to the classical Bourbaki notion of Yoneda-representable functors. We will indicate this difference somewhat vaguely in this section, in order to motivate some of the definitional choices we make later in paper. Namely, given two module categories (either Abelian or differential graded), A-Mod and B-Mod there are a few common ways to "represent" dg functors be- tween them. One, which we can call the "Yoneda" philosophy, is to define a functor F : A-Mod -+ B-Mod by choosing some bimodule Y E A-B-bimod, and defining F (X) := Hom(Y, X). Another, which we call the "Morita" philosophy, is to choose a B-A-bimodule, M, and define FM(X) := M OA X. When one has not chosen a generator and starts with two categories C, D determined by some sort of algebraic data, it is still often possible to interpret the notion of a C-D-bimodule as an object of another algebraic category, informally "DP-type object in C" (formally, this bimod- ule category is determined by some universal property, and denoted C 0 Do when it exists). In defining the functors in this paper, we will identify the relevant bimodule categories, and almost exclusively use the "Morita" language of tensor product with a "kernel" bimodule M rather than the Yoneda constuction of Hom(Y, -). Note that our choice is aesthetic: as our categories are smooth, every "Morita-type" object M can (in the dg world) be replaced by a suitably dual "Yoneda-type" object Y := Mv. However, the relevant duality functors are complicated, and using the Yoneda method of defining functors would make our exposition more cumbersome than it should be. 19 20 Chapter 2 Plan of paper We begin by gathering together in section 3 some results about the category of rep- resentations Smg that at this point can be considered classical. In section 4 we study homological algebra on the compactified category Sm. We begin by recalling basic properties of the compactified category from [BK2], the most important ones being its geometric enrichment over the smooth compact variety X//W for X an n- dimensional toric variety over C compactifying the spectrum of the spherical center, Spec(Z,,h) t T//W. We move between three different points of view of Sm intro- duced in [BK2]. One point of view is to consider Sm as a collection of compatible representations of the topological algebras W-pQ, which can be thought of as noncom- mutative affine charts. A second is a microlocal modification of the first, where we only consider punctured completions RpQ of the Rp with respect to certain closed strata. The final one is a picture of Sm as sheaves of modules over a sheaf of al- gebras A over X//W (which we get after choosing an appropriate generator). The most important results of this section are Lemma 7, giving a formula for higher Hom and derived 0 between compactified representations and Lemma 8, which character- izes colimit-compatible (dg) functors Dbgm -÷ C in terms of the data of compatible collections of objects XpQ with action by the topological algebras 7 -pQ. We call such data {XpQ} "kernels" for functors. In the next section, 5, we recall some com- binatorial models for the category of equivariant cosheaves on the building and its derived category in Section 5, with the main sources being [BThes] and [Cul. We 21 also introduce a class of cosheaves we call constant cosheaves on orbifolds, which are orbifold pushforwards of constant sheaves on "6tale subsets" of the orbifold B/G. The remainder of the paper defines and studies various functors DbSm a Db CoShG using appropriate kernels {XpQ}. In section 6, we define the "absolute localization" functor Loc : D bSm -+ Db CoShG which we glue as a homotopy limit of the functors Loc-Q indexed by pairs of parabolics. The functors LocpQ are deduced from constant orbifold cosheaves on quotients of certain special contractible subsets of B which we call corridors. We check that Loc satisfies commutativity of the diagram -o- DbE Lm DbCoShG Db SMG, and show that the stalks of this functor are profinite-dimensional, i.e. the stalk over x becomes finite-dimensional upon taking invariants with respect to any open subgroup of the stabilizer G(x). Now replacing the functors LocpQ by invariants with respect to the "Schneider- Stuhler coefficient system" Ge) ~ G(x, e)O, Moy and Prasad [MP] associate a subgroup G(x, r) C G, normal in the stabilizer G(x). We say that (for some integer e), a representation V E SMG has depth < e if it is generated by the subspaces VG(x,e). It follows from work of Bernstein that the category of all finitely-generated representations of depth < e is Noetherian and a direct summand 25 in the category Smfg(G). When e is an integer, the groups G(x, e) can be taken to be the Schneider-Stuhler coefficient system G( of [SS], which is constant on cells of B. Given a parabolic subgroup P C G, it has a normal unitary radical Up C P C G, and the quotient P/Up is a Levi subgroup, which we will denote Lp. We have a pair of exact adjoint functors rp : SMG SmL iP called the Levi restriction and induction, such that rp(V) := VU with evident Lp- action. We say that a representation V is cuspidal if rp(V) = 0 for any parabolic P ; G, and admissible if it has finite length (in terms of a Jordan-Hblder composition series). Jacquet induction and restriction preserve both the properties of admissibility and of having depth < e. We define the Bernstein center Z := HH(SmG) to be the center of the cate- gory SMG . Given any representation V C SmG, it has a central support subvariety Supp(V) C Spec(Z). The category of representations with central support at a given point x E Spec(Z) is not necessarily semisimple, but is always Artinian, with at most |WI irreducibles (for |WI the size of the Weyl group). The depth of a representation depends only on its singular support, and the variety Spec(Z) is decomposed into a disjoint union by depth. The component Spec(Z 0 V Xi E Xprinc}. Write A+ = A+ n Ap. There is some ambiguity (depending on convention) on the relationship of polarization on the root lattice (i.e. choice of positive cone) to the polarization data B C G. We choose the convention that guarantees that for any rank-one parabolic Pi D B, the action of L+ on the p-adic affine line UB/Up, (viewed as a totally disconnected space with a Haar measure) is expanding. 28 4.2 Definition of Sm Here we will recall the definition and some properties of the compactified category Sm from [BK2]. First, a bit more notation. To a pair of embedded standard parabolics ' c Q we will associate an intermediate cone A+ c A+ C Ap as follows. Definition 3. Write AQ+ := {A E Ap I (A, xi) ;> 0 V xi E X4.,}. Evidently, A ~ A+. Definition 4. For P c Q parabolics in G, define L+, resp. LQ+ to be the preimage in Lp of the semigroups A+, resp. AQ+, in the unramified quotient LPIL. The category Sm will be "glued" out of smooth representation categories of the semigroups LQ+ above. Definition 5. Define R-pQ to be algebra of locally constant, compactly supported functions on the topological semigroup LQ+ Definition 6. Define SmpQ to be the category of smooth representations of the algebra 7i7PQ. One should think of the SmpQ as a system of "6tale" (or, more precisely, flat) opens of Sm, and the open corresponding to the pair (P, Q) can be considered to "contain" (P', Q') if P' C P C Q C Q'. Definition 7. Write K for the poset of pairs (P, Q) of standard parabolics in G satisfying P C Q, with order (P', Q') - (P, Q) when P c P c Q C Q'. Now for any pair (P', Q') - (P, Q) we have a functor j"Q' : SmpQ - SmPQ, defined as a composition of the following two functors. Definition 8. For any triple P' C P c Q, we define the functor j'P' : SmpQ - SmpQ taking VpQ to the coinvariants (Vp Q)u,, where we view VpQ as a representa- tion of PQ+, the subsemigroup of P which is the preimage of LQ+, restrict it to the preimage (P')Q+ of P' in LS,+, then quotient out by Up to obtain a representation of LQ,+. 29 Definition 9. For any triple P C Q C Q', we define the functor jpQ' : SmpQ SmpQ, taking a representation VpQ of 7 ipQ to its extension of scalars VPQ pO-(7QH2pQ,. Definition 10. Now for a quadruple P' C P C Q C Q' of parabolics, we write PQ' := jP'Q' 0 jP' : SmJQ -> SmpfQ1. Note that this functor is canonically equivalent to the composition in the opposite order, jP'' o jP . More generally, any chain of compositions of functors of this sort with the same range and domain will be canonically equivalent. This is encoded in the following lemma. Lemma 2 (Bezrukavnikov, Kazhdan). The categories Sm-pQ and the functors JpQ extend to a strict representation of the poset K in categories. L e. there are canon- ical isomorphisms of functors jpQ"j ' " jp"" : SmpQ -+ Smp/Q/, and these isomorphisms are compatible in an evident sense. We define Sm to be the limit of this diagram of functors parametrized by K in the category of categories. Namely, Definition 11 (Bezrukavnikov, Kazhdan). An object V of Sm is a collection of objects VpQ of SmpQ along with compatible isomorphisms jT"''VpQ VI Q,. A morphism f :V -+ V' is a collection of morphisms fpQ : VpQ -+ V Q such that jp'Q'f-pQ = fpi ,. 4.3 Root toric variety and the geometric center of Sm Let Tc be the algebraic torus over C with character lattice A (which is dual to the character lattice of the maximal torus T C G). The collection of dual hyperplanes in AR to the roots in A form a toric fan. Write X = kc for the corresponding toric variety over C, with open orbit Itc C kc. Then X is smooth W-equivariant (with action induced from W-action on the fan). The W-action preserves the toric 30 stratification, and induced a stratification on the scheme-theoretic quotient X//W. This stratification on k/W then has components parametrized by faces of the Weyl chamber (as it is a fundamental domain for the W-action on the fan), which are indexed by standard parabolics P C G. For P a parabolic, let Wp be the intersection W n P for W C G an embedding of the Weyl group that normalizes some T C B. Then Wp acts on the lattice Ap, as well as on the semigroup A'. Write X- for the spectrum Spec(A+), which is an affine toric subvariety X- C X, with closed toric stratum isomorphic to the torus tP := Spec(Ap). The Xp are then an affine cover of X, and the J?- are an affine stratification of X. Definition 12. Write XpQ for the quotient XQ//Wp and SpQ for the closed stratum tQ//WP C XPQ. Then the XpQ give a finite flat cover of X. (This cover has the flat analogue of the Nisnevich property, which we will see in the next section). Now it follows from work of Bernstein that the identity functor in the category Sm(G) has action by the ring of functions C[AIW r O(//W) on the scheme-theoretic quotient by this action. Equivalently, the Hom functor in Sm(G) is enriched to a functor Hom : SMG X SmG -+ Qcoh(LTC'//Wp) with composition o : Hom(V, W) 0 Hom(U, V) -+ Hom(U, W) fibered over the base LTP'//Wp, and with canonical isomorphism Hom(V, W) a r (LTP//W, Hom(V, W)). An extension of Bernstein's arguments implies that the category of smooth representations of the semigroup ?-pQ is enriched over the com- mutative ring O(XpQ). In the same sense, the paper [BK2] shows that Sm is fibered over the smooth projective scheme X. Namely, Lemma 3 ([BK2]). 1. The category Si is enriched over Xc//W, i.e. there is a functor Hom :Sm 0 Sm -+ Qcoh(Xc//W) with a composition natural transfor- mation o : Hom(V, W) 0 Hom(U, V) -+ Hom(U, W) fibered over the base. 2. Ordinary Hom in Sm is the composition of Hom with global sections, i.e. Hom;(VjW) F(Xc//W, Hom(V, W)). 3. Given an object V of Sm and.F of Qcoh(Xc //W), there is an object VO.F E Sm 31 with a natural adjunction equivalence Hom-(V 0 F, W) 2 HomQcoh(XC//W)(F, Hom(V, W)). 4. In an itale neighborhood of the boundary stratum Sp, the pullback of the inner Hom Hom(V, W) to any XpQ agrees with HomxPQ (VpQ, WpQ) (and in partic- ular, the fiber of Hom(V, W) over the open stratum T//W is Hom(V, W)). 4.3.1 Local projectivity and extension In order to rightfully call Smfg a "compactified" category, it would be nice to know that any object V E Smfg(G) can be extended to an object V E Smfg, at least in a dg sense. This is in fact true on the level of abelian categories, but in order to simplify our life a little, we prove it in a simpler setting of locally projective objects, which we show to dg span all of Dbm. Definition 13. We say that an object V E Sm is locally projective if every VpQ is projective as an object of SmpQ. Lemma 4. Every object in Sm has a finite locally projective resolution. Proof. Every SmpQ has projective resolution of length < n. Now given any object V such that each VpQ has projective resolution of length k > 0 and a map F -+ V from a locally projective object F which is surjective on every PQ-component, the kernel ker(P -4 V) locally has projective resolutions of length < k - 1. Thus by induction, it is enough to show that any object V E Sm admits a surjective map from a locally projective P. This is shown in 1BK2]. Now we prove the following lemma. Lemma 5. For any locally projective V E Smfg(G), there is a (not necessarily canon- ical) object V with the underlying representation VGG = V- Proof. We proceed by induction. Suppose we have constructed a collection of com- patible (in the sense of Definition 11) objects VpQ for all Po C P C Q, with VGG = V- 32 Then we can automatically extend it to a compatible collection of objects VpQ for all Q D PO, by taking VpQ := (VO Q)UI,. Now it suffices to extend this collection of com- patible representations by an object of type Vp,0 whose localizations produce Vp0 Q for Q 2 Po. Now note that as (by assumption) the Vp 0Q are finitely-generated and projective, hence torsion-free, we can choose collections of generators xiOQ E VpOQ. Now we define Vp0p0 to be subspace of the L'-span of x 00 E VPoG which are con- tained in all VpOQ C V-PoG. By Noetherianness of the finitely-generated representation categories, this module is finitely-generated. (In fact, the module is independent of choice of generators when the codimension of Po > 2 by the S2 property of the center). Corollary 6. These two lemmas imply that the K-theory map K0 (Smfg) -÷ KO(Smfg(G)) is surjective. 4.3.2 Internal and external tensor product Define the category Sm for the category of collections of collections of right rep- resentations VpQ of JpQ with opposite compatibility conditions. Then given a pair of objects V E Sm,V E SEmR, we can define the complex VOW E Q coh(X) and V 0 W := F(X, VOW). The functor 0 is left exact, and we can define its derived _.L__ ..b - L - --L functor VOW E Db coh(X). We can then define the functor V 0 W RE(VOW) on the derived category which is a dg functor in each component. When V, W are locally finitely generated, VOW e Db coh(X) is a perfect complex of coherent sheaves, hence V 0 W is a finite complex of finite vector spaces. 4.4 Formal charts and higher Hom Lemma 7. 1. For a pair of objects V, W E Sm, the derived Hom space is computed by the limit in the derived category RHom(V, W) a holim(p,Q)g (RHomw-pQ (VpQ, WpQ)). 33 Further, this quasiisomorphism is compatible with the fibered structure of Sm over XC//W. Namely, RHom(V, W) c limj,' (RHomQ(VpQ, WQ)) , where we take j : XpQ X the finite flat map of the previous section. 2. For a pair of objects V z Sm and W G SML, we have - L - W 0 V 2 holim WpQ O®,,H VpQ, and, similarly, the inner derived Hom _L _ L VOW a holimyr WpQ_0Vp Q, L where WpQOVpQ are viewed as derived pushforwards from coh(XpQ) to coh(X). In order to prove this lemma we will give an alternative glueing of the compacti- fied category Sm out of formal representation categories SmpQ fibered over punctured formal neighborhoods of closed strata in X. Write XpQ := S p n XpQ for the for- mal neighborhood of S-p in Xpp intersected with XpQ. This is an n-dimensional formal scheme which is a product of one-dimensional tori, disks, and formal disks. Now we observe that the rings 7-LpQ are canonically fibered along the XPQ, and we can base change to get rings 7 (pQ, with representation categories SmpQ. It is then straightforward to see that an object V of Sm is equivalent to a compatible system of representations VpQ of RpQ. Now the spaces XpQ form a formal cover of X closed un- der intersection, so that for two sheaves F, F' on X, we can compute RHom(F, F') via the "ech complex" RHom(7, F') a holimg (Hom (7| 2 , F'|2 )), and simi- L larly for F & F'. This implies from our fibered property that also given two objects V, W in Sm, we have RHom(V, W) a holimg (Home (VpQ, WPQ)) , and (for V an object of Sm ) we have V 0 W 2 holimr VPghHQ WPQ1 . Now we observe that, 34 fixing ', both the colimits holimQD-P (Homji ( ,PQ, IWpi)) and holimQ:p (HomHpQ(VQ, WpQ)) compute the same complex which is the complex H,*,, ((Xp, X- \ Sp), Hom(Vpp, Wpp)), computing the relative coherent cohomology of the sheaf Homx, (Vpp, Wpp) relative to the complement to the closed stratum. This means that we can introduce filtrations on the complexes holimN RHomPQ (VpQ, WpQ) and R Hom(V, W) a holimg RHom (VPQ, WPQ) compatible relative to the obvious map R Hom(V, W) -+ holimg Homwe (VpQ, WpQ), which induce isomorphisms on associated graded components. The arguments for V O V' are analogous. This proves the lemma. 4.5 Noncommutative pushforwards We've defined the forgetful functors j*>Q : Sm -+ SmpQ; these are exact, hence have obvious derived analogues J*pQ : DbSm -+ Sm-pQ. These functors are noncommutative analogues of affine pull-back and so it makes sense to look for a right adjoint functor RjyfQ. It is proved in [BK2I that it is possible to define a sheaf of algebras A over Xc//W such that Sm is equivalent to the category of sheaves of modules Sm(A). Now tpQ induces a sheaf of algebras ApQ over XpQ such that the category of sheaves of representations of ApQ is equivalent to Sm-pQ. Write jPQApQ for the algebra W considered as an algebra over X via the map of algebraic varieties j7Q : XpQ -+ X. Then it follows from formal arguments that the functor J* Q interpreted as a functor A-Mod -+ jpQApQ-Mod is given by tensor product with some bimodule MpQ flat over both A and jpQApQ. From this it follows formally that we have a well-defined right adjoint functor 3P' to 3*, whose derived functor will give a left adjoint in the 35 derived category, R3jQ. We will not write down a formula for the affine components of (Rj1'Q(VpQ))piQ here (as it is rather involved), but rather just use the existence of this adjoint functor. 4.6 Dg functors out of Sm Now we are ready to characterize DG functors from Sm to an arbitrary dg category C. Lemma 8. Suppose that C is a dg category with all colimits. Then a colimit-preserving functor DbSm'-+ C is equivalent to a collection of objects ApQ of C with right actions by 'HpQ, together with compatible identifications j 'Q' ApQ '= AplQ g, where the functor J ''(A) := (A ®HpQ HpQ))ul, is defined as a colimit in the category C. Proof. Define Funocai for the category of collections ApQ as above and Fun(D~m, C) for the category of (dg) colimit-compatible functors. Then we have a functor a: Funocal -+ Fun(DbSgm,C) given by a({XpQ}) : V - holimXpQ ®7f7PQ V and /3 Fun(DbSBm, C) -+ Funiocal given by /(F)pQ := F(J(NpQ)) (which have obvious right WpQ-action as pQ-modules). It follows from the previous two subsections that a, # are inverse to each other. Notation. We call the data XpQ like in Lemma 8 the kernel of the functor F a({XpQ}). The notation comes from the theory of Fourier-Mukai kernels, since in fact, the data of {XpQ} above is most naturally an object of the tensor product category Sm N C. In Section 6, we will write down a kernel {2ocpQ} which we will use to define the localization functor Loc. 36 Chapter 5 Algebra on the building In this section we write down some standard results about the derived category of G-equivariant cosheaves on the Bruhat-Tits building. Our main sources are [BThes] and [Cu]. 5.1 Models for sheaves and cosheaves Definition 14. Write CoShG for the category of cosheaves on the building B which are constructible with respect to the cellular stratification and equivariant with respect to the G-action on B. Equivalently, this is the opposite category of the category of sheaves with values in Vect P. Given any point x E o C B and cosheaf V E CoShG, the costalk Vx has action by the parahoric subgroup G,. Because the strata of the cellular stratification are contractible, the data of these stalks together with the specialization morphisms on costalks is sufficient to reconstruct V. More precisely, choose a top-dimensional cell E C B. Note that as we are studying sheaves which are constant on cells and the stabilizer G(o-) coincides with the stabilizer G(x) for any x E -, we can unambiguously write V, for the costalk of V at an arbitrary x E a. Definition 15. Write PE for the (non-additive) category with objects cells - C E 37 and morphisms HomE (a, a') : ~ -,o'co 0, ' Co with compositions given by embeddings of subgroups and the group structure on G(U). This category is generated by the automorphisms G(u) = AutpE (a) together with "specialization" morphisms t,,, : a-+ a' for a' C a (in fact, it's enough to take the two cells to be of consecutive dimension). Then we have Lemma 9. There is an equivalence of categories between CoShG and the category of left modules PE - Mod taking a cosheaf V to the representation RV(O-) := V, with AutpE(a) action induced by equivariance and action of t,, given by cospecialization morphisms of stalks of cosheaves. We will abuse notation and go between these two interpretations freely. The most important category for us will be the category CoShG of equivariant cosheaves above. However, it will also be useful for us to have similar "representation-theoretic" models for the categories CoSh (non-equivariant cosheaves) as well as the categories ShG, Sh of equivariant and non-equivariant sheaves on B. We define another poset category. Definition 16. Define the category PB to be the poset of closed cells of B, ordered by reverse containment. Now the same arguments as above give us the following equivalences. Lemma 10. With this definition, we have 1. The category of nonequivariant cosheaves CoShB is equivalent to the category of representations of the category PB 2. The category of equivariant sheaves Sh Gis equivalent to the category of repre- sentations of the opposite category PEOP. 3. The category of nonequivariant sheaves ShB is equivalent to the category of representations of PB P. 38 In particular, as the pairs of categories Sh, CoSh and ShG, CoShG can be in- terpreted as representation categories of opposite rings, we obtain tensor product functors 0 : Sh x CoSh -+ Vect and 0 : ShG x CoShG - Vect, as well as left L derived versions L. 5.2 Projective and injective objects We will be interested in the derived category Db CoShG. It will be convenient for us to have a notion of derived tensor product between sheaves and cosheaves. Namely, for a sheaf V E ShB and a cosheaf V' E CoShB, write V OPB V' for the tensor product of V, V' as right and left PB-modules. We define tensor product V 0pE V' similarly G G Lfor V E ShG, V E CoShG, and write V 0 V' for the derived functor. By standard homological-algebraic arguments, this derived tensor product can be computed in terms of a projective resolution of either side. We will be especially interested in the L case V = C the constant sheaf, in which case as we will see (C Op V' returns the homology of the cosheaf V'. First, we recall from [BThes] a classification of projective objects in ShG Definition 17. Given a cell o- C B and a vector space V, write *,(V) for the constant sheaf with fiber V on the stellar neighborhood of a. This definition has an equivariant analogue, Definition 18. Given a cell o- and a representation V, of G(-), write *,(V,) E PE - Mod for the sheaf with *, (V)a' = VIG(a') if a' D a and 0 otherwise, where for any pair of cells a' D a the cospecialization morphism *,(V,), ~ *,(V,),, is the identity map. Lemma 11 ([BThesI). The sheaves *,(V) for V irreducible are a complete collection of indecomposable projectives in ShG. Their dual cosheaves, *a(V) for V irreducible form a complete collection of indecomposable injectives in CoShG 39 Lemma 12 (fBThes]). Any sheaf V E Sh has a projective resolution ®~ocE *oVo, - $cDT, 1o,-Ii-=1 *O'VT G...r Ii-IrJ=n *oVT 0 and, analogously, every sheaf V E SliG has a projective resolution @a$UDTVO IoiJ-kI=1 *U1VT E~o... T Ioi--rI=n *oV- - 0 This in particular tells us that Sh and ShG have projective dimension n. Ad- L ditionally, it gives us a formula for a complex V 9 W as follows indexed by pairs - D T: EcE Voa OG, Wo, +D- 1 7r1= 1 Vr ®G(r) W -- .-- 9 -|r=n HomG(o-) Va ®G(r) WOI.. quasi-iso L V D W, with an analogous formula in the non-equivariant setting. Putting in V = CB the constant sheaf, we recover the standard complex RIFe(V) computing the homology of B with coefficients in W, with respect to the baricentric subdivision of our cellular decomposition. Putting in V = CJB/G, the constant sheaf viewed as an object of ShG, we recover a complex computing RF(V)hG, the derived G-coinvariants in the homology of W. 5.3 Constant sheaves on orbifold subsets Here we will introduce a class of sheaves corresponding to "tale sub-orbifolds" S/H of B/G. We will use the notations B, G for the building and the group G, although the same analysis will apply to an arbitrary polyhedrally stratified locally finite CW 40 complex B with smooth action by a totally disconnected topological group G with compact open stabilizers. Suppose that S C B is a (closed, cellular) subset and H C G a closed subgroup fixing S. Then we define the G-equivariant topological space G X H S = Gx, where H acts diagonally. We define the "action map" #3: G x H S -4 BH' via (g, x) '-+ gx, and define CS/H := A!(-GxS), the "constant cosheaf on the orbifold S/H", to be the !-pushforward of the constant cosheaf on G x u S via 3. This is the cosheaf whose costalk over a point x C B is the vector space of compactly supported functions on G/G(x) n U. (Here G(x) is the stabilizer of x in G, equivalently the stabilizer of a small symmetric open neighborhood of x). We have the following important observation. Proposition 13. RF(_Cs/u) 7 G x'H RVe(S). This follows from the fact that RPF(V) := Rpt!(V), for pt : B -+ * the map to a point, hence (Q(CxHG)) .pt(C(S x HG)) C* C(S X H G). (Here we write C* to denote the complex of chains.) The terminology of constant cosheaf is motivated by the fact that CS/H E CoShG corepresents the functor of invariants in cochains, RHom(CS/H, V) C* (S V)hH for arbitrary V E CoSh. In particular, if we have Si C S2 and H1 C H2 , then we have a canonical map t : /H 1 - 2 /H2 corresponding to the constant section on Si of the constant sheaf on S2. In fact, this construction can be extended. Let G(s,52) c G be the collection of all -y E G with y(S1) C S2. Definition 19. Given two subsets S 1 , S2 C B invariant with respect to H1, H2 C G, respectively, define the "geometric Hom" Homgeom (SI /H1,7 S2 /H2) := (H2\G(s, S2>) H1 41 to be the set of right H1-invariant points in the quotient H2\GS 1/ H1,5 2/H 2 (with evident right H1-action). Then the space Cc Homgeom(SI/H1 , S2/H 2) of compactly supported locally constant functions on Homgeom(S/Hi, 2/H 2) (viewed as a complex concentrated in degree 0) maps to HomC o ShG(CS1/H 1 , CS 2 /H 2 ) HO RHomDbCOShG(CS,/H1, CS2/H 2 ) . Note that Homgeom(S/H1, S2/H2) defines a category structure on pairs S/H (with H C G acting on S C B), and the map C; Homgeom(Si/Hi, S2/H2) 4 HomCoShG(CS I/H1 , CS1/H 2 ) is compatible with this category structure. 42 Chapter 6 Definition of the localization functor 6.1 Corridors There is a tradition of making papers on Bruhat-Tits theory read like manuals on real estate. Buildings contain apartments that consist of alcoves. There is however a problem with buildings that until now has not been resolved: there is no a priori way of getting from one apartment to another. Here we will finally propose a solution for the long-suffering tenants. We will introduce a notion of corridors, parametrized by standard parabolic subgroups Q, each of which connects apartments in a Q-conjugacy class along a Weyl chamber corresponding to Q. Fix a basepoint of the building, xo E B, fixed by a maximal compact K C G. Let A be the apartment in B (viewed as an abstract affine space). Write 73 : B - A for the projection to the quotient A 2 1/UB. For a standard parabolic Q ;? B, write A' for the kernel of the composition A C T -+ Q -+ L -+ X*(L/[L, L]). Then A' 0 R acts on A. Write AQ := A/At & R and 7rQ : B -+ AQ for the evident composed projection. Write A- C AQ for the cone of all points strictly smaller than 7rQ(xo) in the usual poset structure on the coweight lattice of G. Definition 20. The standard corridor of type Q is the preimage IDQ := ,r-1 (A-) C B. Example. Let G = SL 2(Q2 ). Then 0 DG is the whole building B. 43 UL : * Dt is the contractible graph that looks like this. I Figure 6-1: DL3 for SL 2 From the "hyperbolic" point of view, corridors should be thought of as cylinders in the hyperbolic geometry, tangent to the boundary (in the polyhedral compactification, see [Lal or section 7.1) at a hyperbolic subspace (which will always be a building for a Levi subgroup). For example, ID, in the example above is a disk that meets the boundary at a point: DR _ Figure 6-2: Hyperbolic corridor In particular, we have the following theorem about the geometry of corridors. Theorem 14. 1. D- is convex and contractible. 2. The normalizer of the standard corridor Dp is LU-p. 44 3. Suppose y E L g+ such that it is not in any LQ'+ for P' C P. Then the union Un>o-DQ '== Dp. 4. For two corridors D-p, IDQ and -y E G we have -yD- C DQ if and only if P C Q and Y E Q+. Proof. We give a brief sketch of the proofs of these combinatorial results. (P1) Convexity of DQ is equivalent to the convexity of the closure UDQ C B in the polyhedral compactification (FLal). It remains to observe that the compactifica- tion UQ is the convex hull of the boundary component &Q corresponding to Q and the boundary, 7rQ'(0) c B. A nonempty convex set in a hyperbolic metric is contractible. (P2) We first prove that Lo UQ does indeed normalize DQ. Let A- C H be a lift of A- in B (necessarily contained in B). Choose any point x E AU nlQ 1 (AQ OR) on the closed boundary component of A- corresponding to a maximal compact, and let Qx be the stabilizer of x in P. By standard arguments involving parabolic orbits in characteristic p, the projection of a cell near x to AQ only depends on its Px-orbit, hence in a stellar neighborhood of x, the image of Px(D) under lrQ is contained in AQ. By an inductive argument, we deduce that such Px normalizes D. On the other hand, the maximal torus of L0 is contained in the preimage of AQ in the maximal torus T C G, hence also normalizes ID. We conclude using a version of the Cartan decomposition for LO. In the other direction, let k be the rank of L,,. Then D contains exactly one k-dimensional component of the polyhedral compactification of B (see [La], or section 7.1), which is the unique component normalized by Q. Thus anything that fixes ID must be contained in Q. Let -y E Q be an element. By arguments analogous to the above, we see that the image 7rQ(yD)is a shift of A' by the image of y in the lattice L/Lo C AQ, hence -y must be in L0 to stabilize D. (P3) This follows by pullback to B from the evident covering of the affine cone AQ x A' by translates of A- x A'. 45 (P4) Note that given a pair P C Q and a corridor D := yAp (some y E G), the corridor D' := yAQ is the unique minimal conjugate of AQ that contains D (as Q := -yQ-y 1 is the unique parabolic conjugate to Q which contains yP-y- 1 , and D' can then be recovered as LO UQ - D). The result now follows. QO 6.2 The localization kernel Note that each DQ (being a preimage of a subset of B/Us) is invariant with respect to the unitary group UB, hence also invariant with respect to all UP (which are contained in U3). Now our localization kernel Loc will be constructed out of the equivariant cosheaves OC PQ := C1Q/U in the terminology of section 5.3. Namely, observe that for two arbitrary parabolics Q, Q', Theorem 14 implies that the set of elements sending DQ to DQ, is Gin~n,) = L+,UQ,lQcQ 0, Q Q. This means that Homgeom(DQ/Up, IDQ,/Up) = (Up,\L+,U Q,) = ((up, n LQ,)\LQ)Lfup so long as P' C P c Q c Q'. If P = Q = 2 = Q', then we have L C L+UQ, which is bi-invariant with respect to Up, and so L + C Homgeo(DQ/Up, DQI/UP,). This gives us the desired Lp-action. Further, we have tautologically for P C Q that Loc-pQ 2 (LocpQ) u, . The identity class 1 - Up, c L+,UQ, is right Up-invariant (since Up, D Up), hence gives a class t 'Q' E Homgem(Dp/Q, De/Q'), inducing a map t (CocpQ)U,, -+ ocpQ, which is visibly L+-equivariant. In order 46 to check that the ocpQ indeed define a kernel, we need to check that the map OCPQ ®+ 1 24'-* ocpQl adjoint to tpQ' is an isomorphism. This follows from part 3 of Theorem 14. Having checked conditions of Lemma 8, we get a functor Loc : Sm -+ CoShG defined as follows. Definition 21. Define Loc(V) := holimpQ ocpQ OW,, VpQ = holimpQ CD,/UQ ® HPQ VpQ E Db CoShG. We introduce also some notation for the "affine components" of Loc, namely Definition 22. We define the functor LocpQ : SmpQ -+ CoShG by LocpQ V PQ '- VPQ OpQ IDQ/UJ, for VpQ a reprsentation of W7pQ. Recall here that _Q is defined as the !-pushforward of the constant sheaf on DQ x u, G under the action map # : G x U, DQ -+ B. This means that we can describe the stalks of IDQ/Up alternatively via the following definition. Definition 23. Define {; = - E G I - - DQ -> al -} \GHGQ=UUpU\G. This is an open subset of G/Up left invariant with respect to G(U-) and right invariant with respect to L+Q C Homgeom(D/Up,DQ/Up). Proposition 15. We then have P-oc p := C (HpQ). Definition 24. Define (abusing notation somewhat) Zoc' for the G(O-)-equivariant object of the right compactified category (S-m-R)G(,) to be the object with affine compo- nents 2 oc'Q, and evident componentwise G(-)-action. 47 From our definition of kernels, we now have: Proposition 16. The stalk Loc(V), Z oc" 0 V. Having defined the functor, we begin verifying its properties. Lemma 17. For any sheaf V E Sm, we have canonically RFc Loc(V) e V. Proof. We will prove a corresponding statement independently for each component LocpQ. Proposition 18. RFe LocQ VQ VPc OL+ WLp ®Lp c (Up\G). Proof. It suffices to check this in the universal case, with VpQ = HpQ, in which case it follows from Proposition 13 and the contractibility of DQ, (Theorem 14, part 1). E Note that this proposition in particular implies that, given an object V of the com- pactified category, RFc(LocpQ(V)) a RFe(LocpQ,(V)) for any Q' D Q (as VpQ ®L + HLp is independent of Q). Now since all our functors are dg functors and com- mute with finite homotopy limits, we can compute RF,(Loc(V)) as the limit of RFc(LocpQ(VpQ)), giving RC (Loc(V)) = holimp,Q VPQ 0L4+ Gc (G/Up). Thus decomposing the partially ordered set K of pairs (P, Q) into subcategories (W, -) C K, we are taking the homotopy limit along a diagram which is constant along each (M, -). Since each of these categories has a terminal object: namely, (P, G), the nerve of the corresponding subcategories is contractible, and the limit computation can be simplified to holimp V(p,G) ®Lp C (G/Up). But the subcategory (P, G) c K, in turn, has a terminal object, namely, (G, G) leaving us with RFe(Loc(V)) e V(G,G) OW W V 48 Example. For SL 2 , the category K has three objects, and looks like this: (BIB) >- (BIG) -< (GIG),I and the colimit computation above then identifies the homotopy limit of the fibered product diagram V V13 OTCcO (G / U) ~VU OT C (G/IUL3) with V (note that the colimit of any diagram of the form A -+ A +- B is B). This concludes the proof. The next lemma establishes that Loc(V) almost has finite-dimensional stalks when V E Smg. In the next chapter, we will see how to get rid of the "almost". Lemma 19. So long as V is locally finitely generated, Loc(V) has stalks that have finite-dimensional invariants with respect to open compact subgroups. Proof. Since G(-) is compact, taking invariants with respect to an open subgroup is an exact functor, and hence for J C G(o), we have the left invariants J Loc(V) C V ® J(2oco). Now it is sufficient to show (see Section 4.3.2) that J(2oco) is locally finitely-generated. In order to see this, we observe that J2oc'Q = Ce (J\HpQ) is the space of compactly supported functions on the subset J\HPQ of the discrete double quotient J\G/Up. Now in the special case J = G(a), the double quotient J\HpQ c G(a)\G/U can be identified with the collection of cells of a' C A of the apartment which are W-conjugate to a and satisfy a-a xO (in the partial order induced by polarization on A = B/Uy). Hence it is generated over A+ by finitely many classes (corresponding to the minimal cells in each A-conjugacy class of W a). Let these generators be {x} E G(-)\G/U. Then their finitely many preimages in J\G/U will 49 give a generating set for J( 2 oc"). This gives us finite generation of (2ocpQ)J, and completes our proof. E 50 Chapter 7 The truncated localization functor Fix an integer e > 1, which we will assume to be chosen larger than the depth of our compactified representation V. The paper [SS] defines a conjugation invariant system of open normal subgroups G) < G, indexed by cells o C B, with the property that G) c G$,) for r C -. This allows us to define a "truncation" functor (e) CoSh(e) - CoSh(e) defined as follows: Definition 25. For V E CoShG define I(e)V e CoSh(e) to be the cosheaf whose stalks are invariants, with respect to the ("Schneider-Stuhler") system of subgroups above. Definition 26. Define Loc(e) to be the composition I(e) o Loc : Sm -+ Sh. By Lemma 19, the functors Loc(e) have finite-dimensional cohomology of stalks. This section will be devoted to proving the following theorem. Theorem 20. Suppose that V E Sm has depth < e. Then the compactly supported global sections, R'c(Loc(V)(e)) In fact, we will prove a stronger result. Definition 27. Define Loc( := I(e) LocpQ : Sm -+ CoSh. Then we have 51 Theorem 21. Suppose VpQ C SmpQ has depth < e. Then the compactly supported global sections, RFc (LocN(VPQ)) 2 RFc (LocpQ(VpQ)). 7.1 Building combinatorics We will give here some reminders about the theory of buildings and the polyhedral compactification of [LaJ. This subsection and the next will be inspired by construc- tions and notation in the paper [MS]. We will take a combinatorial point of view based on the Weyl partial order on the coweight lattice. Namely, for an algebraic group G, write T for its torus, with lattice of characters (the weight lattice) X*(T) and lattice of coweights X,, (T). We will choose a uniformizer W E Gm (K), and write A C T for the lattice A r X(T) of coweights embedded in the K-point group T := T(K) via multipowers of the uniformizer. Write AR := A & R and choose a polarization on G. Let x1,..., x, E AV be the collection of simple roots. Let A+ C A be the sublattice of all element which pair positively with the xi. This is naturally identified (via the metric) with a Weyl chamber. We define a partial order on A with a -< /3 if 3 - a E A'. Now to any torus T C G there corresponds an apartment AT C G. If we choose a containment T C B in a Borel (equivalently, a polarization), then we get a partial order on A with a >- b when we have a containment of stabilizers U(a) D U(b) in the unipotent radical U C B. This partial order satisfies wva a> a if and only if A E A+ (where the parametrized embedding A C T is determined by the polarization as above). In particular, if we choose in addition a x E A, we get a canonical identification A 2 AR compatible with partial order. We will use the shorthand notation polarized apartment to denote an apartment with choice of partial order corresponding to a pair T C B as above. Definition 28 (Meyer and Solleveld). The convex hull of two cells U, T C B, denoted U,-T, is the intersection of all apartments containing both - and T. More generally, the convex hull of a subset F C B is the union of all convex hulls of pairs of points of F. This notion generalizes in an obvious way to a subset F C lB 52 Remark 1. The idea behind this terminology is to replace the notion of a geodesic line segment, which is the collection of points on a shortest path between a, b, by a 'partially ordered geodesic line segment", which is the collection of points x in a parametrized apartment between a, b E B which satisfy a - x -< b in the Weyl partial order corresponding to the parametrization. Definition 29. Given a point x E A in a polarized apartment, we say that a subset R C A with the data of a partial order is a geodesic ray out of x in the given polarization if R is cofinal with minimal point x in the partial order, i.e. if any y E R satisfies y >- x and for any pair y, y' E R, there is z E R with z > y, z > y'. Remark 2. We are not using the usual notion of metric on the Bruhat- Tits building, and a geodesic ray is in general not one-dimensional. For an arbitrary pair of points x, y, there is then a unique geodesic ray x- with minimal point x and maximal point y. If o, -' are a pair of closed simplices, then there is a unique vertex x E o-, y E o' such that z 3 contains both o- and o-'. We define 0-, o-' to be this subset (with induced partial order). Remark 3. The notion of a geodesic ray allows us to give an alternative definition of the polyhedral compactification of B (see [La]) as follows. Define 9 to be the quotient of the collection of closed geodesic rays R C B by the equivalence relation that R - R' if R n R' is cofinal in both R and R'. In particular, the convex hulls x,-y and x', y (with partial order such that y is maximal) are equivalent as their intersection contains y. This gives the embedding B C B. 7.2 Consistent systems of idempotents Definition 30 (Meyer and Solleveld). Let V be a vector space with action of G. We say that a system of idempotents E, E Endc V indexed by cells of the building is consistent if it satisfies the following three properties. (a) (local commutativity) E,, E, commute if a,,r are in the closure of the same face 53 (b) (local multiplicativity) The idempotent corresponding to a cell is the product of those corresponding to its vertices, i.e. E, = -1,] o Ex. (c) (convexity) For any triple of cells T, U, 0' with T C o and - in the convex hull of T, O', we have the identity EE,, = EEo,,. Lemma 22 (Meyer and Solleveld). For any depth e > 1, the idempotents E(): 6G() form a consistent system of idempotents. L We mention that in the proof of [MS], property (b) above is reduced to the fol- lowing group identity, which will be useful to us as well: Proposition 23 (Meyer and Solleveld). G!e) -Ge : G . Ee Given a system E = {Ea} of consistent idempotents, write VE for the coefficient system with the vector space VE over cell o. Note that this admits a map to the constant coefficient system V. Lemma 24 (Meyer and Solleveld). For any consistent system of idempotents {E,} and any closed convex subset F C B, the derived global sections RF (VE) are quasiiso- morphic to the vector space ' yEa C V concentrated in degree 0. This identification is consistent with the embedding of coefficient systems RF (VE) C RF() = V [0]. El Choose a parabolic P D B. Now let Vp = iprp Lp = C 0 (G/Up). We have two algebras acting on Vp. First, 1-G acts via the usual representation structure. Secondly, the commutative algebra R := C (G/U) of all locally constant functions acts on compactly supported functions by multipli- cation. The two actions combine into an action of the crossed product algebra Ap:= R#G. Now choose another parabolic Q D P. Observe that the subsets H+' giving local action of the localization functor correspond to idempotent functions 6 Q+ on R, and 54 that these are preserved by the subgroups G,, hence commute with the idempotents Ee) = 6(G(e)) (hence their products are idempotent). To unburden notation, write 6, 6 PQ I. e) and The idempotents 4b, act on the space Vp iPLp of compactly supported functions, and have image Loc'p (7 -tpQ) (compactly supported, G()-equivariant functions on H Q C G/Up). Up to a universality argument, it suffices for us to prove the following. Proposition 25. The idempotents E(e) form a consistent system of idempotents. Proof. It will be convenient for us to give a formula for products of functions of the form 6,. Namely, given an element -y E G/Up, write DI := 7-yDQ for the corresponding corridor (of type Q). Recall that the space H'Q is the collection of all -y E G/U such that - C DQ. Because corridors are convex and closed, we can multiply idempotents of the form 6, in the following way. Suppose that o-, ... , o- are a collection of cells (of arbitrary dimension). Write E for the convex hull of the closed cells 5:. Write H = { E G/U I DQ - E}. By convexity of corridors, we have H PQ in this case is the intersection of all HPQ C G/Up. Write 6E for the corresponding characteristic function. We deduce that we have the following formula. Lemma 26. We have k = = with E the convex hull of the closed cells uj as above. E In order to prove proposition 25, we need to check the three properties of Definition 30 for the 4D,. By Meyer and Solleveld's Lemma 22, we have consistency of the S, and the lemma 26 applied to the vertices of a single cell cell gives us the local multiplicativity property for the system of idempotents 6,. The other two properties are obvious from commutativity of the 6g, giving us consistency of the system {Yo} 55 as well. Note that this is not yet good enough to give us the desired consistency of the {b,} = ,6,} as E, may not commute with 6,, for a, a' far apart. First, we observe that local commutativity and multiplicativity follows from the corresponding properties of the systems {E}, {6,} by checking that idempotents of the two systems mutually commute at nearby vertices. Claim. the idempotents E,4, E :,, and 6,, pairwise commute for T, T' both in the clo- sure of a cell a. The only pairs for which we still need to check this are (E, 7) and (4, 6,1). Now we have by construction that GI C G(') C G (see [SS], 1.2, where these groups are called Ue). Since G, normalizes HQ, the idempotents S,4 and 67 commute and we are done WLOG. E It remains for us to check the convexity condition of definition 30. Note that, fixing a Haar measure, the crossed product A = R#G can be identified with locally constant functions on G/U x G which are supported over a bounded subset of G. Product is computed via the multiplication kernel x#^yd(G/U)dG - x'#-y'd(G/U)dG:= 6x,,,/ -x#y1 1'd(G/U)dG. Suppose we have a triple T C a, and T' such that a C TT' is in the convex hull (of the open cells). Write E := T, T' for the convex hull of the closures of the cells, which coincides with FT'. We compute I'dydq ____ . d '#dr ' (1 ( d) d y'dre' 6(n, (77') )#YY' (7.2)iEG(e),DT 97,Y'EG '7,Dtp' IG)G . |G? Where we are using the notation 6(rq, y') for the delta measure on the diagonal q = (r/)y. Now note that for -y E G, the conditions D, E T, Doq 3 T' and q = (r')7 together are equivalent to ]D, 3 T,1D, D3 yT' and q' = 9f-4 , which can further be reduced to 56 D77 D ^/E (as T = yT). This lets us rewrite bb,= f#y' (7.3)4D7W'(eD'r'/UD~y~ (Ge) I (e = EG(e),,7'EG *-1, ,EG/UDOD-yE IGy)| - I GIrl Now note that multiplying ^y in the above expression on the right by any element o c G that fixes E and is contained in the product G(')Ge) will not change the result. In particular, this is true for any yo E G (e) n GE. Write Go:= G(,) n GE and averaging over -yo E Go as above, we can safely introduce a new variable -Yo in the integral above: