The Schwartz space of a smooth semi-algebraic stack
TTHE SCHWARTZ SPACE OF A SMOOTHSEMI-ALGEBRAIC STACK
YIANNIS SAKELLARIDISA
BSTRACT . Schwartz functions, or measures, are defined on any smoothsemi-algebraic (“Nash”) manifold, and are known to form a cosheaf forthe semi-algebraic restricted topology. We extend this definition to smoothsemi-algebraic stacks, which are defined as geometric stacks in the cate-gory of Nash manifolds.Moreover, when those are obtained from algebraic quotient stacks ofthe form
X/G , with X a smooth affine variety and G a reductive groupdefined over a number field k , we define, whenever possible, an “evalua-tion map” at each semisimple k -point of the stack, without using trunca-tion methods. This corresponds to a regularization of the sum of those or-bital integrals whose semisimple part corresponds to the chosen k -point.These evaluation maps produce, in principle, a distribution whichgeneralizes the Arthur–Selberg trace formula and Jacquet’s relative traceformula, although the former, and many instances of the latter, cannotactually be defined by the purely geometric methods of this paper. Inany case, the stack-theoretic point of view provides an explanation forthe pure inner forms that appear in many versions of the Langlands, andrelative Langlands, conjectures. To Joseph Bernstein,in admiration and gratitude. C ONTENTS
1. Introduction 22. Nash stacks 103. Local Schwartz spaces 204. Local and global stalks for affine reductive group quotients 305. Equivariant toroidal compactifications, and orbital integrals onlinear spaces 436. Evaluation maps 62Appendix A. From algebraic to Nash stacks: a presentation-freeapproach 71Appendix B. Homology and coshseaves in non-abelian categories 76Appendix C. Asymptotically finite functions 83References 89
Mathematics Subject Classification.
Primary 58H05; Secondary 22A22, 11F70.
Key words and phrases. stacks, Schwartz spaces, orbital integrals. a r X i v : . [ m a t h . AG ] M a y YIANNIS SAKELLARIDIS
References 961. I
NTRODUCTION
Overview.
It is an insight of Joseph Bernstein [12] that several prob-lems in representation theory and, in particular, the Arthur–Selberg traceformula and its generalizations (such as Jacquet’s relative trace formula)should be understood in terms of quotient stacks of the form X = X/G ,where X is an affine variety and G a reductive group. For the Arthur–Selberg trace formula this stack is the adjoint quotient of a reductive group;for the relative trace formula it can be a more general stack of the form H \ G/H (specializing to the adjoint quotient of H when G = H × H and H = H = the diagonal copy of H in G ).One piece of evidence for the relevance of algebraic stacks is the appear-ance, in all nice formulations of the Langlands conjectures (and their rela-tive variants), of pure inner forms of the groups that one originally wantsto consider. Compare, for example, with the local Langlands conjecturesas stated by Vogan [44], or the local Gan–Gross–Prasad conjectures [23],which have now been proven in most cases [45, 36, 15, 24, 9].Eventually, one should have a local and a global Langlands conjectureattached to (at least) stacks of the form ( X × X ) /G diag , where X is some“nice” (e.g., spherical) G -variety. (In the group case X = H , G = H × H ,this specializes again to the adjoint quotient of H .) Both conjectures shoulddescribe, in terms of Langlands parameters, the “spectral decomposition”of a distinguished functional on the local and global Schwartz space of thestack. Locally, this functional should be, essentially, the L -inner productof two functions on X , while globally it should be the pertinent “trace for-mula”. It is not my intention in this article to provide a formulation of suchconjectures, but to start building the theoretical framework necessary forsuch a formulation. It should also be mentioned that there is a more ad-vanced version of the local Langlands conjecture due to Kaletha [29] whichincludes non-pure inner forms. It is clear that one needs to go beyond thepoint of view of the present article to incorporate Kaletha’s conjecture.Let X denote a smooth algebraic stack over a number field k or a local,locally compact field F . For non-smooth stacks we do not know the appro-priate definitions of Schwartz spaces etc., even when the stack is a variety— however, this is an important problem, cf. [17]. The goal of this article isto define: • locally, the Schwartz space S ( X ( F )) , generalizing the well-knownSchwartz space of smooth, rapidly decaying (or of compact sup-port, in the non-Archimedean case) measures on the F -points of asmooth variety; HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 3 • globally, a notion of “evaluation” of a Schwartz function at a closed k -point x : spec k → X , where the space of Schwartz functions is bydefinition a restricted tensor product of the local Schwartz spaces,divided by a Tamagawa measure. I do this under a certain set ofrestrictions on the algebraic stack.For smooth stacks of the form X = X/G , an equivalent definition of thelocal Schwartz space S ( X ( F )) was suggested to me by Joseph Bernstein:without loss of generality, G (cid:39) GL n , and then we can set S ( X ( F )) := S ( X ( F )) G ( F ) , (1.1)the coinvariant space of the space of Schwartz measures on X ( F ) . (Fortopological spaces, we define coinvariants by taking the quotient by the closure of the span of vectors of the form g · v − v ; a more refined notionis provided by derived categories that we will encounter below.) The re-quirement G (cid:39) GL n ( F ) is in order to have trivial Galois cohomology, i.e.,so that the F -points of X surject onto the F -points of X — otherwise thedefinition would depend on the presentation of the stack. This explainsthe appearance of pure inner forms: for another presentation X = W/H ,where H is an arbitrary reductive group, choose a faithful representation H (cid:44) → G = GL n and set X = W × H G . Then X = X/G = W/H , but in termsof the quotient
W/H , an F -point of X corresponds to a torsor T (equiva-lently: a pure inner form) of H over F , together with an H -equivariantmap: T → W . As we will see, the G ( F ) -coinvariants of S ( X ( F )) can beidentified with (cid:77) T S ( W T ( F )) H T ( F ) , where T runs over all isomorphism classes of T -torsors, W T = W × H T and H T is the inner form Aut H ( T ) .In this article I follow a more general approach for the definition ofSchwartz spaces, which turns out to specialize to Bernstein’s in the caseof quotient stacks. To define the Schwartz space for a general smooth alge-braic stack, I extend and exploit the following well-known fact: Schwartz spaces form a cosheaf.
This is true in the semi-algebraic (restricted) topology on the F -points ofa smooth variety or, more generally, on a smooth semi-algebraic (“Nash”)manifold, and has been systematically exploited, e.g., in [21, 1]. It is, how-ever, true in a stronger sense, namely: in the smooth Grothendieck topologyon the category of Nash manifolds. This is the observation that allows usto define spaces of Schwartz measures as a cosheaf on the smooth topologyover a smooth semi-algebraic stack.These Schwartz spaces are nuclear Fr´echet spaces in the Archimedeancase, and the push-forward maps with respect to smooth morphisms ofvarieties are strict , i.e., have closed image. Taking the functor of globalsections over our stack X ( F ) , we obtain a well-defined element S • ( X ( F )) YIANNIS SAKELLARIDIS in the derived category of nuclear Fr´echet spaces. (For the notions of de-rived categories of non-abelian categories see Appendix B.) For a smoothquotient stack X = X/G , the zeroth homology essentially coincides withBernstein’s definition that we saw above, and of course higher homologygroups correspond to the higher derived functors of coinvariants.In the non-Archimedean case, the same statements are true in the de-rived category of vector spaces without topology. It should be mentioned,however, that in positive characteristic there are serious issues with semi-algebraic geometry. On the other hand, one does not need the semi-algebraicstructure to define Schwartz spaces in the non-Archimedean case: the usualnotion of compactly supported smooth measures or functions makes senseon F -analytic manifolds. Thus, in positive characteristic one should readany mention of “Nash” manifolds in this paper as referring to F -analyticmanifolds, and any mention of semi-algebraic restricted topology as refer-ring to the usual topology on F -points. In the non-Archimedean case incharacteristic zero, the reader can choose either of the two approaches. Al-though the definitions make sense for F -analytic manifolds, there is some-thing to be gained by restricting to the semi-algebraic case, namely: onepreserves a notion of “polynomial growth”. Moreover, one might wish toreplace the usual Schwartz spaces of compactly supported smooth mea-sures by the Fr´echet spaces of “almost smooth” measures defined in [39,Appendix A].The second goal, defining an “evaluation map”, is performed only forsmooth quotient stacks of the form X = X/G and corresponds to definingregularized orbital integrals for the G ( A k ) -action on S ( X ( A k )) . There is lit-tle stacky here, although a lot of work goes into showing that the construc-tions are independent of choices and give rise to well-defined functionalson the Schwartz space of a stack. Once we choose a closed (“semisimple”)point x : spec k → X , corresponding to a G ( k ) -orbit on X ( k ) (by appro-priately choosing the presentation X = X/G among pure inner twists), thegoal is to define a regularization for the sum of G ( A k ) -orbital integrals cor-responding to G ( k ) -orbits on X ( k ) whose “semisimple” part is isomorphicto x . In the literature, this has been done almost invariably by generalizingthe methods of truncation [5] in the monumental work of Arthur (which,in turn, is a vast generalization of Selberg’s).There is a very general approach to truncated orbital integrals by J. Levy[34] (who, however, does not produce invariant distributions), and severalvariants of the truncation method which are better suited for applicationsto the relative trace formula by Jacquet, Lapid, Rogawski, Ichino, Yamana,Zydor and others [28, 26, 48, 46, 47]. Joseph Bernstein sometimes expressedthe wish to see this done without truncation, and this is what I do here, ina restricted setting that does not include the adjoint quotient of the group(but for reasons that cannot be addressed in a purely geometric way). Itdoes include, however, many other interesting examples of relative trace HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 5 formulas, such as those related to the Gross–Prasad conjectures (see Exam-ple 6.4.4). Its scope is not restricted to the trace formula and its variants;see Example 5.7.4 for an application to the Kudla-Rallis regularized periodof the theta correspondence.It is well-known, of course, that truncation at first does not produce aninvariant distribution. What I do here, before attempting to integrate, con-sists of two steps. First, replace the problem with a linear one, namelyreplace the quotient
X/G , by the ´etale neighborhood
V /H of the givensemisimple point x , provided by Luna’s ´etale slice theorem. Here H =Aut( x ) and V is the H -representation obtained by considering the actionof H on the normal bundle to the G -orbit corresponding to x . In the usualtrace formula, this is nothing other than replacing the adjoint quotient ofthe group by its Lie algebra version (or rather, the adjoint Lie algebra quo-tient for the centralizer of the given semisimple point).The second step is the crucial one, and it consists in analyzing the as-ymptotic behavior of the function: Σ V f : h (cid:55)→ (cid:88) γ ∈ V ( k ) f ( γh ) , where f is a Schwartz function on the vector space V ( A k ) . The above func-tion is an “asymptotically finite” function on [ H ] = H ( k ) \ H ( A k ) , i.e., itcoincides, up to a Schwartz function, with functions which have specifiedbehavior of multiplicative type close to infinity. To describe this asymp-totic behavior one needs a quite natural class of compactifications of [ H ] which I term “equivariant toroidal”, because they are described by datasimilar to those of toric varieties. The “asymptotically finite” functions arethen described as sections of a certain cosheaf on such a compactification,which depends on some characters, or exponents , that can easily be read offfrom the weights of the representation V . Although the method and argu-ments here are not fundamentally different from those used in truncationmethods, the approach is completely general and provides a conceptuallyclear answer for when one should not expect to have an invariant, regu-larized orbital integral out of purely geometric considerations: when the“exponents” of Σ V f are critical, i.e., equal to the modular character whoseinverse describes the decay of volumes at infinity.In cases with critical exponents, no reasonable invariant distribution canbe defined, to the best of my understanding, by purely geometric meth-ods: this would amount to finding a continuous invariant extension, to theFr´echet space of functions on R × + that are of rapid decay in a neighborhoodof ∞ but constant (up to a rapidly decaying function) in a neighborhood ofzero, of the distribution defined by a multiplicative Haar measure. Such acontinuous invariant extension does not exist as follows, e.g., from Tate’s However, these compactifications are not related to the toroidal compactifications ofShimura varieties but, rather, to the “reductive Borel–Serre” compactification.
YIANNIS SAKELLARIDIS thesis. The critical case includes the adjoint quotient of a group; as we knowfrom Arthur [6, 7], a combination of geometric and spectral considerationscan give rise to an invariant distribution on the Schwartz space in this case.1.2.
Outline of the paper.
In Section 2 I introduce “Nash stacks” over alocal field F in characteristic zero. They are generalizations of Nash mani-folds, i.e., F -analytic manifolds with a restricted topology and a finite opencover by analytic submanifolds of F n described by polynomial equalitiesand inequalities (and some generalizations of those). The main advantageof Nash, over arbitrary F -analytic, manifolds is that they come equippedwith a notion of “polynomial growth” for functions, which allows one todefine spaces of Schwartz functions and measures. Nash stacks are a gen-eralization, which allows one to talk of quotients X/G of Nash manifoldsby Nash groups, as if G were acting freely and properly on X . As men-tioned above, in positive characteristic we simply work with F -analyticmanifolds, losing this notion of polynomial growth, because some state-ments of semi-algebraic geometry that we need are not known. Ratherthan considering two different cases when the arguments are identical, Ileave it to the reader to translate all statements from the semi-algebraic tothe analytic setting.The reader who is not familiar with stacks should be reminded of thequotient X/G of algebraic varieties, where X = Res E/F G m , the restrictionof scalars of the multiplicative group from a quadratic extension E of a field F to F , and G is the kernel of the norm map from E × to F × . The quotient isan algebraic variety isomorphic to G m (over F ), in such a way that the map X → X/G is the norm map:
Res
E/F G m → G m . However, an F -point of thequotient does not necessarily correspond to a G ( F ) -orbit on X ( F ) = E × ,since the norm map is not surjective on F -points: E × → F × ; rather, itcorresponds to a subvariety of X , defined over F , which is a G -torsor ( G -principal homogeneous space), although it may not have any F -points.Generalizing this observation, for any group variety G acting on a vari-ety X over F , the Y -points of the algebraic stack X/G (where Y is spec F ,or any scheme over it) correspond to (´etale) G -torsors T over Y , togetherwith a G -equivariant map: T → X . This is not a set but a category fiberedover the category of F -schemes Y , and objects in the fiber over spec F (i.e.,“ F -points”) may have a non-trivial group of automorphisms when the map T → X is not an immersion.In § N ofNash manifolds. Again, one should not try to describe “points” of X , sincethey do not form a set; instead, one should describe what is a morphismfrom a Nash F -manifold Y to X — thus obtaining a category fibered over N . For stacks of the form X = X/G with X a Nash manifold and G a Nashgroup acting on it, a morphism from Y to X corresponds to a G -torsor T over Y (in the category of Nash manifolds), together with a G -equivariantmorphism T → X of Nash manifolds. HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 7
General Nash stacks are defined as “geometric stacks in the category N of Nash manifolds”, that is: stacks fibered over this category (endowedwith the semi-algebraic, equivalently the smooth, Grothendieck topology)and admitting a smooth (i.e., submersive) surjective map (“presentation”)from a Nash stack.I also discuss how to obtain a Nash stack from a smooth algebraic stack X over F — this is completely achieved in the case of stacks which are quo-tients of smooth varieties by linear groups, Proposition 2.4.1. The idea hereis to use the groupoid R X ( F ) ⇒ X ( F ) , where X → X is a presentationby a smooth scheme and R X = X × X X , but there is a problem with thisapproach: the Nash groupoid obtained depends on the choice of algebraicpresentation. The reason is easy to see and lies at the heart of the motiva-tion for this paper: as we saw above in the example of the norm map, aquotient algebraic stack X/G has more F -points than can be accounted forby the F -points of X . An F -point of X/G represented by a torsor T and anequivariant map T → X can be accounted for by the F -points of the “pureinner” twist: X T = X × G T , which is a space with an action of the innerform G T = Aut G ( T ) . Thus, we only obtain a Nash stack when there is an“ F -surjective presentation” X → X of the algebraic stack X , i.e., one forwhich the map of isomorphism classes X ( F ) → X ( F ) is surjective. A moregeneral and presentation-free approach to construct a stack is sketched inAppendix A, but I do not know if it always produces a Nash stack.In Section 3, I introduce the Schwartz space of a Nash stack, generaliz-ing the space of (complex-valued) Schwartz measures on a Nash manifold.Measures are better suited than functions for this purpose, because theyhave natural push-forwards, and push-forwards of Schwartz measures un-der smooth (submersive) semi-algebraic maps are also Schwartz. As men-tioned above, the main observation that makes the definition possible isthat Schwartz measures form a cosheaf for the smooth topology (Propo-sition 3.1.2). Thus, we can postulate that the collection of all Schwartzspaces S ( X ) , for all smooth covers X → X of a given Nash stack X “is”its Schwartz cosheaf G X . By the Schwartz space S ( X ) we mean the “globalsections” of this cosheaf, i.e., the (coarse) zeroth homology of the canoni-cal element S • ( X ) in the derived category of nuclear Fr´echet spaces (vectorspaces without topology in the non-Archimedean case) that one obtainsfrom ˇCech homology of this cosheaf. A main result of this section is Theo-rem 3.3.1, which says that this “complex” can be computed from any pre-sentation of the Nash stack, which is a consequence of the fact that theSchwartz cosheaves of Nash manifolds are acyclic for the smooth topology,Proposition 3.1.4. Proposition 3.4.1 and its corollary confirm that, for quo-tient stacks, this definition coincides with the one suggested by Bernstein.As the referee has pointed out, there is a need for extending the notion ofSchwartz space and, in particular, Theorem 3.3.1 to non-trivial “bundles”over a Nash stack in order to include, for example, G -coinvariants (where YIANNIS SAKELLARIDIS G is a Nash group) of the Schwartz sections of a G -equivariant vector bun-dle over a Nash manifold X . Such “quotients” with X/G a (real or p -adic)Nash manifold appear, for example, in Bernstein’s “Frobenius reciprocity”for distributions [14], and extensions of this idea by Baruch [10] and Aizen-bud and Gourevitch [2]. While clearly very important for applications, thestudy of Schwartz sections of non-trivial “bundles” on stacks is beyond thescope of the present article; I hope to return to it in the near future.In the following sections the goal is to define the global “evaluationmaps” for algebraic stacks of the form X/G defined over a global field,where X is smooth affine and G is reductive. When X is a smooth al-gebraic variety defined over a global field k , one has a global Schwartzspace F ( X ( A k )) of functions on the adelic points of X . The difference be-tween measures and functions is not very significant globally, where wehave Tamagawa measures, although it can only be bridged without extradata at the level of “stalks” over a k -point of the invariant-theoretic quo-tient c = X (cid:12) G . Ignoring this technical difficulty for now (for which wewill need the whole Section 4 to carefully develop the appropriate notionsof stalks), we have a global Schwartz space S ( X ( A k )) for a smooth quotientstack X = X/G , which should be the right space of “test functions” for thetrace formula and its generalizations. The “relative trace formula” for X isdefined in § RTF X : f (cid:55)→ (cid:88) x ev x , (1.2)where x runs over isomorphism classes of closed (“semisimple”) k -pointsinto X , and ev x is a certain “evaluation” functional at x .The evaluation functional is the analog of (regularized) orbital integrals:We may assume without loss of generality that the point x corresponds to asemisimple G ( k ) -orbit on X ( k ) , and our evaluation map is a regularizationof the sum of G ( A k ) -orbital integrals corresponding to all G ( k ) -orbits on X ( k ) whose semisimple part (cf. Proposition 4.1.3) is isomorphic to x . Inthe present paper I only define these evaluation maps when the (reductive)stabilizer H of the given semisimple k -point is connected, for reasons ofsimplicity. More importantly, as mentioned earlier, these evaluation mapsare defined whenever there are no “critical exponents”, i.e., no logarithmicdivergence.The definition is given in Section 6 with the help of the linearized versionof the quotient stack, i.e., with the help of Luna’s ´etale slice theorem. In theworld of the trace formula, this linearized version is known as the “Liealgebra version”. Luna’s ´etale slice Theorem 4.1.1 allows us to replace X ,in the neighborhood of the given semisimple k -point x corresponding toa closed G -orbit C on X , by the quotient N C X/G , the normal bundle of C divided by the action of G . Thus, we arrive at stacks of the form V /H ,where V is a representation of a reductive group H (the automorphismgroup of x ), and we have isomorphisms between the stalk of the global HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 9
Schwartz space S ( X ( A k )) around x and the stalk of S ( V ( A k )) around thepoint corresponding to ∈ V . These isomorphisms depend on choices,though, and a lot of care is taken to show that the definitions given areindependent of those choices. Luna’s slice theorem and its corollaries arediscussed in Section 4.In Section 5, the main goal is to define the functional “evaluation at zero”for V /H ; this is the core of the argument, and has nothing to do with stacksand their Schwartz spaces, hence can be read independently. Having donethat, we check in § V /H , therefore we have a well-defined map ev x on theglobal Schwartz space of X (Definition 6.2.3).The regularization technique used here rests upon a description of theasymptotic behavior of the function: Σ V f ( h ) := (cid:88) γ ∈ V ( k ) f ( γh ) on [ H ] = H ( k ) \ H ( A k ) , instead of a description of its truncated orbital inte-grals. The main result, Theorem 5.7.1 says that this function is “asymptot-ically finite”, where “finite” means that the function is, up to a rapidly de-caying one, a generalized eigenfunction with respect to some multiplicativegroup actions. To describe such “multiplicative group actions” one needs atheory of “equivariant toroidal compactifications”. These are blow-ups ofthe “reductive Borel–Serre compactification” which are described by sometoric-type invariants, more precisely: a fan in the Weyl chamber of anti-dominant coweight for H . I think that this description of the function Σ V f is very natural, and it gives rise to a natural notion of regularized orbitalintegral (essentially as the analytic continuation of a Mellin transform). Inote that it is easier to describe the asymptotic behavior of this function, thanit is to describe the asymptotic behavior of its truncated orbital integrals:the behavior of the function is partitioned in cones of the anti-dominantchamber that are dual to the weights of H acting on V , while for the be-havior of truncated orbital integrals one needs several projections of thoseweights onto the faces corresponding to proper parabolics. The argument,which has also been used with truncation methods, essentially rests on thefact that the “nilpotent cone” of V consists of those elements v for whichthere is a cocharacter λ into H with lim t → v · λ ( t ) = 0 ; and that alongeach cocharacter, the function becomes Σ V f becomes multiplicative up toa rapidly decaying function, as can be seen from a simple application of thePoisson summation formula on the vector space V .1.3. Acknowledgements.
This article is dedicated to Joseph Bernstein, indeep appreciation of his mathematical genius and gratitude for everythingthat he has taught me. It is impossible to overstate the impact that his ideas,and his generosity in sharing them, have had on the world of mathematics.It has been the greatest privilege of my mathematical life to have had him as my post-doctoral mentor, besides frequent mathematical conversationsthat I still enjoy with him.I would also like to thank Pierre-Henri Chaudouard for inviting me toParis in January 2015, during which some of the ideas in this paper wereworked out, and for many conversations on the trace formula which provedvery fruitful for the present paper, including references to the work of Ja-son Levy and explanations on the thesis of Michal Zydor. Finally, I thankBrian Conrad, Franc¸ois Loeser and Akshay Venkatesh for helpful feedbackon some questions, Dmitry Gourevitch for several corrections on an earlierdraft, and the referee for a careful reading and many insightful comments.This work was supported by NSF grants DMS-1101471 and DMS-1502270.1.4.
General notation.
Here is a general guide to the notation most fre-quently used. • k is used for a number field, A k for its ring of adeles, F for a local,locally compact field. • S is used for “Schwartz” spaces of complex-valued measures, F isused for “Schwartz” spaces of functions. These are always sectionsof certain cosheaves, and they do not have to be “Schwartz” in theusual sense of rapid decay: in Section 5, I also define extensions ofthese cosheaves, where the measures/functions have “asymptoti-cally finite” behavior. That means that, instead of being of rapiddecay at infinity, they extend over a compactification to sections ofvector bundles determined by characters of tori. • The quotient algebraic stack of an algebraic variety X by a group G ,or the quotient Nash stack of a Nash manifold X by a Nash group G , will simply be denoted by X/G ; it is more usual in the literatureto denote it by [ X/G ] . If X is an affine variety (over, say, the field k ), the invariant-theoretic quotient spec k [ X ] G will be denoted by X (cid:12) G . • I typically use calligraphic letters ( X , Y ...) to denote algebraic stacks,and gothic letters ( X , Y ...) for Nash stacks (or X ( F ) , Y ( F ) , when theNash stack is obtained by taking F -points of an algebraic stack). Fora stack X over some category (such as schemes or Nash manifolds),and an object U in that category, the groupoid of X over U is de-noted by X U , and its set of isomorphism classes by X ( U ) . • If H is an algebraic group defined over k , the automorphic quotientspace H ( k ) \ H ( A k ) is denoted by [ H ] .2. N ASH STACKS
In this section we work over a local field F .2.1. General definition.
When F has characteristic zero, we let N be thecategory of Nash (smooth semi-algebraic) F -manifolds. I point the reader HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 11 to [21, 1] for a discussion and references on Nash manifolds over R . For def-initions that include the complex case, cf. [42, 30]. In the non-Archimedeancase, I do not know any reference for Nash manifolds, but their construc-tion can clearly be performed in exactly the same manner by glueing a finite number of smooth semi-algebraic submanifolds of A n ( F ) = F n as in [1, § submersive (“smooth”) semi-algebraic maps: the image of such a map is also semi-algebraic. Since thisstatement is, to the best of my knowledge, not known, one needs to trans-late all statements to the category of F -analytic manifolds; hence, N willstand for this category, and every reference of “Nash” manifolds should bereplaced by “ F -analytic” manifolds. The restricted topology of open semi-algebraic subsets, to be used for Nash manifolds, should be replaced by theusual topology on F -analytic manifolds.Back to characteristic zero, I summarize the definitions of Nash mani-folds and related notions:2.1.1. Definition. A semi-algebraic subset of A n ( F ) is any subset obtained byboolean combinations (finite intersections, finite unions, complements) ofsubsets of the form { x ∈ F n |∃ y f ( x ) = y m } where f ∈ F [ X , . . . , X n ] , m ∈ N . The above sets generate the closed setsfor a restricted topology on A n ( F ) , the semi-algebraic topology .I remind the reader that “restricted” refers to the fact that infinite unionsof open sets do not need to be open. Any references to “closed”, “open”sets and to “sheaves” will refer to this topology, unless otherwise specified.Notice that we get all polynomial equalities by setting m = 0 , and inthe real case we get all polynomial inequalities from taking complementsof the condition: ∃ y f ( x ) = y . In the complex case, on the other hand,the allowed inequalities can all be written in the form f ( x ) (cid:54) = 0 , and hencein that case semi-algebraic subsets of A n ( F ) are precisely the constructiblesubsets. For inequalities derived from the defining condition in the non-Archimedean case, s. [20, Lemma 2.1].2.1.2. Definition. A smooth semi-algebraic or Nash submanifold M of A n ( F ) isa closed semi-algebraic subset which is also an F -analytic submanifold. Itis equipped with the induced semi-algebraic restricted topology, and thesheaf O M of smooth semi-algebraic or Nash functions , i.e., those F -valued I will often be using A n , the symbol for affine space, to emphasize the semi-algebraicstructure that is deduced from the algebraic one. functions which are F -analytic and whose graph is a semi-algebraic sub-set of M × A ( F ) .2.1.3. Definition. A Nash F -manifold is an F -analytic manifold M equippedwith a restricted topology and a sheaf O M of F -valued functions, which ad-mits a finite open cover M = (cid:83) i M i such that each ( M i , O M ) is isomorphicto a Nash submanifold of A n ( F ) together with its sheaf of Nash functions.A morphism: X → Y of Nash manifolds is a Nash map between them, i.e.,an F -analytic map whose graph is a semi-algebraic subset of X × Y .In the complex case, Nash manifolds are simply smooth algebraic vari-eties (of finite type over C ), and Nash maps/functions are algebraic.We now proceed to define Nash stacks. The approach is analogous to thedefinition of differentiable stacks in [11].We consider the category N of Nash F -manifolds, where the morphismsare Nash maps, as a site, equipped with the Grothendieck topology gener-ated by covering families { U i → X } i ∈ I such that:(i) the set of indices I is finite;(ii) the (Nash) maps U i → X are ´etale, that is: local diffeomorphisms;(iii) the total map (cid:70) U i → X is surjective.In any case other than the complex case, the second condition can bereplaced by the condition that the maps are open embeddings, since ev-ery ´etale morphism becomes an open embedding over elements of a finiteopen cover. In fact, it is true, more generally, over R or p -adic fields (see[2, Appendix A] for R , and it can similarly be proven for p -adic fields) thatevery smooth — that is: submersive — semi-algebraic map between Nashmanifolds admits local Nash sections over a finite open cover of the image.In the complex (algebraic) case, any smooth map is known to admit sec-tions locally in the ´etale topology. Hence, in every case, this Grothendiecktopology is equivalent to the Grothendieck topology generated by smoothcovers, i.e., the one obtained by replacing the second condition above bythe requirement that the maps U i → X are smooth.For a covering family as above, and an ordered set of indices ( i , i , . . . , i r ) we will be denoting U i × U × · · · × U U i r by U i i ··· i r .Let X → N be a category fibered in groupoids. Recall that this meansthat:(i) for every arrow V → U in N , and every object x of X lying over U ,there exists an arrow y → x in X lying over V → U ;(ii) for every diagram of the form W → V → U in N and arrows in X : z → x lying over W → U and y → x lying over V → U , there existsa unique arrow z → y lying over W → V , such that the composition z → y → x equals the given z → x .The full subcategory of objects in X over a given object U in N (the fiber of X over U ) is denoted by X U , and its set of isomorphism classes by X ( U ) . Anobject u ∈ X U will sometimes be denoted by u U or ( U, u ) , and thought of as HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 13 a -morphism in the -category of categories fibered in groupoids over N : u : U → X . The axioms imply that for any morphism V → U in N and any x U ∈ ob( X U ) , the base change x V → x U , x V ∈ ob( X V ) is defined up to isomor-phism. Categories fibered in groupoids over N form a -category, with -morphisms being the functors which are compatible with the “structuremorphism” to N , and -morphisms being isomorphisms between thosefunctors. Fiber products are defined in a standard way [33, § (2.2.2)] for anypair of -morphisms X → Y ← X (cid:48) , with objects of X × Y X (cid:48) over U ∈ ob( N ) being the triples: ( x, x (cid:48) , g ) , with x ∈ ob( X U ) , x (cid:48) ∈ ob( X (cid:48) U ) and g an arrow(isomorphism) between the image of x and the image of x (cid:48) in Y U .Recall that X is called a stack if “both the isomorphisms and the objectsare glued from local data” (with respect to the given Grothendieck topol-ogy), that is:(i) For every object U in N and two objects x, y in X U , the pre-sheaf ofisomorphisms between x and y : Isom( x, y ) : N U → ( Set )( V → U ) (cid:55)→ Hom X V ( x V , y V ) is a sheaf. That is, any isomorphism φ : x → y is uniquely de-termined by its restrictions φ i : x U i → y U i to any covering family { U i → U } i , and vice versa any family of such isomorphisms φ i with φ i | U ij = φ j | U ij : x U ij → y U ij determines an isomorphism φ : x → y .(This condition defines a pre-stack .)(ii) Given { U i → U } i a covering family, x i ∈ ob( X U i ) and morphisms: φ ij : x i,U ij → x j,U ij whose restrictions to U ijk satisfy the cocycle con-dition: φ jk ◦ φ ij = φ ik , there exists an object x in X U and a family ofisomorphisms φ i : x U i → x i , such that φ j | U ij = φ ji ◦ ( φ i | U ij ) . Theobject x and the isomorphisms φ i are necessarily unique up to uniqueisomorphism, by (i).We say that a stack as above is representable if there is a Nash manifold X ,viewed as the stack Hom( • , X ) over N , together with an equivalence: X ∼ −→ X . We say that a -morphism of stacks: X → Z is smooth , or a representablesubmersion , if for every morphism: U → Z , where U ∈ ob( N ) viewed as astack, the fiber product X × Z U is representable by a Nash manifold, andthe induced morphism: X × Z U → U is a submersion.2.1.4. Remark.
As in [11], we first define representable submersions, beforedefining representable morphisms in general; the reason is that the abovedefinition cannot be used for a general morphism: X → Z of Nash man-ifolds. Indeed, for another morphism U → Z , the fiber product X × Z U is not necessarily defined as a Nash manifold. It is, however, if X → Z is The definition of this pre-sheaf requires choices of base changes, cf. [41, Tags 02Z9 and02XJ]. The conditions of it being a sheaf are independent of choices, however. smooth, i.e., a submersion. (This is a difficulty that does not exist, for exam-ple, in the category of schemes, but does exist in other geometric categories,such as differentiable manifolds.)A -morphism f : X → Y of stacks over N is called an epimorphism if every y ∈ Y U , where U is a Nash stack, lifts ´etale-locally to X , that is:there is a covering family (consisting, without loss of generality, of a singleelement) ( U (cid:48) → U ) and an object x ∈ X U (cid:48) such that for all V → U (cid:48) , f ( x V ) (cid:39) y V in Y V .2.1.5. Definition. A Nash , or smooth semi-algebraic stack is a stack X as abovewhich admits a representable, epimorphic submersion from a Nash mani-fold: X → X (2.1)(where X is a Nash manifold).We will call such an epimorphism a presentation of X .We call a -morphism: X → Z of Nash stacks representable if for one,equivalently any, presentation V → Z the stack X × Z V is representable.A smooth, ´etale etc. morphism of stacks is a representable morphism suchthat for every U → Z the morphism X × Z U → U has the said property.2.1.6. Lemma.
Let X → Z be a smooth (submersive) -morphism, and Y → Z any -morphism of Nash stacks.Then the fiber product X × Z Y is a Nash stack.Proof. Indeed, if Y → Y is a presentation of Y , then it is easy to see that X × Z Y , which by assumption is a Nash manifold, gives a presentation of X × Z Y . (cid:3) Nash groupoids.
Assume that [ R X ⇒ X ] is a groupoid object in N .This consists of a pair of smooth (submersive) morphisms between Nashmanifolds (denoted s and t for “source” and “target”), with a “multipli-cation”: R X × t,X,s R X → R X , an identity section and an inverse map (allmaps assumed to be Nash), satisfying axioms analogous to the usual groupaxioms.There is a Nash stack X associated to such a groupoid object, defined asfollows: the fiber of X over U ∈ ob( N ) has as objects all R X -torsors over U in the sense of [37, § T → U , together with a Cartesian map of groupoids [ T × U T ⇒ T ] → [ R X ⇒ X ] . I will usually denote such an object by the pair of maps ( T → U, T → X ) ,and sometimes just by T → U , or by T .2.2.1. Proposition.
Let X be a Nash stack, and X → X a presentation. Set R X = X × X X . Then X is canonically (up to -isomorphism) isomorphic to theNash stack associated to the groupoid object [ R X ⇒ X ] . HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 15
Proof.
The proof is identical to the one in the case of algebraic or differen-tiable stacks. I recall the construction of the -morphisms:On the one hand, we have an -morphism from X to [ R X ⇒ X ] by as-signing to every u ∈ X U the fiber product T = U × u, X X . Since X → X isan epimorphism, it follows that T is an R X -torsor over U .Vice versa, even without the epimorphism assumption we have an -morphism in the opposite direction: for U ∈ ob( N ) and an R X -torsor T → U choose, over an ´etale cover U (cid:48) → U , a section U (cid:48) → T . The compositionwith the map to X and then to X gives an object in X U (cid:48) , which by the stackaxiom can be seen do descend to an object in X U . (cid:3) Remark.
In the complex case, F = C , there is no difference betweenNash and smooth algebraic stacks of finite type over C . This follows, forexample, from the description of a Nash stack in terms of groupoid objectsand the fact that all Nash manifolds are algebraic, and all Nash maps arealgebraic.The following closely related lemma will be useful later:2.2.3. Lemma.
Let X (cid:48) → X be a smooth epimorphism of Nash manifolds, andlet [ R X ⇒ X ] be a groupoid object. Let [ R X (cid:48) ⇒ X (cid:48) ] be obtained by base change,that is: R X (cid:48) = X (cid:48) × X,s R X × t,X X (cid:48) . Then the Nash stacks defined by the twogroupoids are canonically equivalent.Proof. Again, I will just describe the -morphisms in terms of objects.Given U ∈ ob( N ) and an R X -torsor T → U , we obtain an R X (cid:48) -torsor T (cid:48) → U by base change: T (cid:48) = T × X X (cid:48) .Vice versa, given an R X (cid:48) -torsor T (cid:48) → U assume, at first, that it admits asection U → T (cid:48) . Let α : U → X (cid:48) be the map obtained by composing with T (cid:48) → X (cid:48) , and β the map obtained by further composing with X (cid:48) → X . We let T = U × β,X,s R X . It is an R X -torsor over U . I claim that, canonically, T (cid:48) (cid:39) T × X,s R X (cid:48) . (2.2)Indeed, the R X (cid:48) -action and the section define an isomorphism U × α,X (cid:48) ,s R X (cid:48) ∼ −→ T (cid:48) . Moreover, R X (cid:48) = X (cid:48) × X,s R X × t,X X (cid:48) because the same holds as schemes,and the claim now follows easily.For the general case, the map T (cid:48) → U always admits a section locally inthe ´etale (or semi-algebraic, if F (cid:54) = C ) topology, and the canonical isomor-phisms (2.2) (or even just the map T (cid:48) → T ) suffice to glue them, becauseevery local trivialization for T (cid:48) induces one for T . (cid:3) From smooth algebraic to Nash stacks.
Let X be a smooth algebraicstack over F . By this I will mean, throughout, that it is an algebraic stackover F which admits a presentation as a smooth groupoid [ R X ⇒ X ] ,where X and R X = X × X X are smooth schemes of finite type over F .I would like to attach to it a Nash stack X , also denoted by X ( F ) . (The no-tation X ( F ) really stands for the isomorphism classes in the groupoid X F ofpoints spec F → X , but we may think of this set as having the richer struc-ture of a Nash stack, just as we think of the set X ( F ) as a Nash manifold,when X is a smooth algebraic variety over F ; thus, this notation shouldnot cause confusion.) I do not quite achieve that in full generality, becauseI do not know if there is always a presentation of an algebraic stack over F to which all F -points lift. This is true in many cases of interest, such asquotients of smooth varieties by linear algebraic groups, and in any casewe can always give to X ( F ) the structure of a stack over N , whether thereis such a presentation or not.To do that, we need to define what is a (“semi-algebraic”) morphismfrom a Nash manifold U to X . I follow the standard approach of [37, § X as a quotient of a smooth groupoid.However, in contrast to [37], we are not working over an algebraicallyclosed field; as a consequence, once we have properly defined the stack X , for an epimorphism X → X of algebraic stacks, where X is a smoothvariety over F , it does not necessarily induce an epimorphism X ( F ) → X of stacks over N . Thus, we cannot work with an arbitrary presentation X → X and hope that every morphism U → X (where U is a Nash mani-fold) will produce a surjective map: ( U × X X ( F )) → U .For example, for X = spec F/G with the standard presentation: spec F π −→ spec F/G , where G is an algebraic group with H ( F, G ) (cid:54) = 0 , we shouldclearly have a morphism of Nash stacks: pt α −→ X induced from the al-gebraic morphism: spec F α −→ spec F/G corresponding to a non-trivial G -torsor; however, the fiber product pt × α, X ,π pt is empty, because this is the case for the F -points of the algebraic fiber prod-uct spec F × α, X ,π spec F. Thus, to define the Nash stack X we will either have to choose a presen-tation which is “surjective on F -points”, or take some kind of limit, overall possible presentations X → X of the algebraic stack X , of the Nashstacks associated to the groupoids [( X × X X )( F ) ⇒ X ( F )] . I follow thefirst approach, which is simpler although not very nice, because it relies onchoosing a presentation (if a presentation which is surjective on F -pointsexists). Of course, it turns out that the resulting stack does not dependon the F -surjective presentation chosen. In Appendix A, I will outline amore canonical approach which produces a stack over N for every smooth HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 17 algebraic stack of finite type over N ; this poses the interesting problem ofdetermining, in general, when the resulting stack is Nash.Hence, let X = a smooth algebraic stack over F , and assume that it ad-mits a presentation X → X with the property that X ( F ) → X ( F ) (= the setof isomorphism classes of objects in X F ) is surjective. We will call such apresentation “ F -surjective ”.2.3.1. Lemma.
Let X → X , X (cid:48) → X be two algebraic presentations of X suchthat the induced maps X ( F ) → X ( F ) , X (cid:48) ( F ) → X ( F ) (2.3) are surjective.Let R X = X × X X , R X (cid:48) = X (cid:48) × X X (cid:48) (fiber products as algebraic stacks). Thenthe Nash groupoids [ R X ( F ) ⇒ X ( F )] , [ R X (cid:48) ( F ) ⇒ X (cid:48) ( F )] are canonically equivalent (up to 2-isomorphism).Proof. The equivalence implied in the lemma is induced from the obviousmorphisms from the third groupoid [ R X (cid:48)(cid:48) ( F ) ⇒ X (cid:48)(cid:48) ( F )] , where X (cid:48)(cid:48) = X × X X (cid:48) . Notice that since both maps (2.3) are surjective, the same holds forthe corresponding map from X (cid:48)(cid:48) ( F ) , and also for the maps from X (cid:48)(cid:48) ( F ) to X ( F ) , X (cid:48) ( F ) .This reduces us to the case when we have smooth, F -surjective epimor-phisms X (cid:48) → X → X , in which case the claim is Lemma 2.2.3. (cid:3) We can now define:2.3.2.
Definition.
Given a smooth algebraic stack X of finite type over F ,which admits an F -surjective presentation, the associated Nash stack X = X ( F ) is “the” Nash groupoid [ R X ( F ) ⇒ X ( F )] , where X → X is any F -surjective presentation.The association X (cid:55)→ X ( F ) is a functor from the 2-category of smoothstacks of finite type over F which admit F -surjective presentations to thecategory of Nash stacks over F . Indeed, as in the proof of Lemma 2.2.3,given a morphism X → Y of two algebraic stacks, and F -surjective presen-tations X → X , Y → Y , a morphism U → X ( F ) lifts ´etale-locally to X ( F ) ,and then in turn to ( X × Y Y )( F ) . As in the lemma, it can be seen that thisgives rise to a well-defined morphism U → Y ( F ) .The functor preserves fiber products: Proposition.
Let
X → Z ← Y be morphisms of smooth algebraic stacks offinite type over F , all of which admit F -surjective presentations. Then we have ( X × Z Y )( F ) (cid:39) X ( F ) × Z ( F ) Y ( F ) canonically, up to -isomorphism.Proof. The -morphism from ( X × Z Y )( F ) to X ( F ) × Z ( F ) Y ( F ) is clear.Let us construct its quasi-inverse. First, we notice that the proposition isclearly true iff all three stacks are schemes. We will reduce the general caseto that. Denote by X → X , Y → Y , Z → Z three F -surjective presentations.Recall that an object in X ( F ) × Z ( F ) × Y ( F ) over U ∈ ob( N ) consists ofmorphisms x : U → X ( F ) , y : U → Y ( F ) together with an isomorphism oftheir compositions with the maps to Z ( F ) .By definition of the functor on -morphisms, the composition U → X ( F ) →Z ( F ) is obtained, ´etale-locally, by a map α : U → ( X × Z Z )( F ) . (We assume that all statements that hold ´etale-locally hold already over ourNash manifold U , in order to simplify notation.) Similarly for the compo-sition U → X ( F ) → Z ( F ) : β : U → ( Y × Z Z )( F ) . Now, these maps depend on choices of sections, and it is not necessarilythe case that composing with projection to Z ( F ) the two maps will coin-cide. (Denote these compositions by ¯ α , ¯ β .) However, the isomorphism oftheir compositions with the maps to Z ( F ) is an isomorphism between the R Z ( F ) -torsors T := U × ¯ α,Z ( F ) ,s R Z ( F ) (cid:39) U × ¯ β,Z ( F ) ,s R Z ( F ) , where these isomorphisms are over Z ( F ) with respect to the “target” map t : R Z ( F ) → Z ( F ) .Therefore, the compositions of the two sequences of maps below coin-cide: T α × Z ( F ) ,s R Z ( F ) −−−−−−−−−−→ ( X × Z Z )( F ) × Z ( F ) ,s R Z ( F ) = ( X × Z R Z )( F ) t −→ t −→ ( X × Z Z )( F ) → Z ( F ) ,T β × Z ( F ) ,s R Z ( F ) −−−−−−−−−−→ ( Y × Z Z )( F ) × Z ( F ) ,s R Z ( F ) = ( Y × Z R Z )( F ) t −→ t −→ ( Y × Z Z )( F ) → Z ( F ) . Hence we get a map T → ( X × Z Z )( F ) × Z ( F ) ( Y × Z Z )( F ) = ( X × Z Y × Z Z )( F ) . Taking a section, ´etale-locally, of the map T → U , and composing withthe map ( X × Z Y × Z Z )( F ) → ( X × Z Y )( F ) , we obtain that the map HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 19 U → X ( F ) × Z ( F ) Y ( F ) factors ´etale-locally through ( X × Z Y )( F ) , andhence factors through ( X × Z Y )( F ) . (cid:3) Quotients by linear groups.
Consider the case of an algebraic quo-tient stack X = X/G , where X is a smooth variety over a local field F and G is a (smooth) linear algebraic group acting on it.The Nash stack associated to the Nash groupoid [ X ( F ) × G ( F ) ⇒ X ( F )] will be denoted by X ( F ) /G ( F ) .Let T be a G -torsor, and set X T = X × G T , G T = Aut G ( T ) (an inner formof G ). Then X T carries a G T -action, and we have a canonical equivalenceof algebraic stacks: X T /G T (cid:39) X/G. (2.4)2.4.1.
Proposition.
The algebraic stack X admits an F -surjective presentationand hence X = X ( F ) is a Nash stack. It is canonically equivalent to the Nashstack X ρ ( F ) / GL N ( F ) , for any faithful representation ρ : G (cid:44) → GL N (where N ∈ Z > ) and X ρ = X × G GL N .Equivalently, via (2.4) it is canonically equivalent with the disjoint union (cid:71) T X T ( F ) /G T ( F ) , (2.5) where T runs over a set of representatives for the isomorphism classes of G -torsorsover F . Notice that X T ( F ) could be empty.2.4.2. Remark.
In characteristic zero, the set H ( F, G ) classifying G -torsors isfinite, thus the above disjoint union is finite and, hence, a Nash manifold.In positive characteristic, any linear group can be embedded in a groupwith finite (or even trivial) cohomology, but since we are working with F -analytic manifolds in this case, we do not need the above disjoint union tobe finite. Proof. If ρ : G (cid:44) → GL N is a faithful representation and X ρ is as in thestatement of the proposition, we have a canonical equivalence of algebraicstacks: X = X/G = X ρ / GL N . (2.6)Because GL N has trivial Galois cohomology (Hilbert 90), the set X ρ ( F ) sur-jects onto the set of isomorphism classes of F -points of X . Therefore, X is awell-defined Nash stack.Alternatively, we observe that the presentation ˜ X := (cid:70) T X T → X issurjective on F -points, and hence we get that X is equivalent to the stack Throughout the paper we will be using the same notation for a right G -torsor and thecorresponding left G -torsor obtained by inverting the G -action; for example, T × G T iscanonically the trivial G -torsor. defined by the Nash groupoid [ R ˜ X ( F ) ⇒ ˜ X ( F )] , where R ˜ X = ˜ X × X ˜ X .For non-isomorphic torsors T , T , the images of X T ( F ) and X T ( F ) in X ( F ) are disjoint (consist of different isomorphism classes), and thereforewe have R ˜ X ( F ) = (cid:71) T R X T ( F ) , where R X T = X T × X X T . (cid:3)
3. L
OCAL S CHWARTZ SPACES
Schwartz measures on Nash manifolds.
In what follows we will needto do some homological algebra in the quasi-abelian category of nuclearFr´echet spaces; I point the reader to [40, 18, 22] for presentations of thetopic. The basic notions have been summarized in Appendix B. We recallthat a complex of nuclear Fr´echet spaces is strictly exact if the set-theoretic images and kernels coincide; in particular, such a complex is made up of strict morphisms, i.e., morphisms with closed image. Derived categoriesof a quasi-abelian category are well-defined, by localizing the homotopycategory of complexes with respect to the null system of strictly exact com-plexes. Two complexes are said to be (strictly) quasi-isomorphic if they havethe same image in the derived category; equivalently, if they are dominatedby a third one by maps whose mapping cones are strictly exact. In the non-Archimedean case, of course, the corresponding spaces are simply vectorspaces without topology, and any mention of Fr´echet spaces should be ig-nored. The space of Schwartz functions on Nash manifolds is well-known, andhas been studied as a cosheaf on the restricted topology of semi-algebraicopen sets in [21, 1]. Again, in Appendix B, I recall the basic notions that wewill be using about cosheaves. In the Archimedean case, the sections of theSchwartz cosheaf over an affine open semi-algebraic set U are those func-tions f on U such that Df is bounded for every Nash differential operatoron U . For the notions of Nash differential operators, densities, measuresetc. cf. [1, § flabby : extension maps are strictand injective. Hence (cf. Lemma B.2.3), it is acyclic for the semi-algebraictopology.3.1.1. Remark.
For a finite extension F (cid:48) /F of local fields, and a Nash F (cid:48) -manifold X , the space of Schwartz functions on X does not change if weconsider X as a Nash F -manifold, and more generally its cosheaf of Schwartzfunctions for the F (cid:48) -semi-algebraic topology is simply the restriction of thecorresponding cosheaf for the F -semi-algebraic topology. This, togetherwith an acyclicity result that we will prove (Proposition 3.1.4), will render Unless one wants to work with almost smooth functions, cf. [39, Appendix A].
HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 21 it indifferent for Schwartz spaces whether a Nash F (cid:48) -manifold, or stack, isconsidered as an F (cid:48) -manifold/stack or an F -manifold/stack.Functions do not have well-defined push-forwards, and therefore wewill instead be working with spaces of Schwartz measures : these are prod-ucts of Schwartz functions by Nash measures. In the non-Archimedeancase, Schwartz functions are by definition locally constant and compactlysupported, and hence so are Schwartz measures. We will be denoting thespace of Schwartz measures on a Nash manifold X by S ( X ) . I will nowshow that Schwartz measures actually form an acyclic cosheaf in the smooth topology over a Nash manifold.Let N ∞ be the category whose objects are Nash manifolds (that is, theobjects of N ) and whose morphisms are only those morphisms of Nashmanifolds which are smooth (=submersive). Consider N ∞ as a site, withthe Grothendieck topology generated by smooth covers, that is collectionsof smooth morphisms { p i : U i → X } such that (cid:70) i p i : (cid:70) i U i → X is surjec-tive.3.1.2. Proposition.
The spaces of Schwartz measures form a cosheaf of nuclearFr´echet spaces on N ∞ , with strict push-forward maps. That is, every smooth mor-phism π : X → Y of Nash manifolds induces, functorially, a strict morphism: π ! : S ( X ) → S ( Y ) , and if π is surjective, it induces a coequalizer diagram: S ( X × Y X ) s ! ⇒ t ! S ( X ) π ! −→ S ( Y ) . (3.1)The proof will rely on the following:3.1.3. Proposition.
Let F (cid:54) = C . Let f : X → Y be a smooth morphism of Nash F -manifolds. We can cover X by a finite number of open subsets U i such that foreach i the restriction of f to U i factors through an open embedding e i : U i (cid:44) → Y × A n ( F ) and the projection to Y .Moreover, we may choose the data so that the image of e i contains the zerosection: f ( U i ) × { } ; in particular, there are smooth local sections of f : f ( U i ) → U i .Finally, choosing the data this way, there is a(n F -valued) Nash function T i on f ( U i ) , for every i , with the property that the tubular neighborhood B T i := { ( y, v ) ∈ f ( U i ) × F n | | T i ( y ) | · (cid:107) v (cid:107) ≤ } , (3.2) where (cid:107) • (cid:107) is some fixed norm on F n , is contained in e i ( U i ) . Notice that the first statement is the analog of the ´etale factorization ofsmooth morphisms in algebraic geometry, though here we can replace ´etaleby an open embedding, because (as a special case of this proposition) ev-ery ´etale map admits semi-algebraic local sections (when F (cid:54) = C ). We noticethat the case F = C is excluded, because the semi-algebraic topology is notsufficiently fine in this case; most importantly, sets of the form | x | < c in A ( C ) are not semi-algebraic; however, we will be able to use this proposi-tion by restriction of scalars from C to R , when we need it. Proof.
Starting from the last statement, the existence of such tubular neigh-borhoods is [1, Theorem 3.6.2] over R .Using this, Aizenbud and Gourevitch prove the existence of local sec-tions in [2, 2.4.3]. In the course of the proof of this result, namely in theproof of [2, Proposition A.0.4], they show the existence of local open em-beddings U i (cid:44) → Y × A n ( F ) .The proofs in the non-Archimedean case are similar, but easier. (cid:3) Proof of Proposition 3.1.2.
We may, and will, assume that F (cid:54) = C , by con-sidering any complex Nash manifolds X and Y as in the statement of theproposition as Nash R -manifolds.Since it is known that Schwartz spaces form a cosheaf (clearly with strictpush-forward maps) for the restricted topology of semi-algebraic open cov-ers, we may replace X by a cover U i such as in the previous proposition,and Y by the image of U i . Thus, we may assume that Y × { } ⊂ X ⊂ Y × A n ( F ) , with the map f being the standard projection to Y . We also fix a Nashfunction as in the proposition (here denoted T ), such that X ⊃ B T . We willdenote the set B T by X T , in order to remember that it is a subset of X . Wemay assume that T is bounded away from zero, so that its absolute valueis a smooth function on Y .For a direct product of Nash manifolds Z × U it is known that S ( Z × U ) = S ( Z ) ˆ ⊗S ( U ) . Thus, the push-forward of measures in S ( Z × U ) to Z clearlylands in S ( Z ) , and the map S ( Z × U ) → S ( Z ) is a strict epimorphism.In our setting, since S ( X ) ⊂ S ( Y × A n ( F )) , we get that push-forward ofSchwartz measures on X is a continuous map into Schwartz measures on Y . On the other hand, the Nash function T essentially allows us to treata subset of X as a product space. Namely, we can fix a Schwartz measure µ on the open unit ball in F n with total mass , and then notice that for agiven f ∈ S ( Y ) the measure ( y, a ) (cid:55)→ f ( y ) µ ( T ( y ) a ) ∈ S ( Y × A n ( F )) actually belongs to S ( X T ) and its push-forward to Y is f . Thus, the map S ( X ) → S ( Y ) is a strict epimorphism.Now consider the diagram (3.1). We have π ! ◦ s ! = π ! ◦ t ! since the push-forward of measures corresponding to a commutative diagram of spaces iscommutative. We will show that any measure f in the kernel of S ( X ) →S ( Y ) is of the form ( s ! − t ! ) F , for F ∈ S ( X × Y X ) .It is known [3, Theorem A.1.1] that a Schwartz function can be writtenas a product of two Schwartz functions. In fact, there is a positive Schwartzfunction φ on Y and an f ∈ S ( X ) such that f = π ∗ φ · f . Notice that π ! f is also zero. HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 23
Choose a nowhere vanishing, positive smooth measure of polynomialgrowth µ on Y , and let S (cid:48) ( X ) be “the quotient of elements of S ( X ) by µ ”,by which I just mean that push-forwards of elements of S (cid:48) ( X ) to Y willbe thought of as Schwartz functions by dividing by µ . (There is a canonicalway to think of S (cid:48) ( X ) as some space of densities on X which are “measuresin the vertical direction and functions in the horizontal”.)Choose h ∈ S (cid:48) ( X ) with π ! h = φ as Schwartz functions on Y . The product F ( x , x ) = f ( x ) h ( x ) is a well-defined Schwartz measure on X × Y X ,and we have ( s ! − t ! ) F ( x ) = f ( x ) π ! h ( π ( x )) − π ! f ( π ( x )) h ( π ( x )) . We have π ! f = 0 and π ! h = φ , therefore we get: ( s ! − t ! ) F ( x ) = f ( x ) φ ( π ( x )) = f ( x ) . This proves the cosheaf condition. (cid:3)
Consider, for every X ∈ ob( N ) , the smooth site X ∞ over X , whose ob-jects are all smooth morphisms Y → X from other Nash manifolds, a mor-phism from ( Y → X ) to ( Z → X ) is a smooth map Y → Z which commuteswith the maps to X , and covering families are finite families { U i → X } i which are jointly surjective. The site N ∞ is therefore the smooth site of X = a point.The cosheaf of Schwartz functions on N ∞ induces, by restriction, a cosheafon X ∞ . We will be denoting this cosheaf by S X , to distinguish it from thespace S ( X ) of its global sections. Since S X ( U ) = S ( U ) for every smooth U → X , i.e., this space depends only on U and not on X , we may also dropthe index X and denote this cosheaf by S , without creating any confusion.The following is the “Poincar´e lemma” stating that this cosheaf is (strictly)acyclic over any Nash manifold.3.1.4. Proposition.
For every smooth morphism π : X → Y of Nash mani-folds, denote by [ X ] nY the n -fold fiber product X × Y X × · · · × Y X and let ∂ n : S ([ X ] n +1 Y ) → S ([ X ] nY ) be the alternating sum of push-forwards: f (cid:55)→ n (cid:88) i =0 ( − i π i, ! f, where π i : [ X ] n +1 Y → [ X ] nY is the omission of the i -th variable (counting fromzero).For any smooth morphism X → Y the sequence · · · ∂ n −→ S ([ X ] nY ) ∂ n − −−−→ · · · S ( X × Y X ) ∂ −→ S ( X ) → is strictly exact.Proof. The proof is very similar to that of Proposition 3.1.2: Let f ∈ S ([ X ] nY ) be in the kernel of ∂ n − and write it as f ( x ) = φ ( π ( x )) f ( x ) ( x ∈ [ X ] n ) where φ ∈ S ( Y ) and f is also in the kernel of ∂ n − . (We denoteby π the map to Y , for any fiber power of X over Y .)Choose h ∈ S ( X ) with π ! h = φ .Let F ∈ S ([ X ] n +1 Y ) be the product: F ( x , x ) = h ( x ) ⊗ f ( x ) , ( x , x ) ∈ [ X ] n +1 Y . Then ∂ n F ( x ) = π ! h ( π ( x )) f ( x ) − h ⊗ ∂ n − f ( x ) = φ ( π ( x )) f ( x ) − f ( x ) . (cid:3) As we will see, acyclicity fails when we replace Nash manifolds by gen-eral Nash stacks.3.2.
The Schwartz cosheaf on a Nash stack.
Let X be a Nash stack. The smooth site X ∞ on X has as objects those objects ( U, u ) ∈ ob( X ) which rep-resent smooth -morphisms of Nash stacks u : U → X . A morphism from ( U, u ) to ( V, v ) will be a morphism: u → v in X which is smooth; this isequivalent to requiring that the induced morphism: U → V be smooth(since u → v is automatically a base change of the latter).This category is endowed with the coverage whose elements over ( U, u ) are finite collections of smooth morphisms { φ i : u i → u } i such that theinduced morphism of Nash manifolds (cid:71) i φ i : (cid:71) i U i → U is surjective.The assignment ( U, u ) (cid:55)→ S X (( U, u )) := S ( U ) is a pre-cosheaf on thissite, i.e., a functor that associates to every morphism π : ( U, u ) → ( V, v ) thepush-forward map π ! : S ( U ) → S ( V ) .3.2.1. Proposition.
The pre-cosheaf S X is a cosheaf on X ∞ . Moreover, for every(representable) smooth -morphism of Nash stacks φ : X → Z there is a canonicalidentification of cosheaves: S X (cid:39) φ − S Z . (3.3)The last statement allows us to denote the cosheaf S X simply by S . Proof.
The cosheaf property follows from the cosheaf property that has al-ready been established for Nash manifolds, Proposition 3.1.2.A smooth -morphism of Nash stacks is, in particular, a functor betweencategories fibered over N , which preserves the classes of smooth objects.Hence, any ( U, u ) ∈ ob( X ∞ ) induces ( U, u (cid:48) ) ∈ ob( Z ∞ ) .The identification (3.3) is simply the identification: S X (( U, u )) = S ( U ) = S Z (( U, u (cid:48) )) . (cid:3) The fact that S (( U, u )) is equal to the Schwartz space of the Nash man-ifold U does not mean that there is no new information contained in thestack. Indeed, we have a “copy” of S ( U ) for every u : U → X , and thecopies of u, u (cid:48) are identified with each other only when there is a morphism HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 25 u → u (cid:48) ; this includes the automorphisms of u , which induce the identity on S ( U ) .3.3. Resolutions.
Consider now a presentation of a Nash stack X (cid:16) X . It induces a simplicial object in Nash manifolds: [ X ] • X : · · · ... ⇒ ... [ X ] n X ... ⇒ ... · · · X × X X ⇒ X, where [ X ] n X is the fiber product of n copies of X over X .By taking Schwartz spaces, this induces a simplicial object in the cate-gory of nuclear Fr´echet spaces (with strict morphisms) and the correspond-ing long sequence (as in Proposition 3.1.4): · · · ∂ n −→ S ([ X ] n X ) ∂ n − −−−→ · · · S ( X × X X ) ∂ −→ S ( X ) ∂ −→ . (3.4)3.3.1. Theorem.
For any two presentations of X , the complexes (3.4) are canon-ically strictly quasi-isomorphic. In particular, we get a well-defined object S • ( X ) (up to unique isomorphism) of the derived category D − ( F ) of nuclear Fr´echetspaces (vector spaces without topology, in the non-Archimedean case) concentratedin non-positive degrees. The association X (cid:55)→ S • ( X ) is functorial with respect tosmooth -morphisms of Nash stacks. The quotient S ( X ) /∂ S ( X ) will simply be denoted by S ( X ) and called“the Schwartz space of X ” or the space of “global sections” of its Schwartzcosheaf. Notice that this is simply a coarse version of the “zeroth homol-ogy” of the complex S • ( X ) because, at least a priori, the image of S ( X ) by ∂ may not be closed. Proof.
The quasi-isomorphism for any two presentations follows formallyfrom the acyclicity of Schwartz spaces for Nash manifolds, Proposition3.1.4, by the usual argument for ˇCech cohomology: For any two presenta-tions X → X , X (cid:48) → X , we form the double complex K ij = S ([ X ] i X × X [ X (cid:48) ] j X ) .Notice that [ X ] i X × X [ X (cid:48) ] j X is the i -fold fiber product of X × X [ X (cid:48) ] j X over [ X (cid:48) ] j X .Thus, the rows and columns of the double complex (except for the zerothrow and zeroth column) are strictly exact by Proposition 3.1.4, and by thestandard argument we get a quasi-isomorphism between the zeroth rowand the zeroth column.Smooth -morphisms X → Z preserve the corresponding smooth sites,and hence functoriality is clear. (cid:3) Remark.
It immediately follows from the definitions and Remark 3.1.1that for a finite extension F (cid:48) /F of local fields and a Nash stack X over F (cid:48) ,the cosheaf S X is simply the restriction to all smooth morphisms X → X of Nash F (cid:48) -manifolds of the corresponding cosheaf when X is considered as aNash stack defined over F .Moreover, by applying Theorem 3.3.1 to a presentation X → X of Nash F (cid:48) -manifolds, we see that the element S • ( X ) in the derived category andthe space S ( X ) are independent of whether we consider X as a Nash F (cid:48) -manifold or a Nash F -manifold.3.4. The Schwartz space of a quotient stack.
Now let X be the quotient ofa Nash manifold M by a Nash group H , i.e., the Nash stack associated tothe groupoid object [ M × H ⇒ M ] in N .In the Archimedean case, we consider the category M H of smooth F -representations of H , in the language of Bernstein and Kr ¨otz cf. [13]. (Theyare called smooth representations of moderate growth elsewhere.) Thismeans that the representation is a countable inverse limit of Banach spacerepresentations, and (topologically) equal to the space of its smooth vec-tors. Again, in the non-Archimedean case we just consider smooth rep-resentations, without topology; since this case is easier, we discuss theArchimedean case here, and the adaptation to the non-Archimedean caseis obvious.The category of smooth F -representations of H is equivalent, by [13,Proposition 2.20], to the category of non-degenerate continuous represen-tations of the (nuclear Fr´echet) algebra S ( H ) of Schwartz measures on H on Fr´echet spaces. ( Non-degenerate means that S ( H ) V = V , where V is thespace of the representation.) For any such module V , the action of S ( H ) gives rise to a topological quotient map: S ( H ) ˆ ⊗ V → V , where ˆ ⊗ denotesthe projective tensor product.The functor of coinvariants V (cid:55)→ V H is the functor from M H to F = Fr´echet spaces that takes V to its quotient by the closure of the span of vec-tors of the form v − g · v , v ∈ V, g ∈ H . Because of the closure operation,and the lack of exactness, this is a very rough version of the operation of“modding out by the H -action” (like the “Schwartz space” S ( X ) that weassigned to a Nash stack was a very rough version of the notion of “globalsections”). A finer version is obtained by the homological algebra of “totalderived functors” of Deligne, explained in [18, § S ( H ) -modules, we need to adapt some results from [43], which iswritten for unital algebras, to the category of non-degenerate modules ofa non-unital algebra. I do this in Appendix B, and present here only theresult:An object V ∈ ob( M H ) is quasi-isomorphic (for the appropriate exactstructure, explained in the appendix) to the complex · · · → S ( H n ) ˆ ⊗ V → · · · → S ( H ) ˆ ⊗ V → . (3.5) HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 27
Notice that S ( H n +1 ) = S ( H ) ˆ ⊗S ( H n ) , and the action of S ( H ) is on the copyof S ( H ) on the left, by definition. The boundary maps are f ⊗· · ·⊗ f n ⊗ v (cid:55)→ n − (cid:88) i =0 ( − i f ⊗· · ·⊗ f i f i +1 ⊗· · ·⊗ v +( − n f ⊗· · ·⊗ f n − ⊗ f n v, where we denote convolution of measures simply as multiplication.The corresponding “total derived functor”: L H : D − ( M H ) → D − ( F ) , where D − denotes the derived categories of complexes which are strictlyexact in sufficiently large (positive) degrees, is obtained by applying thecoinvariant functor to (3.5). The functor of coinvariants is recovered as the“rough” version of the zeroth homology of the resulting complex, i.e., asthe quotient of ( S ( H ) ˆ ⊗ V ) H by the closure of the image of ( S ( H ) ˆ ⊗ V ) H .Now we return to the Nash stack X associated to the groupoid object [ M × H ⇒ M ] .3.4.1. Proposition.
The “complex” S • ( X ) is canonically strictly quasi-isomorphicto the derived coinvariant complex of the H -module S ( M ) : S • ( X ) (cid:39) L H ( S ( M )) . In particular, the Schwartz space S ( X ) is canonically isomorphic to the coinvariantspace S ( M ) H .Proof. We have a presentation: M → X . It is easy to see that [ M ] n +1 X = M × H n with the i -th map ( i = 0 , . . . , n ) to [ M ] n X being ( x, g n , . . . , g ) (cid:55)→ ( x, g n . . . , g ) when i = 0 , ( x, g n , . . . , g i +1 , g i , . . . , g ) (cid:55)→ ( x, . . . , g i +1 g i , . . . , g ) when < i < n , and ( x, g n , . . . , g ) (cid:55)→ ( xg n , g n − , . . . , g ) when i = n .On the other hand, setting R = M × H we can identify [ R ] n +1 M with M × H n +1 , the copies of H numbered in a decreasing order from n to , insuch a way that the i -th map to S ([ R ] nM ) is forgetting the i -th copy of H .By Proposition 3.1.4, the sequence resulting from taking alternating sumsof these maps: · · · → S ( M × H n +1 ) → S ( M × H n ) → · · · → S ( M × H ) → (3.6)is quasi-isomorphic to → S ( M ) → . Moreover, it is H -equivariantly so,when H acts diagonally on the right on all coordinates of M × H n .By a change of variables, we can easily see that (3.6) coincides with (3.5)for V = S ( M ) . Applying the functor of H -coinvariants we get the a com-plex in the strict quasi-isomorphism class of L H ( S ( M )) . On the other hand, the push-forward under the map [ R ] n +1 M = M × H n +1 → M × H n = [ M ] n X given by ( x, g n , . . . , g ) (cid:55)→ ( xg − n , g n g − n − , . . . , g g − ) identifies S ([ M ] n X ) asthe coinvariant space S ([ R ] n +1 M ) H , and the boundary maps for the formerdescend to boundary maps for the latter. This proves the proposition. (cid:3) Corollary.
Let X = X/G , where X is a smooth variety under the action ofa linear algebraic group G . Let X = X ( F ) be the Nash stack associated to it. Then S • ( X ) is canonically strictly quasi-isomorphic to the direct sum (cid:77) T L G T ( F ) S ( X T ( F )) , (3.7) where T runs over a set of representatives for all isomorphism classes of G -torsorsover F , G T = Aut G ( T ) and X T = X × G T .Proof. Immediate corollary of Proposition 2.4.1 and Proposition 3.4.1. (cid:3)
This implies, of course, a corresponding decomposition for the Schwartzspace S ( X ) .3.4.3. Example.
Let X be the quotient of the space X = G by the adjoint G -action, where G is an algebraic group. We have S ( X ( F )) = (cid:77) T ∈ H ( F,G ) S ( G T ( F )) G T ( F ) , by remembering that the first cohomology set H ( F, G ) parametrizes iso-morphism classes of G -torsors over F . The coinvariants here are taken withrespect to the adjoint action.If G is quasi-split, the local Langlands conjectures in the form describedby Vogan [44] describe local L -packets as unions of packets over all pureinner forms of G , i.e., over all elements of H ( F, G ) .3.4.4. Example.
Consider the algebraic quotient stack X = G \ H/G , where G = SO(( V, q )) the special orthogonal group of a non-degenerate quadraticspace ( V, q ) , and H = SO(( V, q )) × SO(( V ⊕ G a , q ⊕ , where by ‘ ’ wedenote some non-degenerate quadratic form on the one-dimensional vec-tor space G a , and G is embedded “diagonally” in H . Equivalently, X is thequotient of SO(( V ⊕ G a , q ⊕ by SO((
V, q )) -conjugacy. This is the quotientstack associated to the Gross–Prasad conjectures [25].In this case, isomorphism classes of G -torsors T are parametrized by iso-morphism classes of quadratic spaces ( V (cid:48) , q (cid:48) ) of the same dimension anddeterminant as V , and viewing X as the quotient of X = G \ H by G , forsuch a G -torsor we have G T = SO(( V (cid:48) , q (cid:48) )) and X T = G T \ H T , where H T = SO(( V (cid:48) , q (cid:48) )) × SO(( V (cid:48) ⊕ G a , q (cid:48) ⊕ . Thus, (3.7) gives a sum of coin-variant complexes parametrized by such classes of torsors. HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 29
Notice that we can also present X in terms of the group G × G acting onthe space H . In this case, we would apparently get more summands in (3.7)parametrized by pairs T × T of G -torsors; nonetheless, only when T = T would the set of F -points of the space be non-empty.3.5. Stalks.
For restricted topological spaces, I use the notion of stalks forcosheaves as in [39, § B.4], that is: the stalk of a cosheaf F of vector spacesor Fr´echet spaces on a restricted topological space X over a closed subset Z is the cosheaf over X which to an open U ⊂ X assigns the quotient of F ( U ) by the closure of the image of F ( U (cid:114) Z ) . (In [39] it was assumed thatthe image of F ( U (cid:114) Z ) is closed, and this will be the case for the sheavesconsidered here, even over the smooth site of Nash manifolds.) Clearly,this cosheaf is supported on Z , in the sense that for U ⊂ X (cid:114) Z , F Z ( U ) = 0 .The image of an element of F ( X ) in F Z ( X ) will be called its germ at Z .Let X be a Nash stack. An open Nash substack U of X is a strictly fullsubcategory U ⊂ X such that U is a Nash stack and U → X is an open im-mersion; that is, the morphism is representable, and for one, equivalentlyany, presentation X → X , the morphism U × X X → X is an open immersion(embedding) of Nash manifolds.Let Z be the “complement of X ”, a “closed substack” of X . There is anobvious way to define this as a stack over N , but it will not necessarily bea Nash stack, since it is not necessarily a smooth image of a smooth semi-algebraic manifold. In any case, for us the notion of “closed substack” willbe just symbolic, the rigorous notion being its open complement.We define the stalk of S at Z to be the cosheaf S ( • ) Z over X ∞ defined by S ( U, u ) Z := S ( U ) / S ( U × u, X U ) , for every ( U, u ) ∈ ob X .Notice that U × u, X U is an open Nash submanifold of U , and therefore itsspace of Schwartz measures is a closed subspace of S ( U ) , so the quotientmakes sense.3.5.1. Proposition.
The association ( U, u ) (cid:55)→ S (( U, u )) Z defined above is indeeda cosheaf, with strict push-forward maps, on the smooth site of X .Proof. Let π : ( V, v ) → ( U, u ) be a smooth map over X . If U (cid:48) := U × u, X U ,then V (cid:48) := V × v, X U = π − ( U (cid:48) ) . Therefore, π ! ( S ( V )) = S ( π ( V )) ,π ! ( S ( V (cid:48) )) = S ( π ( V ) ∩ U (cid:48) ) ⊂ S ( U (cid:48) ) , and therefore the push-forward map S (( V, v )) Z → S (( U, u )) Z is well-defined and strict, since S ( π ( V )) + S ( U (cid:48) ) = S ( π ( V ) ∪ U (cid:48) ) is closedin S ( U ) . Now assume that V → U is surjective. We verify that we have a coequal-izer diagram (omitting, for simplicity, the maps to X from the notation): S ( V × U V ) Z ⇒ S ( V ) Z → S ( U ) Z . This follows immediately from the corresponding coequalizer diagrams: S ( V × U V ) ⇒ S ( V ) → S ( U ) , S ( V (cid:48) × U V (cid:48) ) ⇒ S ( V (cid:48) ) → S ( U (cid:48) ) , and the fact that ( V × U V ) × X U = V (cid:48) × U V (cid:48) . (cid:3) The stalk, as a cosheaf, has a lot of representation-theoretic interest. Forthe purposes of this paper, however, I will restrict myself to an ad hoc defi-nition of its global sections as S ( X ) Z := S ( X ) /ι ( S ( U )) , (3.8)where ι : S ( U ) → S ( X ) is the natural morphism.Notice the following:3.5.2. Lemma.
Let X → X be any presentation, R X = X × X X , with the“source” and “target” maps s, t : R X → X . Then S Z ( X ) can alternatively bedescribed as S ( X ) Z = S ( X ) Z / ( s ! − t ! ) S ( R X ) Z . (3.9) Proof.
This simply follows from the definitions:If U := X × X U → U is the corresponding presentation of U , then both(3.8) and (3.9) amount to the formula: S ( X ) Z = S ( X ) / S ( U ) + ( s ! − t ! ) S ( R X ) . (cid:3)
4. L
OCAL AND GLOBAL STALKS FOR AFFINE REDUCTIVE GROUPQUOTIENTS
Luna’s ´etale slice theorem.
We now focus on the case X = an alge-braic quotient stack of the form X/G , where X is a smooth affine varietyand G is a reductive group acting on it.The variety and the group may be defined over a number field k , inwhich case the Nash stack obtained from a completion F = k v of k (de-noted by X in § X v . (This notation is compatiblewith the notation X v = X ( k v ) for a smooth variety, if we consider the Nashmanifold X ( k v ) as a Nash stack.) Let π : X → c be the invariant-theoreticquotient attached to X , i.e., c is the affine variety X (cid:12) G = spec k [ X ] G . (Itdoes not depend on the presentation X = X/G chosen.)It is well-known that there is a bijection between geometric points of c and closed geometric orbits of G on X . The fiber of X over a point ξ ∈ c ( k ) contains, as a closed substack, a gerbe X ξ corresponding to the closedgeometric orbit of G on the preimage of X in ξ . By definition, a gerbe HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 31 (over k ) is special type of stack characterized by the fact that the morphisms X ξ → spec k and X ξ → X ξ × spec k X ξ are epimorphisms. In simple terms,over the algebraic closure it becomes a stack of the form pt /H , where H isan algebraic group (reductive, in our case), but the automorphism group H may not, in general, be defined over k . Note that we don’t denote by X ξ thewhole fiber over ξ , but just its closed sub-gerbe.The gerbe is called neutral if it admits a section ˜ ξ : spec k → X ξ , that is: ˜ ξ ∈ ob( X ξ,k ) , in which case it is isomorphic to spec k/H , where H = Aut( ˜ ξ ) .The group H depends, of course, on the chosen k -point ˜ ξ , and the isomor-phism class of this k -point determines H up to H ( k ) -conjugacy. All theisomorphism classes in X ξ,k are obtained from the chosen one by choosingthe isomorphism class of an H -torsor, in which case H gets replaced by the H -automorphism group of the torsor. We will call a point ξ ∈ c ( k ) neutral if X ξ is such. In the general case, the gerbe X ξ becomes neutral after pass-ing to a finite extension. For a discussion of these gerbes in the case of theadjoint quotient of a reductive group, cf. [31].Assume that ξ ∈ c ( k ) is a neutral point, and let ˜ ξ : spec k → X ξ be asection over ξ . Without loss of generality (as we may replace the presenta-tion X/G by the presentation X T /G T , where T denotes a G -torsor, or evenreplace G by GL N , see § G ( k ) -orbit C in X ( k ) . Let V denote the quotient stack of the normal bundle of C by G . Itcan be identified with the quotient V /H , where V is the fiber of the nor-mal bundle over a point x ∈ C , and H = G x is the stabilizer of this point.The isomorphism class of ˜ ξ determines the pair ( H, V ) up to the simulta-neous action of an element of H ( k ) on H (by conjugation) and on V . Onthe other hand, choosing a different isomorphism class for ˜ ξ will have thefollowing effect: H gets replaced by an inner twist H (cid:48) corresponding to aGalois -cocycle into H (¯ k ) , and V is replaced by the representation V (cid:48) of H (cid:48) obtained by twisting by this cocycle.In particular, the quotient V is canonically determined up to a canonical -isomorphism by ξ , and does not depend on the choice of ˜ ξ . On the otherhand, the pair ( H, V ) is determined up to simultaneous H ( k ) -action by theisomorphism class of ˜ ξ . Whenever we have chosen a k -point ˜ ξ , we will beusing the presentations X = X/G , V = V /H corresponding to this k -pointas above (i.e., with ˜ ξ corresponding to a G ( k ) -orbit C on X ( k ) and V beingthe fiber of the normal bundle over a x ∈ C ), without further mention.We now recall Luna’s slice theorem [35], in its generalization by Alper[4], in order to use it in our local and global analysis of Schwartz spaces.It holds over an arbitrary field (or more general, locally noetherian base),which here we denote by k :4.1.1. Theorem.
Let ξ ∈ c ( k ) be a neutral point, and choose a section ˜ ξ : spec k →X over ξ , corresponding to a G ( k ) -orbit C in X ( k ) . Let V = V /H be as above.
Identifying the stabilizer of a point x ∈ C with H , there is an H -stable sub-scheme W (cid:44) → X , smooth and affine over k , containing x , and an H -equivariantmorphism of pointed schemes: ( W, x ) → ( V, , such that the induced diagram ofpointed stacks W/H (cid:118) (cid:118) π W (cid:15) (cid:15) (cid:39) (cid:39) V = V /H π V (cid:15) (cid:15) c W = W (cid:12) H (cid:118) (cid:118) (cid:39) (cid:39) X = X/G π (cid:15) (cid:15) c V = V (cid:12) H c = X (cid:12) G (4.1) is Cartesian with ´etale diagonals. Remark.
Although the stack V is completely determined by ξ , as men-tioned above, the identification of ´etale neighborhoods of V and X of theabove theorem is not canonical.We can make it a little more canonical by recalling that V is, up to the H -action, identified with the fiber of the normal bundle over x . We canalso identify the tangent space of W at x with V (through the canonicalquotient map from the tangent space to x of X to the normal space of x G ), and then require that the ´etale map W → V induce the identity ontangent spaces. This is implicit in the construction of [35, III.1.Lemme], butwill not be used here.The way that ´etale neighborhoods of ˜ ξ ∈ ob( X k ) and ∈ ob(( V /H ) k ) areidentified is important (we freely use for the zero point in V ( k ) but alsofor its images in ( V /H ) k and ( V (cid:12) H )( k ) ), and we will insist on clarifying thedependence of various constructions on choices. Let W, W (cid:48) be two H -stablesubvarieties of X as in Theorem 4.1.1, together with ´etale, H -equivariantmorphisms: W → V , W (cid:48) → V . This gives rise to a diagram with ´etalediagonal maps: Y (cid:126) (cid:126) (cid:15) (cid:15) (cid:32) (cid:32) V (cid:15) (cid:15) c Y (cid:126) (cid:126) (cid:33) (cid:33) V (cid:15) (cid:15) c V c V , (4.2)where Y = W/H × X W (cid:48) /H , c Y = W (cid:12) H × c W (cid:48) (cid:12) H .Again, this is a diagram of “pointed stacks”, in the sense that there isa distinguished isomorphism class of points ξ (cid:48) ∈ ob( Y k ) mapping to theisomorphism class of ∈ ob( V k ) and, moreover, the diagram induces the HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 33 identity on the (neutral) gerbe V ⊂ V (as easily follows from the construc-tion of this diagram by Theorem 4.1.1 — namely, the fact that ∈ V ( k ) corresponds to a G ( k ) -orbit on X ( k ) ).We should make sure that, whenever we make a definition for X usingLuna’s ´etale slice theorem, the objects that we define are invariant withrespect to correspondences of the form (4.2) defined over k . However, thediagram (4.2) does not capture all the information present; for example, it isnot clear how k -points of W and W (cid:48) are related, and when they “appear” inthe fiber product Y = W/H × X W (cid:48) /H (i.e., when they have the same imagein X ). In order to encode such information, we complete this diagram tothe following diagram with Cartesian squares: P (cid:125) (cid:125) (cid:33) (cid:33) W (cid:126) (cid:126) (cid:33) (cid:33) W (cid:125) (cid:125) (cid:32) (cid:32) V (cid:32) (cid:32) Y (cid:126) (cid:126) (cid:15) (cid:15) (cid:32) (cid:32) V (cid:126) (cid:126) V (cid:15) (cid:15) c Y (cid:125) (cid:125) (cid:33) (cid:33) V (cid:15) (cid:15) c V c V . (4.3)The varieties W , W and P are defined by the Cartesian property, i.e., W and W are the fiber products of Y with V over V with respect to theleft and right maps of (4.2), respectively, and P is their fiber product over Y . Alternatively, in terms of the subvarieties W, W (cid:48) of X used to define Y ,we have P = W × X W (cid:48) . (4.4)The space P carries an action of H × H , and is an H -torsor over W , resp. W , with respect to the second, resp. first copy of H . It contains a distin-guished point x which corresponds to the point x of W and W (cid:48) , whose H × H -orbit is isomorphic to H (acted upon by left and right multiplica-tion). The fiber at x of the normal bundle to its orbit can be identified with V via either the projection to W or the projection to W , and it is easy tosee that these identifications coincide. Applying Luna’s ´etale slice theoremto P , we deduce that there is a subvariety W , containing x and stableunder the diagonal copy of H , and an ´etale map W → V giving rise to adiagram analogous to (4.1). What we will need from this is the followingdiagram with Cartesian squares and ´etale diagonal maps: W (cid:122) (cid:122) (cid:15) (cid:15) (cid:37) (cid:37) W (cid:15) (cid:15) c W (cid:122) (cid:122) (cid:37) (cid:37) W (cid:15) (cid:15) c Y = c W c Y = c W , (4.5)where c W i = W i (cid:12) H .We will return to these diagrams several times. As a first corollary, weget the existence of a well-defined semisimplification for k -point of X .Let ξ ∈ c ( k ) be a neutral point, and choose a closed k -point ˜ ξ : spec k →X over ξ , with stabilizer H . Let N ξ be the preimage of ξ under the map: X → c . Let W be an H -stable subvariety of X as in Luna’s theorem, and N W the preimage of the distinguished point of c W in W . From (4.1) we getan equivalence N W /H ∼ −→ N ξ . Composing with the natural map N W /H → spec k/H ∼ −→ X ξ , we get adiagram N ξ ∼ ← N W /H → spec k/H ∼ −→ X ξ . (4.6)4.1.3. Proposition.
The map from isomorphism classes of N ξ to isomorphismclasses of X ξ induced from (4.6) does not depend on the chosen k -point ˜ ξ , or on thechoice of W and the ´etale map W → V .Proof. Two choices W → V and W (cid:48) → V as before give rise to the abovediagrams, and in particular to (4.5). This, together with the maps W → W , W → W (cid:48) (which are also Cartesian over c Y → c W , c Y → c W (cid:48) ) give rise to adiagram of equivalences N W /H (cid:121) (cid:121) (cid:37) (cid:37) N W /H (cid:37) (cid:37) N W (cid:48) /H (cid:121) (cid:121) N ξ , and both “maps” (at the level of isomorphism classes) N ξ → spec k/H are obtained by inverting those and composing with N W /H → spec k/H .Therefore, they coincide.It is easy to see that a choice of different k -point modifies these maps andthe isomorphism spec k/H ∼ −→ X ξ compatibly; thus we get a well-definedmap from isomorphism classes of objects in N ξ to isomorphism classes ofobjects in X ξ . (cid:3) HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 35
Local Schwartz stalks.
Let F be a local field, and assume that X, G are defined over F . Denote by X the stack X ( F ) , as before. We use thenotation of the previous subsection, with ξ now being a neutral point of c ( F ) . Applying the slice theorem and the fact that ´etale maps become localisomorphisms in the semi-algebraic topology on F -points (when F (cid:54) = C ),we get from Theorem 4.1.1:4.2.1. Corollary.
Assume F (cid:54) = C . There is a semi-algebraic open neighborhood U of ξ in c ( F ) and a semi-algebraic open neighborhood U (cid:48) of the image of ∈ c V ( F ) ,such that the corresponding open Nash substack X U = X × c ( F ) U of X is equivalentto the open Nash substack of V ( F ) lying over U (cid:48) .In particular, setting Z = π − ( ξ ) , Z (cid:48) = π − V (0) , where π : X → c ( F ) , π V : V ( F ) → c V ( F ) , the stalk (see § S X over Z isisomorphic to the stalk of S V ( F ) over Z (cid:48) . From now on we will be denoting Schwartz stalks over Z simply by theindex ξ , and stalks over Z (cid:48) by the index . Note, however, that Z (and thisnotation for the stalk) does not refer to the gerbe denoted by X ξ before, butcontains it.Notice that the isomorphism between the stalks is a tautology, given thefirst statement of the corollary and the fact that the stalks are cosheavessupported away from the complement of Z , resp. of Z (cid:48) . Applying the iso-morphism to the global sections (3.8), we get an isomorphism: S ( X ( F )) ξ (cid:39) S ( V ( F )) . (4.7)This isomorphism, though, depends on the identification of ´etale neighbor-hoods as in (4.1), and it is important to keep this in mind in constructionsthat follow.4.2.2. Remark.
From Remark 3.3.2 it follows that the isomorphism (4.7) isinduced from a diagram of the form (4.1) even when F = C , simply byconsidering X as a Nash R -manifold. Therefore, in what follows we put norestriction on the local field F .Now I describe a way to pass from measures to functions, which will beuseful for the regularization of orbital integrals. We will only work withthe coinvariant space S ( V ( F )) H ( F ) which by Corollary 3.4.2 is canonicallya direct summand of S ( V ( F )) (and hence also provides a direct summand (cid:0) S ( V ( F )) H ( F ) (cid:1) of the stalk S ( V ( F )) ).We would like to divide all Schwartz measures on V ( F ) by a Haar mea-sure on V ( F ) , in order to obtain Schwartz functions on V ( F ) . However,Haar measure on V ( F ) will not be preserved by the correspondences of theform (4.2); therefore we need to consider more general classes of measures.Let δ V be the character by which the group H ( F ) acts on Haar measureon V ( F ) ; it is the absolute value of an algebraic character d V . Considerthe space L of volume forms on V which are H -eigenforms with eigen-character d V ; clearly, they are all multiples of a “Haar” volume form by a polynomial function on c V = V (cid:12) H ; thus, we can think of them as a linebundle over c V (also to be denoted by L ). The absolute value of such vol-ume a volume form, multiplied by any H ( F ) -invariant Nash function on V ( F ) , is a smooth, ( H ( F ) , δ V ) -equivariant measure of polynomial growthon V ( F ) , and all those measures are multiples of the Haar measure by sucha function. We will think of them as “sections” of a “Nash line bundle” L over c V ( F ) ; although c V ( F ) may not be smooth, or, even if it is, H ( F ) -invariant Nash functions on V ( F ) do not necessarily descend to Nash func-tions on c V ( F ) , we will by abuse of language talk about the “fiber” of L over ∈ c V ( F ) to refer to the quotient of the sections of L by those sec-tions of the form f ( v ) dv , where dv is a Haar measure and f ( v ) in an H ( F ) -invariant Nash function on V which vanishes on the preimage of . (Thisfiber is clearly one-dimensional.)Let F denote the cosheaf of Schwartz functions on V ( F ) . Let N ⊂ V bethe preimage of ∈ c V , the “nilpotent cone” of all elements of V which con-tain in their H -orbit. Let F ( N ( F )) denote the space of functions on N ( F ) which are restrictions of elements of F ( V ( F )) . In the non-Archimedeancase, this is isomorphic to F ( V ( F )) N ( F ) , (global sections of) the stalk of F over N ( F ) . In the Archimedean case, though, it is just a quotient of it, sinceit does not remember derivatives in the transverse direction. In any case,we have a map F ( V ( F )) N ( F ) → F ( N ( F )) . The following is obvious:4.2.3.
Lemma.
Multiplication by a δ V -eigenmeasure µ ∈ L , which is non-vani-shing at ∈ c V ( F ) (i.e., has non-zero image in the fiber over ) gives rise to anisomorphism of the coinvariant spaces of stalks (cid:0) F ( V ( F ) N ( F ) (cid:1) ( H ( F ) ,δ − V ) ∼ −→ (cid:0) S ( V ( F )) N ( F ) (cid:1) H ( F ) . (4.8) Moreover, the resulting map (cid:0) S ( V ( F )) N ( F ) (cid:1) H ( F ) → F ( N ( F )) ( H ( F ) ,δ − V ) (4.9) depends only on the image of µ in the fiber of L over ∈ c V ( F ) . Notice that the index ( H ( F ) ,δ − V ) denotes ( H ( F ) , δ − V ) -coinvariants of thespace, i.e., its quotient by the closure of the span of vectors of the form ( v − δ V ( h ) h · v ) (so that H ( F ) acts on the quotient by the character δ − V ). I explainthe statement in words, because the notation has become a bit heavy: The H ( F ) -coinvariants of the stalk of S ( V ( F )) over N ( F ) are identified, via thischoice of measure, with the ( H ( F ) , δ − V ) -coinvariants of the correspondingstalk of Schwartz functions; moreover, the resulting map that one obtainsby restricting the functions to N ( F ) depends only on the (non-zero) imageof the chosen measure in the “fiber” of L at ∈ c V ( F ) .Notice that taking H ( F ) -coinvariants commutes with taking stalks at N ( F ) , by Lemma 3.5.2. In other words, (cid:0) S ( V ( F )) N ( F ) (cid:1) H ( F ) is the same HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 37 as the direct summand (cid:0) S ( V ( F )) H ( F ) (cid:1) of the stalk S ( V ( F )) that we sawbefore. Moreover, by Corollary 3.4.2 the coinvariant space S ( V ( F )) H ( F ) isnot just a subspace, but also canonically a direct summand of S ( V ( F )) : S ( V ( F )) H ( F ) (cid:44) → ← S ( V ( F )) . (4.10)Thus, from (4.9) we get a map S ( V ( F )) → F ( N ( F )) ( H ( F ) ,δ − V ) , (4.11)depending only on the evaluation of µ over .In the next subsection we will see that the direct summand (4.10) (thoughnot, necessarily, the map (4.11)) is preserved at the level of stalks by a dia-gram of the form (4.2).4.3. Global Schwartz stalks.
Now let
X, G be defined over a global field k , and let ξ ∈ c ( k ) be a neutral point. From now on we will assume that thereductive automorphism group associated to ξ is connected. The reason is thatthe presentation would be more complicated if it were not connected, andI do not presently have the experience to know which approach would bemore useful in applications.To demonstrate the issues that arise in the non-connected case, recallthat the semisimple (closed) geometric point over the neutral point ξ cor-responds to a closed substack of X of the form X ξ = spec k/H , for somereductive group H . Its k -points: spec k → spec k/H can be partitionedinto equivalence classes for a relation stronger than geometric equivalence,which we can call “stable equivalence”, whereby the k -point correspondingto the trivial k -torsor is equivalent to all points in the image of spec k/H ( k ) → spec k/H ( k ) (where H is the connected component of H ). Since every k -point is iso-morphic to the trivial one for some presentation spec k/H (cid:39) spec k/H (cid:48) , thisuniquely defines a relation, which is easily checked to be an equivalencerelation on X ξ ( k ) .Note that the definition of stable equivalence can be extended to arbi-trary geometric classes of k -points of X , simply by observing that to anyisomorphism class of k -points: x : spec k → X lying over ξ ∈ c ( k ) one canattach by Proposition 4.1.3 a unique isomorphism class of closed k -points: x s : spec k → X lying over ξ (the “semisimple part of x ”). We can then calltwo points in the same geometric equivalence class stably equivalent if thisis the case for their semisimple parts. In the adjoint quotient of the group,for example, x s is the k -conjugacy class of the semisimple part of the Jordandecomposition of x , and this notion of stable conjugacy is the one of [31].Now, for two stably equivalent closed k -points: spec k → spec k/H , the H -torsors that they define (with respect to this presentation), and hencealso the stabilizer groups, are isomorphic at almost every place, and themap G → H \ G gives an onto map of integral points, for almost every place. This is not the case, in general, with non-stably equivalent k -points, whichcreates the need for some extra bookkeeping, that I will not do here. Thus,from now on we assume that the stabilizer groups of the closed points weare considering are connected.Let ξ ∈ c ( k ) . We define the global Schwartz stalk at ξ as a restricted tensorproduct: S ( X ( A k )) ξ := (cid:48) (cid:79) S ( X v ) ξ , (4.12)with respect to “basic vectors” that will be described below. Recall that S ( X v ) denotes the nuclear Fr´echet space — vector space without topol-ogy in the non-Archimedean case — of “global sections” of the Schwartzcosheaf over X v . Thus, from this point on we are talking about actual (topo-logical) vector spaces, not cosheaves. I repeat that the index ξ that we areusing does not refer to the stalk at the closed gerbe X ξ , but to the stalk overthe whole preimage in X of ξ ∈ c , which includes all geometric points whoseclosure contains the closed point corresponding to ξ .The restricted tensor product is taken with respect to a “basic vector”described as follows: First, fix a section spec k → X over k , in order tohave a pair ( H, V ) as before. Fix a (non-zero) invariant volume form ω H on H (over k ); all local volumes will be computed with respect to its ab-solute power. Moreover, fix a section ω V of the line bundle L over c V de-scribed before Lemma 4.2.3; here, both the line bundle and the section ω V are defined over k . Finally, fix compatible integral models for X, G, V, H over the S -integers o S , where S is some finite set of places, containing theArchimedean ones.Now consider the following vector in S ( V ( k v )) , where v / ∈ S : f v := 1 | ω H | v ( H ( o v )) · V ( o v ) · | ω V | v , (4.13)where V ( o v ) denotes the characteristic function of V ( o v ) .4.3.1. Lemma.
For any two sets of choices as above, the images of the resultingvectors f v in the stalk S ( V ( k v )) coincide for almost every v .Proof. First, for any two sections spec k → X and o S -models of the corre-sponding groups H , the resulting o v -groups H o v will be canonically iso-morphic up to H ( o v ) -conjugacy for almost every v ; we are using here ourassumption that H is connected.Thus, for almost every place the sets H ( o v ) resulting from two differentchoices are identified, and so are the sets V ( o v ) . The Haar measures | ω H | v are equal almost everywhere. The measures | ω V | v do not need to coincide,but their images in the one-dimensional fiber of L over ∈ c V do coincide,and this is enough by Lemma 4.2.3, in the non-Archimedean case, in orderto identify the resulting elements in the stalk. (cid:3) HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 39
The images (“germs”) of the vectors f v in the stalk will be our basic vectorsfor the definition of the global Schwartz stalk S ( V ( A k )) , i.e., S ( V ( A k )) := (cid:48) (cid:79) S ( V v ) , where the restricted tensor product is taken with respect to the basic vectors f v above.Before I proceed to use this for defining the global Schwartz stalk of X as in (4.12), I introduce, for later use, a map to a related restricted tensorproduct of spaces of functions. Namely, choose again a global section ω V of the line bundle L over c V (for example, the Haar volume form on V ).Recall that N denotes the nilpotent cone in V . Using the absolute value of ω V , at every place v , we get out of (4.11) maps S ( V ( k v )) → F ( N ( k v )) ( H ( k v ) ,δ − V ) . Putting them all together, we obtain a linear map: E : S ( V ( A k )) → (cid:48) (cid:79) v F ( N ( k v )) ( H ( k v ) ,δ − V ) , (4.14)where the restricted tensor product on the right-hand side is taken withrespect to the images of the functions: | ω H | v ( H ( o v )) · V ( o v ) . Remark.
Notice that this is not the space of restrictions to N ( A k ) ofglobal Schwartz functions on V ( A k ) ; rather, it is a formal multiple of that.The reason is that the partial Euler product of the factors | ω H | v ( H ( o v )) may not make sense, even if we try to interpret it as a special value of a partial L -function. This happens precisely when (the connected reductive group) H has a non-trivial k -character group; but these factors will be formallycancelled out when we attempt to integrate against (non-regularized) Ta-magawa measure.We have the following easy lemma:4.3.3. Lemma.
The map (4.14) does not depend on the volume form ω V chosen.Proof. Indeed, we know that the local factors of this map depend only onthe image of | ω V | v in the fiber over ∈ c V ( k v ) . For any two choices, thosewill coincide at almost every place, and their quotients at the remainingplaces will multiply to , by the product formula. (cid:3) We return to the definition of the stalk S ( X ( A k )) ξ . In order to use thebasic functions for V , via (4.7), to define the basic vectors for the stalk of X ,we need to make sure that the basic vectors are invariant, at almost everyplace, under the isomorphisms of stalks induced by a diagram of the form(4.2). The whole point of introducing the line bundles L , L , instead of just choosing Haar volume forms on V , was precisely to make these definitionsinvariant.4.3.4. Proposition.
Consider a correspondence as in (4.2) defined over k , andthe automorphism: S ( V v ) ∼ −→ S ( V v ) that it induces on stalks. For almost ev-ery place v , this automorphism preserves the basic vector, and for every place v itpreserves the distinguished direct summand of S ( V v ) corresponding to H ( k v ) -coinvariants of S ( V ( k v )) . The proof will use the following lemma that will recur in other proofs,as well:4.3.5.
Lemma.
Consider diagram (4.3) , induced from two different choices of datafor Luna’s theorem. Let N ⊂ V denote the preimage of ∈ c V , and N P ⊂ P thepreimage of the distinguished point of c Y . Consider the map: P → V induced fromeither the left or the right sequence of maps from P to V . Then this map admitsa section N → N P over N . In particular, the map N P → N is surjective on R -points, for every ring R over which this diagram is defined.Proof. Indeed, let us denote by N W i the corresponding preimages of thedistinguished point of c Y in W i . Recall diagram (4.5), obtained by applyingLuna’s slice theorem to P . The maps c W → c Y and c Y → c V being ´etale,and the diagrams (4.3) and (4.5) Cartesian, we get isomorphisms for thepreimages of ∈ c • (where • = W , Y or V ): N W ∼ −→ N W i ∼ −→ N for i = 1 , . (cid:3) Proof of Proposition 4.3.4.
Consider again diagram (4.3). The fact that, byLemma 4.3.5, the (smooth, H -torsor) map N P → N is surjective on k v -points, implies that this will be the same for semi-algebraic neighborhoodsof N P and N , as well (when k v = C we base change to R for this statementto be true), and therefore the map on coinvariant stalks (cid:0) S ( P ( k v )) ( H × H )( k v ) (cid:1) → (cid:0) S ( V ( k v )) H ( k v ) (cid:1) is surjective (both for the “left” and “right” sequence of arrows). Thisshows that the distinguished summand is preserved by the isomorphismof stalks induced from (4.2).The maps W → V , W → V are ´etale and H -equivariant; that meansthat the pull-back of any ( H, d V ) -equivariant volume form on V is againan ( H, d V ) -equivariant volume form on W , resp. W . Volume forms ofthis type on W i form again a line bundle over c Y = W i (cid:12) H , whose globalsections are generated by the pull-back of Haar measure. It will not createany confusion to denote this line bundle again by L .The space P carries an action of H × H , and is an H -torsor over W ,resp. W , with respect to the second, resp. first copy of H . Since the W i carry a nowhere vanishing ( H, d V ) -equivariant volume form, P carries anowhere vanishing, ( H × H, d V × d V ) -equivariant volume form. Let ω P be HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 41 such a volume form. I claim that the basic vector of the stalk S ( V ( A k )) is(at almost every place) the image of the measure: | ω H | v ( H ( o v )) · P ( o v ) · | ω P | v ∈ S ( P ( k v )) (4.15)under either the left or the right maps to V .By Lemma 4.3.5, we get that for any integral models and almost everyplace v , the set N P ( o v ) surjects onto N ( o v ) through either of the left or theright sequence of maps. It is thus clear that the image of (4.15) under eitherthe left or right maps to V coincides as a measure, in a neighborhood of N ( k v ) , with (4.13) at almost every place.This shows that the basic vector is preserved at almost every place underthe correspondence of stalks defined by (4.2). (cid:3) Corollary.
Let x : spec k → X be a semisimple point and let ( H, V ) be thelinearization of X at x , as per Luna’s ´etale slice theorem. Let ξ denote the image of x in c ( k ) .The global Schwartz stalk S ( X ( A k )) ξ := (cid:48) (cid:79) S ( X v ) ξ is well-defined independently of x , and in such a way that, for any diagram of theform (4.1) defined over k , the isomorphism of stalks of Corollary 4.2.1 gives rise toan isomorphism of global stalks S ( V ( A k )) ∼ −→ S ( X ( A k )) ξ . (4.16) Moreover, the direct summand S ( V ( A k )) H ( A k ) , (cid:44) → ← S ( V ( A k )) (4.17) corresponds, under this isomorphism, to a distinguished direct summand: S ( X ( A k )) xξ (cid:44) → ← S ( X ( A k )) ξ , (4.18) which depends only on the isomorphism class of x , not on the choice of diagram (4.1) . The summand S ( X ( A k )) xξ corresponds to considering, locally, small semi-algebraic neighborhoods of the G ( A k ) -orbit containing the G ( k ) -orbit de-fined by x . Locally equivalent k -points give rise to the same summand(4.18). Notice that the identification (4.16) does depend on the choice of di-agram (4.1).4.3.7. Remark.
In many applications, the variety X will be homogeneousfor a larger group ˜ G ⊃ G , with a chosen ˜ G -eigen-volume form ω X inducingmeasures | ω X | v on the points over each completion. What matters, actually,is just that ω X is a nowhere vanishing G -eigen-volume form.For example, in the relative trace formula one deals with a pair of G -homogeneous spaces X , X , and one considers the quotient stack ( X × X ) /G diag . However, the ˜ G = G × G -action on X × X typically admits aneigen-volume form.In this setting, one can define a global Schwartz space (not just the stalk), asthe restricted tensor product of the local ones with respect to the measures: µ X v = 1 | ω G | v ( G ( o v )) 1 X ( o v ) | ω X | v . What would these global Schwartz spaces have to do with our globalSchwartz stalk at a neutral point ξ ∈ c ( k ) ? I claim that, locally at almostevery place, the germ of µ X v in the stalk over ξ coincides with the basicvector of S ( X v ) ξ .Indeed, choosing x and W (over k ) as in Luna’s Theorem 4.1.1, andintegral models outside of a finite set of places, we have an ´etale map: W × H G → X whose image is the preimage X (cid:48) of a Zariski open subsetof c . The point x will belong to X ( o v ) almost everywhere, and because H is assumed connected, the above map will almost everywhere give a sur-jection: W ( o v ) × G ( o v ) (cid:16) X (cid:48) ( o v ) . The same is true for the ´etale map W → V with image V (cid:48) .This means that the germ of the measure X ( o v ) | ω X | v can be lifted to ameasure supported on W ( o v ) (for almost every v ), and similarly for thegerm of V ( o v ) | ω V | v . Now let us check volume forms.A nowhere vanishing volume form on V pulls back to a nowhere van-ishing volume form ω VW on W ; if the former is ( H, d V ) -equivariant, then thesame will be the case for the latter.Similarly, the pull-back to W × H G of a G -eigen-volume form on X witheigencharacter d X can be factored into an H -eigen-volume form ω XW on W and a G -eigen-volume form ω H \ G on H \ G valued in some line bundle. Theline bundle is defined by the H -eigencharacter, but since H is reductive thiseigencharacter has to be equal to the restriction of d X to H . Correspond-ingly, since ( H \ G )( o v ) = H ( o v ) \ G ( o v ) almost everywhere (by the connect-edness of H ), the germ of the measure: | ω G | v ( G ( o v )) 1 X ( o v ) | ω X | v in the stalk of S ( X v ) over ξ has to be equal to the germ of the measure | ω H | v ( H ( o v )) 1 W ( o v ) | ω XW | v . (4.19)Comparing the forms ω VW and ω XW , since they are both non-vanishing ina neighborhood of the H -fixed point x , and are both H -eigenforms, itfollows that their quotient is polynomial and non-vanishing in an H -stableneighborhood of x . The absolute value of such a function will be equalto in an H ( k v ) -stable Nash neighborhood of x , for almost every v , andhence the measure (4.19) will not change if we replace ω XW by ω VW . HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 43
5. E
QUIVARIANT TOROIDAL COMPACTIFICATIONS , AND ORBITALINTEGRALS ON LINEAR SPACES
Overview.
Let H be a connected reductive group over a global field k , and let [ H ] = H ( k ) \ H ( A k ) . In this section I will discuss a certain classof H ( A k ) -equivariant compactifications of [ H ] , which I term equivarianttoroidal . They are not directly related to the toroidal compactifications of [8],as far as I can tell; rather, they are refinements of the reductive Borel–Serrecompactification [16]. I then define a notion of “asymptotically finite” func-tions, an extension of the Schwartz space of rapidly decaying smooth func-tions on [ H ] , where we allow multiplicative behavior “at infinity”, forminga cosheaf over an equivariant toroidal compactification. Then I proceed toprove the main theorem of this section:5.1.1. Theorem.
Let V be a finite-dimensional representation of H over k , andlet N be its “nilpotent” set, i.e., the closed subvariety of all points whose H -orbitclosure contains . For any Schwartz function f on V ( A k ) , the function Σ N f : h (cid:55)→ (cid:88) γ ∈ N ( k ) f ( γh ) (5.1) on [ H ] is asymptotically finite.More precisely, this map represents a continuous map from the space of Schwartzfunctions F ( V ( A k )) to a space F E ([ H ] F ) of asymptotically finite functions on [ H ] with asymptotic behavior explicitly determined by the representation V . Details on the asymptotic behavior mentioned in the theorem will begiven after the discussion of compactifications. This theorem gives a wayof defining a regularized integral (cid:90) ∗ [ H ] Σ N f ( h ) dh, which, for representatives γ i of the H ( k ) -orbits on N ( k ) , is formally equalto the sum (cid:88) i Vol([ H γ i ]) (cid:90) H γi ( A k ) \ H ( A k ) f ( γh ) (5.2)of orbital integrals of all rational nilpotent orbits of H on V . The regular-ization of orbital integrals defined here will be used in the next section, incombination with Luna’s ´etale slice theorem, to define “evaluation maps”(regularized orbital integrals) for more general reductive quotient stacks.This regularization will be possible if and only if the “exponents” of theasymptotic behavior of Σ N f are not “critical”, in some sense. The prototypefor this is the following:Let s ∈ C . Consider the space S (Γ \H ) s of functions on Γ \H , where H isthe complex upper half-plane and Γ = SL ( Z ) , which are smooth and havethe property that f ( x + iy ) ∼ y s for y (cid:29) , where ∼ means that the difference is a function which, togetherwith all its polynomial derivatives, is of rapid decay. Then, the regularizedintegral (cid:90) ∗ Γ \H f ( x + iy ) dxd × y | y | (where d × y denotes multiplicative measure dy | y | ) is well-defined unless s = 1 ,i.e., unless the growth of the function is inverse to that of volume. Theexponent (multiplicative character) y (cid:55)→ y is what I call a critical exponent .The definition of the regularized integral is as follows: fix any large T > ,and define f t ( x + iy ) = (cid:40) f ( x + iy ) , if y ≤ Tf ( x + iy ) | y | − t , if y > T. Then (cid:82) Γ \H f t ( x + iy ) dxd × y | y | is convergent for (cid:60) ( t ) (cid:29) , and admits meromor-phic continuation with only a simple pole at t + 1 = s . Thus, if s (cid:54) = 1 , wecan define the regularized integral as the analytic continuation of the aboveintegral to t = 0 . Conventions and notation for this section.
In this section we will, for sim-plicity, assume that k = Q , which we may, if k is a number field, by restric-tion of scalars. For function fields in positive characteristic, the analogousconstructions can be performed by picking a place that one calls “infinity”.In this case, as in the rest of the paper, one should ignore any mention ofsemi-algebraic topologies, and work with the usual, honest topologies onthe spaces under consideration; rapidly decaying functions become func-tions which are eventually zero, and asymptotic equalities up to rapidlydecaying functions become exact asymptotic equalities in some neighbor-hood of infinity. Hence, in positive characteristic the theory simplifies con-siderably, and I leave the details to the reader.For any torus, say T , we denote by the corresponding gothic lowercaseletter the vector space t := Hom( G m , T ) ⊗ R . If F is a valued field (or ring),we have a well-defined logarithmic map log : T ( F ) → t (5.3)given by (cid:104) log( t ) , χ (cid:105) = log | χ ( t ) | for any χ ∈ Hom( T, G m ) .In this section I define various cosheaves on toric varieties over R and on“equivariant toroidal embeddings” of [ H ] ; these are stratified spaces witha unique open stratum, and carry a “semi-algebraic” restricted topology,on which the cosheaf is defined. Their definition requires some familiaritywith the theory of toric varieties: I remind the reader here that a normalaffine embedding Y of a torus T over a field k is given by a strictly convex,rational polyhedral cone C ⊂ t . The faces of this cone are in bijection with T -orbits on Y , in such a way that cocharacters λ in the relative interior ofa face are those for which lim t → λ ( t ) belongs to the corresponding orbit. HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 45
More general normal embeddings of T are described by fans in t , i.e., col-lections of such cones closed under the operation of passing to a face of acone and with disjoint relative interiors. The notation [ H ] F in the statementof the above theorem refers to a “equivariant toroidal” embedding of [ H ] F described by a fan F , as will be explained.These cosheaves are denoted by the letter F (for functions — at somepoint we multiply by measures and denote them by S ), and by an index E for an “exponent arrangement”. When we want to clarify the space onwhich they are defined, we will also put its open orbit (under a given groupaction) as an index. Thus, on a space Y with open orbit W we will denoteby F E or F W,E the corresponding cosheaf.Note that sections of the cosheaf are not, in general, functions on thespace. They can be functions on its open stratum W , with “asymptoticallymultiplicative” behavior in the neighborhood of the other strata, or, moregenerally, they can be sections of some vector bundle over the open stratumwith this asymptotic behavior (which can be pulled back to functions on aprincipal torus bundle ˜ W → W that are eigenfunctions for the torus).5.2. Asymptotically finite functions on toric varieties.
Cosheaves of multiplicatively finite functions.
Let T be a split torus de-fined over R and let Y be a toric variety for T . We assume that Y is normal,but we do not assume that T acts faithfully on Y : the open T -orbit of Y is aprincipal homogeneous space for a torus quotient T (cid:48) of T ; for convenience,we will assume that T (cid:48) is the quotient of T by a subtorus T , so that the mapon R -points: T ( R ) → T (cid:48) ( R ) is onto. I would advise the reader to considerthe case T = T (cid:48) at first reading.If Y is smooth, then we have well-defined notions of Schwartz functionson Y ( R ) , and Schwartz functions on T (cid:48) ( R ) are those Schwartz functions on Y ( R ) which vanish, together with all their derivatives, on the complementof T (cid:48) ( R ) .We can generalize this description to sections of more general vectorbundles over Y ( R ) which, however, over T (cid:48) ( R ) will coincide with the triv-ial line bundle (i.e., the sections will be functions on T (cid:48) ( R ) ) or, more gen-erally, with the vector bundle whose sections are generalized T ( R ) -eigen-functions on T ( R ) with specified eigencharacters (“exponents”). We do notneed to assume that Y is smooth.First, let E be a finite, non-empty multiset of characters of T ( R ) . It issometimes helpful to think of positive, real-valued characters, which, viathe logarithmic map (5.3) and exponentiation: R → R × + , can be identifiedwith elements of t ∗ = Hom( t , R ) .We will say that a vector in a representation of T ( R ) is a generalized eigen-vector with (multi)set of exponents E if it is annihilated by the operator (cid:89) χ ∈ E ( a − χ ( a )) for all a ∈ T ( R ) , that is: • in the case where all elements of E are equal to the same character χ , it is a generalized eigenvector with eigencharacter χ and degreeless or equal to the multiplicity of χ in E ; • in the general case, it is a sum of such generalized eigenvectors,corresponding to the distinct characters in E .Now fix an orbit Z ⊂ Y . The pointwise stabilizer of Z is a subgroup T Z ⊂ T , containing T . A given multiset E of characters of T Z ( R ) defines acomplex vector bundle over Z ( R ) , as follows:Sections of this vector bundle are functions on T ( R ) whichare generalized eigenfunctions for T Z ( R ) with exponents E .Sections of this vector bundle will be called multiplicatively finite functions on T ( R ) with multiset of exponents E . There are obvious notions of smoothsections and sections of polynomial growth for this vector bundle, whichcoincide with the analogous notions for functions on T ( R ) . Moreover, thereis a notion of Schwartz sections of this vector bundle: a section σ , representedby the function f σ on T ( R ) , is Schwartz if for one, equivalently any, smoothsemi-algebraic lift (s. the following remark) e : Z ( R ) → T ( R ) the function e ∗ f σ is Schwartz on Z ( R ) .Schwartz sections of this vector bundle over Z ( R ) form a strictly flabbycosheaf of nuclear Fr´echet spaces on Z ( R ) which will be denoted by F Z,E .5.2.2.
Remark.
By “lift” Z ( R ) → T ( R ) or Z ( R ) → T (cid:48) ( R ) , here and later, wemean a section of the contraction map T ( R ) → T (cid:48) ( R ) → Z ( R ) that takes anelement of t ∈ T (cid:48) ( R ) to the unique element of Z ( R ) that is contained in the T Z ( R ) -orbit closure of t . Here we are identifying the open orbit in Y withthe torus quotient T (cid:48) of T , but this is only to avoid adding another piece ofnotation; we might as well distinguish between T ( R ) as the space on whichfunctions live and T ( R ) as the group acting on them. Those sections willalways be chosen to be semi-algebraic.For a general toric variety and an orbit Z contained in it, the contractionmap is defined on the affine open neighborhood Y Z of all orbits which con-tain Z in their closure. In what follows, when we choose a semi-algebraicneighborhood of a point in Z ( R ) it will be assumed to be contained in Y Z ( R ) , so that the contraction map is well-defined, and any mention ofa “lift” Z ( R ) → Y ( R ) will implicitly have image in Y Z ( R ) and be a sectionof this contraction map.5.2.3. Definition of the cosheaf of asymptotically finite functions.
Now assumethat we have an assignment Z (cid:55)→ E ( Z ) of multisets of exponents for everygeometric orbit Z ⊂ Y , with the following property:If an orbit W is contained in the closure of an orbit Z , thenthe restrictions of elements of E ( W ) to T Z ( R ) ⊂ T W ( R ) are HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 47 contained, with at least the same multiplicity, in E ( Z ) . (5.4)It is instructive to think of what this condition means combinatorially,when (for simplicity) T (cid:48) = T and all characters are positive real-valued:The embedding Y is described by a fan F of strictly convex, rational poly-hedral cones in t , corresponding bijectively to (geometric) orbits: Z ↔ C Z .The linear span of C Z is precisely the linear span of the cocharacter groupof the torus T Z . For each orbit Z we have a multiset E ( Z ) of characters of T Z ( R ) which, assumed positive real-valued, can be thought of as function-als on the span of C Z . The condition, then, is that if W is contained in theclosure of Z , i.e., if C Z is a face of C W , then all restrictions of elements of E ( W ) to the linear span of C Z are contained (with at least the same multi-plicity) in E ( Z ) .We now define the strictly flabby cosheaf F E (in the restricted semi-algebraic topology over Y ( R ) ) of functions f on T ( R ) which are asymp-totically finite with exponent arrangement E , inductively, as follows: • over the open orbit T (cid:48) ( R ) , the cosheaf F E coincides with the cosheaf F T (cid:48) ,E ( T (cid:48) ) defined above, i.e., Schwartz sections of the vector bundleof T ( R ) -generalized eigenfunctions on T ( R ) with exponents E ( T (cid:48) ) ; • for an arbitrary open set U , assuming that W is an orbit of maximalcodimension among those meeting U , we demand that there is aSchwartz section σ ∈ F W,E ( W ) ( U ∩ W ( R )) , and a semi-algebraic neighborhood V of W ( R ) ∩ U in U , proper(with respect to the contraction map, s. Remark 5.2.2) over W ( R ) ∩ U , with the property that f − f σ coincides, in V ∩ T ( R ) , with therestriction (as a function) of an f (cid:48) ∈ F E ( U (cid:114) W ( R )) .The two conditions, together with the strict flabbiness of the cosheaf (sothat open embeddings of sets give closed embeddings of the spaces of sec-tions) completely define the cosheaf: By strict flabbiness we can alwaysreplace an open set by a cover of open sets which meet a unique orbit ofminimal dimension, and by the second criterion we can remove this orbit,eventually arriving to an open subset of the open orbit.5.2.4. Remark.
Without loss of generality, we may assume that Y is com-plete ( Y ( R ) is compact); if it is not, we can compactify it, and attach theempty multiset of exponents to the new orbits. For example, Schwartzfunctions on T (cid:48) ( R ) are sections over some compactification, when we at-tach the empty set of exponents to every non-open orbit, and the trivialexponent (character of T ( R ) ) to the open orbit T (cid:48) . When different characters in E ( W ) restrict to the same character of T Z ( R ) , it is enoughto assume that its multiplicity in E ( Z ) is at least the maximum of their multiplicities, nottheir sum. There is a natural nuclear Fr´echet space structure on the sections of thiscosheaf, with respect to which extension maps are closed embeddings. Weuse the above inductive definition to define the Fr´echet space structure on F E ( U ) : First, we have a map to the Fr´echet space F W,E ( W ) ( U ∩ W ( R )) ,which provides some continuous seminorms. Secondly, we may fix V asabove and apply the seminorms of the stalk F E ( U (cid:114) W ( R )) / F E ( U (cid:114) V ) to the function f − f σ . Finally, we may apply the seminorms of the stalk F E ( U (cid:114) W ( R )) / F E (˚ V (cid:114) W ( R )) to the function f . This forms a complete setof seminorms for the space F E ( U ) .5.3. Regularized integral on toric varieties.
Let
T, Y be as above, and E an exponent arrangement for Y . Assume now that T = T (cid:48) , i.e., T acts faith-fully on Y . We will say that an exponent χ ∈ E ( Z ) (for some orbit Z , otherthan the open one) is critical if χ is trivial. Assume that E does not containany critical exponents. Clearly, by the compatibility of exponents for differ-ent orbits it is enough to check this condition for orbits Z of codimensionone, where T Z is of dimension one.We denote by S E the cosheaf of measures on T E which are equal to ele-ments of F E times a Haar measure. We can define a regularized integral , thatis, a T ( R ) -invariant functional: S E ( Y ( R )) (cid:51) µ (cid:55)→ (cid:90) ∗ T ( R ) µ ∈ C , essentially as the Mellin transform of the distribution µ evaluated at thetrivial character.Since this distribution is not tempered in the analytic sense (i.e., withrespect to the Harish-Chandra Schwartz space), I explicate the definition:First, any toric variety has a finite open cover by affine toric varieties,and any section µ of S E can be written as a sum of sections over these opensubvarieties. Hence we may, without loss of generality, assume that Y isaffine.We may then choose an algebraic character ω of T which vanishes on thecomplement of T in Y and define: (cid:90) ∗ T ( R ) µ = the analytic continuation of (cid:90) ∗ T ( R ) | ω | s ( t ) µ ( t ) to s = 0 . (5.5)This integral converges for (cid:60) ( s ) (cid:29) (how large depends on the specificexponents).5.3.1. Proposition.
If the exponent arrangement E does not contain critical ex-ponents, the regularized integral µ (cid:55)→ (cid:82) ∗ T ( R ) µ is a well-defined, T ( R ) -invariantcontinuous functional on S E ( Y ( R )) .Proof. We can always choose a smooth blowup ˜ Y → Y of the original va-riety Y . Combinatorially, this corresponds to partitioning the cones of thefan F into (simplicial) sub-cones C with the property that C ∩ Hom( G m , T (cid:48) ) HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 49 is a free monoid, for every C . This way, the relative interior of each newcone C belongs to the relative interior of an old cone D , hence T C ⊂ T D ,and we assign to the new fan ˜ F the exponent arrangement ˜ E obtained fromrestricting characters from T D to T C . Since the exponents of the original ar-rangement were non-critical, we can always choose such a partition wherethe new arrangement ˜ E remains non-critical.Sections of F E over Y ( R ) become, by pull-back, a subspace of the sec-tions of F ˜ E over ˜ Y ( R ) . Therefore, we may assume that Y is smooth. Hence,it is covered by affine open subsets of the form G am × G ba (under the actionof T (cid:39) G a + bm ).We now proceed to the proof in accordance with the definition of thecosheaf F E in § S E ( T ( R )) the regularized integral coincides withthe proper, absolutely convergent integral, and therefore there is nothing toprove in this case. Now fix a T -orbit Z , let Y Z be the affine open of all orbitscontaining it in their closure, as before, and assume that the proposition hasbeen proven for S E (( Y Z (cid:114) Z )( R )) . Let us extend it to S E ( Y Z ( R )) .A measure µ ∈ S E ( Y Z ( R )) is of the form f dt , where dt is a Haar mea-sure on T ( R ) , and f ∈ F E ( Y Z ( R )) . By the definition of F E ( Y Z ( R )) in § V of Z ( R ) , proper over Z ( R ) , and a function f σ on T ( R ) corresponding to a Schwartz section σ of F Z,E ( Z ) ( Z ( R )) , such that f − f σ coincides on V ∩ T ( R ) with the restrictionof an element of S E (( Y Z (cid:114) Z )( R )) . Without loss of generality, if ω , . . . , ω n is a set of characters generating the ideal of Z in (the affine variety) Y Z , wemay assume that V is given by inequalities: | ω i ( y ) | < (cid:15) .In particular, by our induction assumption, the regularized integral (cid:90) ∗ V ∩ T ( R ) ( f − f σ )( t ) dt is well-defined, and continuous with respect to the seminorms of S E ( Y Z ( R )) .Having assumed smoothness, we have Y Z (cid:39) G am × G ba , with Z = G am ×{ } b , and we may assume that the ω i ’s are the coordinate functions for G ba ( n = b ). From the definitions, we clearly have F Z,E ( Z ) ( Z ( R )) (cid:39) F (( R × ) a ) ⊗ W E ( Z ) , where F (( R × ) a ) denotes, as before, the usual space of Schwartz functions,and W E ( Z ) is the finite-dimensional space of generalized eigenfunctions on T Z ( R ) with the given multiset of exponents. Thus, the regularized inte-gral of f σ over V is the product of a usual integral of a Schwartz functionon ( R × ) a with the regularized integral of a generalized eigenfunction on T Z ( R ) ∩ V , which is easily seen to be well-defined in the absence of criticalexponents.Let us now prove invariance of the regularized integral assuming, as inthe definition, that Y is affine. Using the notation of the definition andwriting cont s =0 for the analytic continuation to s = 0 we have, for any a ∈ T ( R ) : (cid:90) ∗ ( a · f − f ) dt = cont s =0 (cid:90) T ( R ) ( f ( at ) − f ( t )) | ω ( t ) | s dt == cont s =0 ( | ω ( a ) | − s − (cid:90) T ( R ) f ( t ) | ω ( t ) | s dt == (cid:0) cont s =0 ( | ω ( a ) | − s − (cid:1) (cid:90) ∗ T ( R ) f ( t ) dt = 0 . (cid:3) Remark.
Depending on the exponent arrangement, asymptotically fi-nite functions may include functions which are eigenfunctions with respectto a subgroup of T ( R ) . But in this case the regularized integral will be zero,unless the corresponding eigencharacter is trivial — hence critical — on amaximal such subgroup, in which case the regularized integral is not de-fined.5.4. Equivariant toroidal compactifications of the automorphic quotient.
Now let H be a connected reductive group over k = Q . In this subsec-tion we will describe compactifications of the automorphic quotient [ H ] = H ( k ) \ H ( A k ) which are refinements of the reductive Borel–Serre compact-ifications of [16]. I do not know of any reference in the literature for thesetoroidal compactifications, and will therefore present them from scratch,though without many details.We denote a = Hom( G m , A ) ⊗ R , where A is the universal maximal splittorus of H , and we denote by a + the anti-dominant cone. Faces of a + corre-spond to conjugacy classes of parabolic subgroups defined over k , and thespan of the face associated to P will be denoted by a P , the correspondingsubtorus of A by A P . (It is the maximal split central subtorus of the Leviquotient of P .)For any class of parabolics P , we have the “boundary degeneration” [ H ] P = M ( k ) N ( A k ) \ H ( A k ) = [ M ] × P ( A k ) H ( A k ) , where N is the unipotent radical of P and M is its reductive quotient. No-tice that any two parabolics defined over k are H ( k ) -conjugate, and self-normalizing. Therefore, the space [ H ] P is defined up to unique isomor-phism by the class of P , not a choice of parabolic in this class, but we willremember that it does not have a distinguished point represented by ∈ H (which would depend on the choice of representative for P ). The same istrue for the space P ( k ) \ H ( A k ) that will be used later.Let F be a fan of strictly convex, rational polyhedral cones, whose sup-port is contained in a + . The relative interior of any cone C of F is containedin the relative interior of a unique face of a + , and hence corresponds to aunique class of parabolics P C ; we will say that the cone C belongs to theclass P C . The corresponding subtorus A C (the subtorus whose cocharacter HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 51 group spans the linear span of C ) belongs to the center of the Levi quotientof P C .We will use the fan to define a topological, H ( A k ) -equivariant compacti-fication [ H ] F of [ H ] = H ( k ) \ H ( A k ) , obeying similar closure relations as fortoric varieties: every cone C ∈ F corresponds to a locally closed stratum( H ( A k ) -orbit) Z C of [ H ] F , we have Z C ⊂ Z C ⇐⇒ C ⊃ C , etc. The stra-tum Z C corresponding to a cone C in the fan is isomorphic to the quotientof [ H ] P C by the action of A C ( R ) ; for example, if C is of full dimension, then A C = A , P C is the minimal parabolic P = M N and Z C is isomorphic tothe (compact) quotient M ( k ) A ( R ) N ( A k ) \ H ( A k ) .5.4.1. Remark.
The compactification is suitable for describing functions thatvary asymptotically according to A C ( R ) -characters (in the vicinity of thestratum Z C ). This is not ideal, but notationally convenient, and sufficientfor our purposes. In our application, actually, these characters will be real-valued and trivial on the maximal compact (finite) subgroup of A C ( R ) andhence will factor through the log map (5.3) A C ( R ) (cid:51) a (cid:55)→ log( a ) ∈ a C, R = Hom( G m , A C ) ⊗ R . There are two alternatives which are philosophically preferable: One isto compactify dividing only with respect to the connected component ofthe identity in A C ( R ) ; this is suitable to encode the minimal informationneeded for the definition of regularized integrals, but would be notation-ally a bit more cumbersome. Another extreme is to compactify dividing by A C ( A k ) , or even by the analogous torus when we take A to be the universalCartan of H , not just its maximal split subtorus. This is suitable for encod-ing the maximal amount of information on asymptotically finite functions,should they be asymptotically equal to A C ( A k ) -eigenfunctions; however,it would not add anything to the applications of the present paper, wherecharacters are trivial on the maximal compact subgroup of A C ( A k ) .We actually use F to define embeddings not only for [ H ] itself but alsofor its “boundary degenerations” [ H ] P . They also have their orbits parame-trized by cones in F , so here we need to make a notational distinction: Thenotation Z C for the stratum associated to C will be reserved for the cor-responding stratum in [ H ] F P for all parabolics P containing P C ; in fact, thestrata Z C for all those embeddings will be identified. On the other hand, ifthe parabolic P does not contain (a parabolic in the class of) P C , then thestratum of [ H ] F P corresponding to C will be different from Z C , and we willnot reserve a symbol for it.Assume first that F is a fan supported entirely on the face of the anti-dominant cone corresponding to a parabolic P with Levi quotient M , whosesplit center A P is canonically a subtorus of A . Then F defines a toric variety Y P for A P , and we set [ M ] F := [ M ] × A P ( R ) Y P ( R ) , (5.6) [ H ] F P := [ M ] F × P ( A k ) H ( A k ) . (5.7)For a general fan F , the H ( A k ) -space [ H ] F P will be defined by the formula(5.7), once [ M ] F is defined. To define [ M ] F , we may assume that M = H ,and that the spaces [ M ] F , [ H ] F P have been defined for all proper parabolics P .We first consider the restriction F H of F to a H (i.e., the sub-fan consistingof all cones which are contained in a H , the span of central cocharacters into H ). By (5.6), it defines an embedding [ H ] F H of [ H ] . Now, all the strata Z C have been defined: If C belongs to H , then Z C ⊂ [ H ] F H . If not, then Z C has been defined as a stratum of [ H ] F P , for all P ⊃ P C . It remains to explainhow to glue those onto [ H ] F H .Equivalently, we should define when a sequence ( z n ) n in (cid:70) D Z D (thisincludes the trivial cone D = { } for which Z C = [ H ] ) converges to a point z of the stratum Z C . (The topology will be separable, of course.) We canrestrict to cones D which are faces of C , since otherwise the orbit Z D willnot contain Z C in its closure. If the sequence belongs to a stratum Z D , with D ⊂ C , D belonging to a proper parabolic Q = P D , then both Z D and Z C can be considered as sub-strata of [ H ] F Q , and convergence is defined there.There remains to consider the case that ( z n ) n ⊂ [ H ] F H .For this, consider the pair of quotient maps: P ( k ) \ H ( A k ) π H (cid:121) (cid:121) π P (cid:38) (cid:38) [ H ] [ H ] P . (5.8)By “the cusp” in [ H ] P we will mean the limit of all one-parameter or-bits of the form M ( k ) N ( A k ) λ ( t ) g , as t → , where λ is a strictly P -anti-dominant cocharacter into the center of the Levi quotient M of P (and N is the unipotent radical of P ). Equivalently, the cusp is the closed orbit in [ H ] F P , when F is the fan of all faces of a + ∩ a P . We will call “neighborhoodof the cusp” in P ( k ) \ H ( A k ) the preimage of any neighborhood of the cuspin [ H ] P .There is a neighborhood of the cusp in P ( k ) \ H ( A k ) where the left ar-row of (5.8) is an isomorphism onto its image — its image will be called a“neighborhood of the P -cusp” in [ H ] . If we apply the operation × A H ( R ) Y H ( R ) HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 53 (recall that this operation on [ H ] produces [ H ] F H ), then we get correspond-ing neighborhoods of the P -cusp in [ H ] F H : P ( k ) \ H ( A k ) × A H ( R ) Y H ( R ) (cid:117) (cid:117) (cid:42) (cid:42) [ H ] F H [ H ] P × A H ( R ) Y H ( R ) . (5.9)Now, [ H ] P × A H ( R ) Y H ( R ) is an open subspace of [ H ] F P P = [ H ] P × A P ( R ) Y P ( R ) , and the latter contains the stratum Z C that we are interested in. Noticethat since C belongs to P , this stratum is “in the cusp”, i.e.: every openneighborhood of the cusp contains an open neighborhood of Z C . (Thisfails to be true for orbits in [ H ] F P P attached to cones which do not meet therelative interior of the P -face of a + .)We declare that the sequence ( z n ) n ⊂ [ H ] F H converges to z ∈ Z C if it canbe eventually lifted (under the left arrow of (5.9)) to a sequence (˜ z n ) n in theneighborhood of the cusp in P ( k ) \ H ( A k ) × A H ( R ) Y H ( R ) , whose image in [ H ] P × A H ( R ) Y H ( R ) converges to z .This completes the description of the compactification [ H ] F .5.4.2. Remark.
Returning to the discussion of strata associated to a cone C in the fan F , we notice once again that, by definition, the stratum Z C isidentified as a stratum of [ H ] F Q , for every Q ⊃ P C . On the other hand, if Q does not contain P , there is a stratum attached to C in [ H ] F Q , but it isnot isomorphic to Z C . Rather, it is isomorphic to the quotient of [ H ] P ∩ Q by A C ( R ) .5.4.3. Remark.
Assume that G is semisimple. Then there is a canonical fan F on a + , consisting of all faces of a + . One can check that in that [ H ] F onegets the reductive Borel–Serre compactification [ H ] RBS of [16], with the onlydifference that here (for notational simplicity) we have defined the com-pactifications (5.6) by using A P ( R ) , while in [16] they just use the identitycomponent A P ( R ) . In other words, to obtain the compactification of [16]one would need to replace in (5.6) the torus A P ( R ) by A P ( R ) , and the toricvariety Y P ( R ) by the closure of A P ( R ) in it. With this minor modifica-tion, it is easy to see that the reductive Borel–Serre compactification is finalamong equivariant toroidal compactifications, that is: for every cone F theidentity map on [ H ] extends to an H ( A k ) -equivariant map [ H ] F → [ H ] RBS .Verifying these claims is just a matter of going through the definitions, andusing the following standard fact in toric geometry: there is a map between Of course, since [16] was not written in the adelic language, literally speaking to obtaintheir compactification from ours one needs to divide by an open compact subgroup of thefinite adeles of H , which amounts to choosing an arithmetic subgroup of H ( R ) . two normal embeddings of the same torus, with fans F and F if and onlyif each cone of F is contained in a cone of F .Although we have described the space [ H ] F as a Hausdorff (it can easilybe checked) topological space, it is a “semi-algebraic” restricted topologythat will be more useful for us. First, notice that quotients of [ H ] by anyopen subgroup J of the finite adeles of H are Nash manifolds, in a waycompatible with each other, covered, e.g., by open Siegel domains in H ( R ) (which are clearly Nash manifolds); thus, we have “open semi-algebraicsubsets” as pull-backs of those of [ H ] /J , for some J .The R -points of the toric variety Y H have a semi-algebraic restrictedtopology, and hence the space [ H ] F H = [ H ] × A H ( R ) Y H ( R ) also inherits a restricted semi-algebraic topology.Similarly, there is a semi-algebraic restricted topology on [ H ] P and [ H ] F P P .In a neighborhood of the P -cusp, the left arrow of (5.8) is an isomorphismand the right arrow is semi-algebraically continuous and open (again, thinkof Siegel domains). Thus, the same is true for (5.9), and this is enoughto define what a semi-algebraic open neighborhood of a point z ∈ Z C is:Start with an open neighborhood in [ H ] F P P , which can be assumed to besufficiently close to the cusp (since C belongs to P ), pull it back and push itforward to [ H ] F H via (5.9). (The intersections of this neighborhood with astratum Z D with D ⊂ C belonging to a proper parabolic Q can be definedin [ H ] F Q .)5.5. Asymptotically finite functions on the automorphic quotient.
We re-main in the setting of [ H ] as above.An exponent arrangement will consist of a fan F as above, i.e., with sup-port in the anti-dominant cone a + , and an assignment, to every cone C of F ,of a multiset E ( C ) (i.e., repetitions allowed) of linear functionals (the “ex-ponents”) χ ∈ a ∗ C = Hom( a C , R ) , where a C denotes the linear span of thecone C in a . As in § C is in the closure of the cone C , then the restrictions of elementsof E ( C ) to a C are contained in E ( C ) (with at least the same multiplicity).More generally, we may, as we did in the toric case, require our functionsto have central (generalized) characters. That is, instead of taking a fan F on a , we may take the fan to be on a / a , where a ⊂ a H corresponds to asplit central torus A of H . In that case, our functions will be sections ofa cosheaf over a toroidal compactification of A ( R ) \ [ H ] , but the exponentsare still allowed to be arbitrary functionals on a , i.e., non-trivial on a . Tokeep notation simple, I will not present this case; the generalizations areimmediate.These linear functionals will be considered as positive characters of thereal, or adelic, points of the corresponding subtorus of A — the subtorus HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 55 A C whose cocharacter group spans a C — as follows: every such functional χ can be written as a linear combination (cid:80) c i χ i , where the χ i are algebraiccharacters of A C and c i ∈ R , and then we define (and denote by the sameletter): χ : A C ( A k ) (cid:51) a (cid:55)→ (cid:89) i | χ i ( t i ) | c i ∈ R × + . (5.10)Equivalently, χ ( a ) = e log a , where log is the map (5.3). Obviously, the dis-cussion here can be extended to arbitrary complex characters, but positiveones are sufficient for our purposes, and notationally simpler since theycan be identified with elements of a ∗ C .We will define a cosheaf F E of “asymptotically finite functions” for therestricted semi-algebraic topology over each of the spaces [ H ] F P , as we didfor toric varieties.First, fix a cone C ⊂ F ; recall that it corresponds to a stratum Z C ⊂ [ H ] F .Let P = P C be the parabolic associated to C ; we have inclusions of tori: A C ⊂ A P ⊂ A . The multiset E ( C ) of characters of A C ( R ) defines a complexvector bundle over Z C , whose sections are functions on [ H ] P which aregeneralized eigenfunctions for A C ( R ) with exponents E ( C ) . As in the toriccase, there are obvious notions of smooth sections, sections of polynomialgrowth, and Schwartz sections of this bundle on Z C . The cosheaf over Z C of Schwartz sections of this vector bundle will be denoted by V Z C ,E ( C ) .Now we define the strictly flabby cosheaf F E (in the semi-algebraic re-stricted topology) over [ H ] F P of functions f which are asymptotically finitewith exponent arrangement E inductively: • We apply any of the definitions that follow to any M , instead of H ,where M is the reductive quotient of any parabolic P , and F H isreplaced by F E ; if F E is defined for [ M ] F , we define it for [ H ] F P bysmooth induction from P ( A k ) to H ( A k ) . • Over an open set U which is contained in [ H ] , the space F E ( U ) co-incides with the space F ( U ) of Schwartz (rapidly decaying togetherwith their polynomial derivatives) functions on U ; more generally,over [ H ] F H , sections of the cosheaf are obtained as push-forwardsvia the quotient map [ H ] × A H ( R ) Y H ( R ) → [ H ] F H = [ H ] × A H ( R ) Y H ( R ) of elements of F ([ H ]) ˆ ⊗S E ( Y H ( R )) , where S E is the corresponding cosheaf of asymptotically finite mea-sures on Y H ( R ) , defined in § E restricted to the sub-fan F H ). To avoid heavy notation, we use the same notation F E for [ H ] , [ M ] and [ H ] P ; thisshould not cause any confusion. • Given a cone C belonging to a parabolic P , we may now by in-duction assume that F E ( U ) has been defined when U is any opensubset:(i) of [ H ] F P ;(ii) of [ H ] F , but not meeting any orbit of codimension ≥ codim( Z C ) .Now, let U be an open neighborhood Z C in [ H ] F , which we mayassume not to meet any orbits of codimension ≥ codim( Z C ) otherthan Z C , and to be sufficiently close to the P -cusp, so that its in-tersection with [ H ] can be identified, via (5.8), with a neighbor-hood U (cid:48) of the cusp in P ( k ) \ H ( A k ) . The image of U (cid:48) in [ H ] P isthe intersection with [ H ] P of an open neighborhood U (cid:48)(cid:48) of Z C in [ H ] F P . For f to be a section of F E ( U ) , we demand that there is aresections f (cid:48) ∈ F E ( U (cid:114) Z C ) , f (cid:48)(cid:48) ∈ F E ( U (cid:48)(cid:48) ) such that the difference π ∗ H f − π ∗ H f (cid:48) − π ∗ P f (cid:48)(cid:48) on U (cid:48) is equal to a rapidly decaying function on U (cid:48) (i.e., an element of F ( U (cid:48) ) ).There is a natural nuclear Fr´echet space structure on the sections of thiscosheaf, which is defined as in the toric case.5.6. Regularized integral of asymptotically finite automorphic functions.
Let ( F , E ) be an exponent arrangement for [ H ] , and let f be a section of F E over [ H ] F . We will say that a character (“exponent”) χ ∈ E ( C ) , for somecone C ∈ F , is critical if it coincides with the modular character of P C , re-stricted to A C ( R ) , and A C is non-trivial. (The modular character is the char-acter by which P C acts on the quotient of its right by its left Haar measure;since A C is in the center of the corresponding Levi, we can equivalentlytake the modular character of the minimal parabolic.)Assume that there are no critical exponents in E . Then we can define on F E ([ H ] F P ) a regularized integral, i.e., a continuous, H ( A k ) -invariant exten-sion of the integral, against some fixed invariant measure on [ H ] P , on thespace of Schwartz functions on [ H ] : F E ([ H ] F P ) (cid:51) f (cid:55)→ (cid:90) ∗ [ H ] P f ( h ) dh ∈ C , as in the toric case. Namely, the definition is parallel to the definition ofsections of F E , and goes as follows: • For a section f ∈ F E ([ H ] F P ) , we fix a factorization of the measure: (cid:90) [ H ] P dh = (cid:90) P \ H ( A k ) (cid:90) [ M ] δ P ( m ) dmdh, Of course, we will be using Tamagawa measure, but see the discussion following 6.1for the cases in which Tamagawa measure needs to be regularized; we assume throughoutcompatible choices on all the spaces [ H ] P , with Tamagawa measure on unipotent radicals. HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 57 and define accordingly: (cid:90) ∗ [ H ] P dh = (cid:90) P \ H ( A k ) (cid:90) ∗ [ M ] δ P ( m ) dmdh once the regularized integral (cid:82) ∗ [ M ] has been defined; we can applyall the definitions that follow to any M , instead of H . • On F E ([ H ] F H ) the regularized integral is induced from the regular-ized integral on S E ( Y H ( R )) : Recall from (5.6) that [ H ] F H is a quo-tient of [ H ] × Y H ( R ) ; we thus have a surjective map F ([ H ]) ˆ ⊗S E ( Y H ( R )) (cid:16) F E ([ H ] F H ) which coincides with taking coinvariants with respect to the A H ( R ) -action, and when pulled back to F ([ H ]) ˆ ⊗S E ( Y H ( R )) the regularizedintegral (cid:82) ∗ [ H ] will be the functional (cid:90) [ H ] dh × (cid:90) ∗ A H ( R ) : F ([ H ]) ˆ ⊗S E ( Y H ( R )) → C . The same definition applies to the space [ M ] F P , and hence to [ H ] F P P since sections over [ H ] F P P are simply obtained by smooth (compact)induction from P ( A k ) to H ( A k ) . • Given a cone C belonging to a parabolic P , we may now by induc-tion assume that the regularized integral has been defined on F E ( U ) when U is any open subset:(i) of [ H ] F P ;(ii) of [ H ] F , but not meeting any orbit of codimension ≥ codim( Z C ) .Now let U be an open neighborhood Z C in [ H ] F , which we mayassume not to meet any orbits of codimension ≥ codim( Z C ) otherthan Z C , and to be sufficiently close to the P -cusp, so that its inter-section with [ H ] can be identified, via (5.8), with a neighborhood U (cid:48) of the cusp in P ( k ) \ H ( A k ) . With U (cid:48) , U (cid:48)(cid:48) , f (cid:48) , f (cid:48)(cid:48) as in the definition ofsections at the end of § (cid:90) ∗ U f dh := (cid:90) ∗ U f (cid:48) dh + (cid:90) ∗ U (cid:48)(cid:48) f (cid:48)(cid:48) dh + (cid:90) U (cid:48) ( π ∗ H f − π ∗ H f (cid:48) − π ∗ P f (cid:48)(cid:48) ) dh. (5.11)The first regularized integral is defined by the induction hypothesis,since f (cid:48) is a section of F E over U (cid:114) Z C , the second is also definedby the induction hypothesis since f (cid:48)(cid:48) lives on [ H ] F P , and the thirdintegrand is of rapid decay.The following can easily be deduced from the corresponding Proposition5.3.1 for toric varieties:5.6.1. Proposition.
The regularized integral (cid:82) ∗ [ H ] dh is a well-defined, H ( A k ) -invariant continuous functional on F E ([ H ] F ) . Regularized orbital integrals for group representations.
Having de-scribed the equivariant toroidal compactifications and asymptotically fi-nite functions on the automorphic quotient, we are ready to formulate andprove the main result of this section, which concerns the sum of “nilpotent”orbital integrals for an algebraic, finite-dimensional group representation V of a reductive group H over k . We assume that H acts faithfully on V .Let N ⊂ V denote the nilpotent cone , i.e., the closed subvariety of elementswhich contain in their H -orbit; equivalently, the preimage of the image of under the quotient map V → V (cid:12) H . For a Schwartz function f on V ( A k ) ,let Σ N f ( h ) = (cid:88) γ ∈ N ( k ) f ( γh ) , a function on [ H ] . We are interested in defining the regularized integral (cid:90) ∗ [ H ] Σ N f ( h ) dh, which is formally equal to the sum (5.2) of orbital integrals for all rationalnilpotent orbits.The character by which a group acts on invariant volume forms (or Haarmeasures) on a vector space will be called the “modular character” of thisvector space.As before, let A be the universal maximal split torus in H , defined asthe split part of the center of the reductive quotient of a minimal parabolic.(For any two choices of minimal parabolics, any element of H conjugatingone to the other induces a canonical isomorphism between these tori.)Fix a minimal parabolic P , a Levi subgroup M , and identify the maxi-mal split torus in the center of M with the universal maximal split torus A of H . The action of A on V splits into a direct sum of eigenspaces; let Φ( V ) be the multiset of eigencharacters, with multiplicity equal to the dimensionof the associated eigenspace. As a subset of the character group of A , itdoes not depend on the minimal parabolic or its Levi subgroup chosen.We consider the partition of the cone a + ⊂ a = Hom( G m , A ) ⊗ R of anti-dominant coweights determined by the following hyperplanes: • the walls of the cone, i.e., hyperplanes perpendicular to (simple)roots; • the hyperplanes perpendicular to elements of the character set Φ( V ) .The intersection of these hyperplanes is trivial, since H is assumed toact faithfully on V . Thus, the cones contained in this partition, togetherwith their faces, form a fan F of rational, strictly convex polyhedral coneson the vector space a whose support is a + . Now we define an exponentarrangement C (cid:55)→ E ( C ) for this fan as follows: for every cone C we let E ( C ) consist of a single element, which is the restriction to the subspace a C HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 59 spanned by C of the character χ C := − (cid:88) χ i , (5.12)where χ i runs over all elements of Φ( V ) which are positive on the relativeinterior of C . Notice that algebraic characters of A are canonically elementsof the dual a ∗ of a , but we can, and will, also consider elements of a ∗ aspositive characters of the real or adelic points of A by (5.10).To understand what the exponents above represent, let us consider thedecomposition V = V λ, − ⊕ V λ, ⊕ V λ, + , (5.13)for any given cocharacter λ : G m → H , where G m acts via λ on each of thethree summands with negative, trivial, or positive weights, respectively.If λ is a cocharacter into A , in the relative interior of the cone C , and weconsider it as a cocharacter into H via the composition: G m → A → M → H , then the pull-back of χ C to G m is equal to the inverse of the modularcharacter for the action of G m on V λ, + via λ .Notice that for every λ in the relative interior of a cone C ∈ F , the de-composition (5.13) is the same, so we may write: V = V C, − ⊕ V C, ⊕ V C, + . (5.14)(We notice, though, that these subspaces vary by the H ( k ) -action when adifferent pair ( P , M ) is chosen.)5.7.1. Theorem.
For any Schwartz function f on V ( A k ) , the function Σ N f : h (cid:55)→ (cid:88) γ ∈ N ( k ) f ( γh ) (5.15) on [ H ] is asymptotically finite.More precisely, this map represents a continuous map from the space of Schwartzfunctions F ( V ( A k )) to the space F E ([ H ] F ) of asymptotically finite functions on [ H ] with the exponent arrangement ( F , E ) as above. We will outline the proof of this theorem below, and complete it in Ap-pendix C.5.7.2.
Example.
As a fast check that the signs of the exponent arrangementthat we defined are correct, let us consider the easy case of G m = GL withits standard representation V , whose weight we will denote by χ . Weidentify a = a + with R by evaluating (linear functionals) at χ , and thenthe fan consists of the three sets { } , R ≥ and R ≤ . The exponents are − χ on R ≥ and on R ≤ . If Φ is a Schwartz function on V ( A k ) = A k , then thefunction Σ N Φ( a ) = (cid:80) γ ∈ V ( k ) Φ( γa ) on A × k is asymptotically equal (up to arapidly decaying function) to Φ(0) when | a | (cid:28) . On the other hand, byPoisson summation formula (as in Tate’s thesis) it is asymptotically equalto | a | − ˇΦ(0) when | a | (cid:29) . This matches the behavior described by theabove exponents. Theorem 5.7.1 and Proposition 5.6.1 allow us to define a regularizednilpotent orbital integral for Schwartz functions on V ( A k ) , when all ex-ponents of the arrangement E are non-critical. I recall from § A C ( R ) is “critical” if it is equal to the restriction to A C ( R ) ofthe modular character of the minimal parabolic (or, equivalently, of P C ).5.7.3. Corollary.
Assume that the exponents of the arrangement E are non-critical.Then the map f (cid:55)→ (cid:90) ∗ [ H ] Σ N f ( h ) dh (5.16) is a well-defined, continuous, H ( A k ) -invariant functional on the space F ( V ( A k )) of Schwartz functions on V . The definition depends on the choice dh of invariant measure on [ H ] .While we have already specified that we will be using Tamagawa measure,there are cases (more precisely, when H has non-trivial, k -rational charac-ters) where the usual regularization of Tamagawa measure (using residuesof zeta functions) is rather arbitrary and non-canonical. In the next section,when we define “evaluation maps” on measures, this ambiguity will beabsorbed in the passage from measures to functions. Beginning of the proof of Theorem 5.7.1.
In preparation for the proof of Theo-rem 5.7.1, I present the crux of the argument, which is also at the heart ofother methods of regularization of orbital integrals, cf. [48, 46].First, we may replace the sum over N ( k ) in the definition of Σ N by asum over V ( k ) . Indeed, consider the quotient map: V → c V = V (cid:12) H ;the preimage of the image of zero is N . Without changing the sum (5.15),we may multiply f by the pull-back of a “Schwartz function” on c V ( A k ) whose support meets c V ( k ) only at the image of zero. (I have put “Schwartzfunction” in quotation marks because c V is not necessarily smooth, but wecan embed it in a larger, smooth variety and consider the restriction of aSchwartz function on that.)Thus, it is enough to prove the theorem when the sum over N ( k ) in 5.15is replaced by the corresponding sum over V ( k ) ; the corresponding func-tion will be denoted by Σ V f .The essence of the argument is its following simplistic version, whichshows that the function has the claimed behavior along a cocharacter λ : G m → H ; let us fix such a cocharacter, and split V as in (5.13). HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 61
Then we have (using Tamagawa measures on all integrals): (cid:88) γ ∈ V ( k ) f ( γh ) = (cid:88) γ − ∈ V λ, − ( k ) (cid:114) { } (cid:88) γ ∈ ( V λ, + V λ, + )( k ) f (( γ − + γ ) h ) ++ (cid:88) γ ∈ V λ, ( k ) (cid:88) γ + ∈ V λ, + ( k ) f (( γ + γ + ) h ) − (cid:90) V λ, + ( A k ) f (( γ + n ) h ) dn ++ (cid:90) V λ, + ( A k ) f (( γ + n ) h ) dn (cid:35) . (5.17)For h = λ ( t ) and t → , the terms on the first line are of rapid decaybecause λ acts on V λ, − with negative weights.The term on the third line is an A × k -eigenfunction, when A × k acts via λ ,with character χ C (when λ factors through A and is in the interior of a cone C ).I claim that the term on the second line is of rapid decay. For this, theround bracket can be re-written, using Fourier transform on V λ, + (whichwe will write as f (cid:55)→ ˆ f ): (cid:88) γ ∗ ∈ V ∨ λ, + ( k ) (cid:114) { } (cid:92) h · f γ ( γ ∗ ) , where f γ is the function v (cid:55)→ f ( γ + v ) on V λ, + ( A k ) , and h · represents itstranslate under h = λ ( t ) . Since the weights by which G m acts on V ∨ λ, + via λ are opposite to the ones by which it acts on V λ, + , hence negative, this sumis rapidly decaying as t → .By making this argument “locally uniform in λ ”, in some sense, we willprove the theorem. Although not much more difficult, essentially, the finalargument needs some careful quantification and the introduction of a no-tion of “derivatives” of exponent arrangements, which does not add muchconceptually. I will therefore finish the proof in Appendix C. (cid:3) Example.
Here is an example that comes from the theta correspon-dence (Howe duality): Let H = SO( W ) , the special orthogonal groupof a non-degenerate quadratic space W of, say, even dimension m . Let V = W ⊗ X , where X is a vector space of dimension n ; in the theta cor-respondence, X arises as a Lagrangian of a symplectic vector space. Theproblem at hand is to define a regularized integral: (cid:90) ∗ [ H ] Σ V f ( h ) dh, (5.18)where f ∈ F ( V ( A k )) and Σ V is defined as in the proof above. This is the Kudla-Rallis period, that appears in the theory of the Siegel-Weil formula [32]. (I thank Atsushi Ichino and Tamotsu Ikeda for drawing my attentionto this example.)For simplicity, let us assume that the quadratic space W (and hence thegroup H ) is split, and let (cid:15) , . . . , (cid:15) m denote the standard basis of charactersof its Borel subgroup, so that the simple roots are (cid:15) − (cid:15) , . . . , (cid:15) m − − (cid:15) m and (cid:15) m − + (cid:15) m . Thus, the anti-dominant cone a + is given by the inequalities (cid:15) ≤ · · · ≤ (cid:15) m − ≤ (cid:15) m ≤ − (cid:15) m − . The multiset Φ( V ) of weights of the representation V consists of theweights of the standard representation: ± (cid:15) , . . . , ± (cid:15) m , each appearing with multiplicity n . The hyperplanes orthogonal to thoseweights divide the cone a + into two simplicial subcones, divided by thenew wall with equation (cid:15) m = 0 .There are m + 1 extremal rays of these cones, whose vectors we will de-note, respectively, by v , . . . , v m and v (cid:48) m , where, in coordinates ( (cid:15) , . . . , (cid:15) m ) : v i = ( x, x, . . . , x ( i -th position) , , . . . , with x ≤ and v (cid:48) m = ( x, x, . . . , x, − x ) with x ≤ .For C = R + v i (or v (cid:48) i with i = m ), the character χ C is equal to v i (cid:55)→ nix .The modular character of H is ρ = (cid:80) i m − i ) (cid:15) i . On the vector v i (or v (cid:48) i with i = m ), we have ρ ( v i ) = 2 xi ( m − i +12 ) .Therefore, the exponents of V are all non-critical if and only if nix (cid:54) = 2 xi ( m − i + 12 ) for all ≤ i ≤ m ⇐⇒ n / ∈ [ m − , m − , in which case the regularized integral (5.18) is defined. This is preciselythe range (including the convergent range) in which Kudla and Rallis de-fined a regularized integral in [32] (by a different method), and proved aregularized Siegel-Weil formula.6. E VALUATION MAPS
We return to the general case of X = X/G , where X is smooth affine and G is reductive, both defined over a global field k . We will freely use thenotation of Section 4. Let x : spec k → X be a closed (semisimple) k -pointlying over a point ξ ∈ c ( k ) . We keep assuming that the (reductive) stabilizergroup of x is connected. We will keep assuming, as we may without loss ofgenerality by changing the presentation if necessary, that x corresponds to a G ( k ) -orbit on X ( k ) ; and, whenever we appeal to Luna’s ´etale slice theorem4.1.1, we will be using the pair ( H, V ) corresponding to the chosen point x ,i.e., the identification of pointed ´etale neighborhoods (4.1) will be such that x corresponds to the image of ∈ V . HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 63
We will attempt to define a canonical evaluation map ev x : S ( X ( A k )) ξ → C , factoring through the direct summand S ( X ( A k )) xξ of (4.18). We define it atfirst for V = V /H , using Theorem 5.7.1, and then verify that it is invariantunder the isomorphisms of stalks induced by a diagram of the form (4.2),thus giving rise to a well-defined functional on the stalk S ( X ( A k )) ξ .Having defined the evaluation maps, we can define the “relative traceformula” for the quotient stack X as the sum of all ev x , for x ranging overall isomorphism classes of semisimple k -points of X .Nonetheless, there is the serious restriction that the “exponents” obtainedby application of Luna’s ´etale slice theorem (see Definition 6.2.1) need to be“non-critical” for this definition to make sense. To the best of my under-standing, this definition cannot be overcome by purely geometric methods.This includes the case of the Arthur–Selberg trace formula (i.e., the adjointquotient of a reductive group), which is why one needs a combination ofgeometric and spectral considerations in this case to produce an invariantdistribution. I will not get into this subject here.6.1. The regular case.
Consider first the case when x : spec k → X , orequivalently its image ξ ∈ c ( k ) is regular , i.e., when it satisfies one of thefollowing equivalent conditions: • the preimage of ξ under the map X → c consists only of the (neutral,by assumption) gerbe X ξ ; • the vector space V of Luna’s slice theorem has trivial H -action.The equivalence of the two can easily be obtained from Luna’s ´etale slicetheorem by observing that for any non-trivial representation V of H therewill be a (geometric) weight space with non-trivial weight, and hence a(geometric) non-zero point whose H -orbit contains in its closure.Now let ( H, V ) be Luna’s ´etale neighborhood corresponding to the point x , and recall the map (4.14): E : S ( V ( A k )) → (cid:48) (cid:79) v F ( N ( k v )) ( H ( k v ) ,δ − V ) , where the restricted tensor product on the right-hand side is taken withrespect to the images of the functions: | ω H | v ( H ( o v )) · V ( o v ) . In the regular case, we have S ( V v ) H ( k v ) = S ( V v ) , (recall that the index H ( k v ) denotes H ( k v ) -coinvariants), the nilpotent set N ⊂ V is trivial (consists just of the point ) and hence the map E hasimage in a restricted tensor product of one-dimensional vector spaces (each of which is canonically equal to the space of functions on a point, althoughtheir restricted tensor product might not be — see Remark 4.3.2).We define the evaluation map for the “zero” point: spec k → V by theformula: ev : S ( V ( A k )) (cid:51) µ (cid:55)→ | ω H | ([ H ]) · E ( µ ) ∈ C , (6.1)where | ω H | ([ H ]) denotes the Tamagawa volume on [ H ] defined by any in-variant volume form on H over k . Notice that we do not need to regular-ize Tamagawa measure, because the above product formally makes sense: E ( µ ) is the quotient of a function by a formal Euler product of volumes,the Tamagawa measure | ω H | is formally defined by the same formal Eu-ler product of volumes, therefore these factors cancel out and (6.1) makessense without regularization of the Euler product! Of course, ev is notdefined (is infinite) if [ H ] is not of finite volume; this should be seen as aspecial case of “critical exponents”, to be discussed in more detail in thenext subsection.This definition is clearly invariant under the isomorphisms of stalks in-duced by a diagram of the form (4.2) defined over k . The resulting evalua-tion maps on S ( X ( A k )) ξ can be intuitively described as follows: First, theyfactor through the direct summand (4.18) associated to x . Secondly, theycorrespond, up to the volumes of stabilizers, to “pushing forward a mea-sure on X ( A k ) to c ( A k ) , dividing by a Tamagawa measure on c ( A k ) , andevaluating”. Of course, when the measures are represented by functionstimes differential forms, the evaluation of their push-forward is nothingelse than an orbital integral of the corresponding functions.6.2. The general case.
Let N ⊂ V denote the nilpotent cone, i.e., the closedsubvariety of elements which contain in their H -orbit; equivalently, thepreimage of the image of under the quotient map V → V (cid:12) H .In § V , a pair ( F , E ) , where F is a fan on the cone a + of anti-dominant cocharacters and E is an exponent arrangement for this fan. If all exponents of this arrange-ment are non-critical, by Corollary 5.7.3 we have a well-defined regularizedintegral: (cid:90) ∗ [ H ] Σ N f ( h ) dh for Schwartz functions on V ( A k ) , where Σ N f ( h ) = (cid:80) γ ∈ N ( k ) f ( γh ) . We willnow adapt this to measures.A slight difference to the setup of § H on V , so let us adapt the definitions of theexponent arrangement to the current setting:Let H (cid:48) be the quotient by which H acts on V , and add a prime to everypiece of notation that was used for H in order to refer to the correspondingobject for H (cid:48) . Thus, we have a well-defined exponent arrangement ( F (cid:48) , E (cid:48) ) on the vector space a (cid:48) , with the fan F (cid:48) supported on the anti-dominant HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 65 chamber. I recall that F (cid:48) was obtained from the hyperplanes perpendicu-lar to the weights of V (and the walls of the Weyl chamber), and E (cid:48) wasobtained by summing all weights that are positive each given cone of thefan (and multiplying by − ); thus, they are both very explicit in terms ofthe representation.If the kernel of a → a (cid:48) contains the cocharacter space of a non-trivialcentral split torus of H , then we stop here and consider it a case of criticalexponents: the orbital integral of Σ N over [ H ] cannot be regularized, sincea central split torus acts trivially on V .If not, then the kernel a of a → a (cid:48) is the (cocharacter space of the) uni-versal maximal split torus of a sum of simple factors of H ; it thus admits acanonical splitting: a = a ⊕ a (cid:48) characterized by the fact that a (cid:48) contains the (cocharacter space of the) uni-versal maximal split torus of the remaining simple factors and the center.We then use this splitting to lift the fan F (cid:48) to a fan F on a , and let E be theexponent arrangement on F obtained by applying the bijection between thetwo fans to E (cid:48) .6.2.1. Definition.
Let d V be the modular character of the vector space V . (Itis a character of H , hence of its universal maximal split torus A .) Let δ V E denote the exponent arrangement E shifted by d V , i.e., its elements (writtenadditively) are sums of elements of E by d V . (As multiplicative charactersof A ( A k ) , they are products of elements of E by the absolute value δ V of d V — hence the notation.) The elements of δ V E ( C ) , as C ranges over all thecones in F , will be referred to as the exponents of the point x ∈ X ( k ) fromwhich the pair ( H, V ) was obtained.Recall again the map (4.14): E : S ( V ( A k )) → (cid:48) (cid:79) v F ( N ( k v )) ( H ( k v ) ,δ − V ) . Given Corollary 5.7.3, we can now define the evaluation map, first forthe k -point corresponding to in the quotient stack V , as follows:6.2.2. Definition.
Let µ be an element of S ( V ( A k )) , and let f be a repre-sentative in (cid:78) (cid:48) v F ( N ( k v )) of its image under the map (4.14). We define theevaluation map corresponding to the zero k -point k → V as ev ( µ ) := (cid:90) ∗ [ H ] Σ N f ( h ) δ V ( h ) | ω H | ( h ) , (6.2)whenever the exponents of V at are not critical.Note that, since the above expression considered as a functional on thevariable f is ( H, δ V ) -equivariant, and since it depends only on the restric-tion of f on N ( A k ) , it does not depend on the choice of representative f . By Corollary 5.7.3, this is a continuous functional on the stalk S ( V ( A k )) , andit clearly factors through the direct summand S ( V ( A k )) H ( A k ) , of (4.17).We recall from § A C is “critical” if it is equal tothe restriction to A C of the modular character of the minimal parabolic (or,equivalently, of P C ).Here, again, the measure | ω H | is formally the one obtained from a globalvolume form, and there are formal cancellations with the volume factorsin the definition of the restricted tensor products, so that in the end weare taking the regularized integral of an actual function against an actual ,non-zero measure; see the discussion following (6.1).Now we extend definition 6.2.2 to the point x of the stack X . It will not,at this point, be clear that the definition is independent of choices; this willbe proven in the next section, Proposition 6.3.1.6.2.3. Definition.
Provided that X has no critical exponents at x , we definethe evaluation map: ev x : S ( X ( A k )) ξ → C as the pull-back of the evaluation map ev under the identification of stalks: S ( X ( A k )) ξ (cid:39) S ( V ( A k )) of (4.16), induced from a Luna diagram (4.1).It is a continuous functional on the stalk S ( X ( A k )) ξ , and it clearly factorsthrough the direct summand S ( X ( A k )) xξ of (4.18).6.3. Invariance under isomorphism of ´etale neighborhoods.
Up to nowwe have defined a distribution ev on S ( V ( A k )) which, in principle, mightnot be invariant under the isomorphism of stalks induced by a diagram ofthe form (4.2). Here we check that when the diagram is defined over k , thenthe distribution ev is invariant under the resulting isomorphisms of stalks.As a corollary, the evaluation map ev x of Definition 6.2.3 is well-defined, inthe absence of critical exponents.6.3.1. Proposition.
Consider a diagram of the form (4.2) over k , arising from twodifferent choices of data for Luna’s theorem, and the automorphism S ( V ( A k )) ∼ −→ S ( V ( A k )) that it induces on the stalk. The functional ev is invariant under this automor-phism.Proof. We will use notation as in diagram (4.3) and in the proof of Proposi-tion 4.3.4.Consider the space of volume forms on P which are d V × d V -equivariantunder the action of H × H — they form sections of a line bundle (cid:101) L over c Y .If N P denotes the preimage in P of the distinguished point of c Y , then as in HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 67
Lemmas 4.2.3 and 4.3.3, any section ω P of (cid:101) L over k gives rise to the samemap E P : S ( Y ( A k )) → (cid:48) (cid:79) v F ( N P ( k v )) ( H v × H v ,δ − V × δ − V ) . (6.3)Here the basic vector for the restricted tensor product on the right side willbe | ω H | v ( H ( o v )) · P ( o v ) . On the other hand, the evaluation map ev that we have defined on S ( V ( A k )) pulls back in two different ways to the corresponding restrictedtensor product (cid:48) (cid:79) v S ( P ( k v )) , (6.4)by the “left” and the “right” sequence of arrows from P to V .I claim that both pullbacks of the evaluation map to S ( Y ( A k )) can beexpressed in terms of (6.3) as follows: µ (cid:55)→ (cid:90) ∗ [ H × H ] (cid:88) γ ∈ N P ( k ) E P ( µ )( γ · ( h , h )) δ V ( h ) δ V ( h ) dh dh . (6.5)Indeed, the map W → V which is Cartesian over c Y → c V identifies N ⊂ V with its preimage N in W over the distinguished point of c Y ,and the preimage of that in P is N P . Recall that in order to define the maps(4.14) and (6.3) we have chosen H -eigen- (resp. H × H -eigen-)volume forms ω V and ω P on V and P (although the global result did not depend on thosechoices). We can pull back ω V to a volume form ω on W . Since P → W isan H -torsor, the conormal bundles to all H -orbits form a subbundle of thecotangent bundle of P , and since ω is non-vanishing, the quotient of ω P by the pull-back of ω is a well-defined section ω (cid:48) P of the top exterior powerof this subbundle. This induces an H -eigen-volume form on every fiber ofthe map P → W .Now recall from Lemma 4.3.5 that the map N P ( k ) → N ( k ) is surjective.In other words, over k -points of N the H -torsor P is trivializable. Choos-ing any k -point to trivialize it, the restriction of ω (cid:48) P on its fiber is identifiedwith a Haar volume form on H . It is now immediate to unfold the integral(6.5) and to see that it corresponds to the pull-back of the evaluation mapvia the left arrows to V . Exactly the same applies, of course, to the rightarrows.Again by Lemma 4.3.5 the image of (6.4) in S ( V ( A k )) is precisely the di-rect summand S ( V ( A k )) H ( A k ) , through which ev factors. Hence, the factthat ev pulls back to the same functional under both the “left” and “right”sequence of arrows shows that ev is invariant under the isomorphism ofstalks induced by (4.2). (cid:3) The “relative trace formula”.
We keep assuming that X is an algebraicstack of the form X = X/G over k , where X is a smooth affine variety and G is a reductive group. We assume that X carries a nowhere vanishing G -eigen-volume form ω X , so that its global Schwartz space S ( X ( A k )) = (cid:48) (cid:79) v S ( X v ) can be defined, as in Remark 4.3.7.6.4.1. Definition.
Assume that the stabilizer subgroups of all semisimplepoints x : spec k → X are connected and their exponents are non-critical.The relative trace formula for X is the following distribution on S ( X ( A k )) : RTF X : f (cid:55)→ (cid:88) x ev x , (6.6)where x runs over isomorphism classes of closed k -points into X , and ev x is the evaluation map of definition 6.2.2.Of course, this has nothing to do with traces, in general.6.4.2. Example.
The simplest example of a relative trace formula where some“stacky” behavior is seen is the one considered by Jacquet in [27], wherethe stack is X = T \ PGL /T , where T is a non-split torus in PGL , definedover k . In this case, since H ( k, T ) injects in H ( k, PGL ) (and the same istrue for the completions k v ), isomorphism classes of T -torsors R inject intoisomorphism classes of quaternion algebras D R , and the image consists ofthose quaternion algebras such that D × R contains T or equivalently: thequaternion algebra splits over the quadratic extension splitting T .Thus, isomorphism classes of k -points of X are in bijection with (cid:71) R T ( k ) \ P D × R ( k ) /T ( k ) , where R runs over isomorphism classes of those quaternion algebras, and P D × R denotes the quotient of D × R by its center, and the relative trace formulacan be written as a sum (cid:88) R (cid:88) x ∈ T ( k ) \ P D × R ( k ) /T ( k ) ev x . (6.7)The local Schwartz spaces are the T ( k v ) × T ( k v ) -coinvariants of the directsum (cid:88) R v S ( P D × R v ( k v )) , the sums ranging over isomorphism classes of T -torsors over k v , and anySchwartz measure on P D × R v ( k v ) can be written as f dg , where f is a Schwartzfunction and dg is a Haar measure on P D × R v ( k v ) .Thus, the evaluation maps ev x of (6.7) can be identified with orbital in-tegrals of those functions f , and this holds both for regular (with trivial HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 69 stabilizers, in this case) and irregular points, as can easily be checked fromthe definitions using the fact that T is globally anisotropic and the nilpotentcones have no k -points.6.4.3. Example.
In the aforementioned paper, Jacquet compares the relativetrace formula for T \ PGL /T with the relative trace formula when T is re-placed by a split torus A , but with the T ( A k ) -action twisted by a quadraticcharacter on one side. Such characters are not part of the formalism of thepresent paper (and they are important, as they show up elsewhere as well— including additive characters in the Kuznetsov formula), but it wouldprobably not be difficult to incorporate them as line bundles over the per-tinent Nash stack.The case of A \ PGL /A (with trivial character) was not considered in[27] because it would lead to no new arithmetic results, but it is interestingfrom the analytic point of view to make sense of such a trace formula, andwas considered in [39, 38]. In this case, at the irregular points the lineariza-tion of the stack (i.e., the space V /H of Luna’s theorem) is of the form V = G a (1) ⊕ G a ( − , where G a ( i ) denotes an one-dimensional vector space where A (cid:39) G m actswith weight a (cid:55)→ a i . By [38, Lemma 2.6.1], the contribution of nilpotentcone to the “relative trace formula” functional applied to a measure Φ dv on V ( A k ) (where dv is Tamagawa measure) is given by lim t → ( ζ (Φ | x , t ) + ζ (Φ | y , − t )) , (6.8)where Φ | x denotes the restriction of Φ to the “ x -axis” G a (1) , Φ | y its restric-tion to G a ( − and ζ the corresponding Tate integrals. One can check thatin this case the function Σ N Φ , in the notation of Theorem 5.7.1, is asymp-totically finite on [ G m ] with exponents t (cid:55)→ | t | − , resp. t (cid:55)→ | t | as | t | → ,resp. | t | → ∞ , and (6.8) is the regularized integral of Σ N Φ .6.4.4. Example.
The exponents are non-critical, and hence the relative traceformula can be defined by the methods of the current paper, for the quo-tient spaces ( X × X ) /G diag when X is the homogeneous space of the Gross–Prasad conjectures ( SO diag n \ SO n × SO n +1 or U diag n \ U n × U n +1 , considered in[48, 47]), and for a variant of the corresponding Jacquet–Rallis relative traceformula for linear groups [46, 47]. I expect that, in the unitary Gross–Prasadcase that has been studied by Zydor, the resulting distribution coincideswith his, but it would be an interesting exercise to show that.Let us check that the exponents are non-critical at the most singularpoints of the morphism X → c , where X denotes the pertinent stack in eachcase and c denotes the invariant-theoretic quotient. In the Gross–Prasadcases, this is the point represented by the element in the presentations: X = SO n +1 / SO n -conj , resp. X = U n +1 /U n -conj . Here SO n +1 , resp. U n +1 , denotes the special orthogonal, resp. unitary groupof the quadratic/hermitian space obtained from that of SO n by adding anorthogonal line (with a non-degenerate quadratic/hermitian form on it).Let H := H n := SO n , resp. U n . The linearization of the stack at is theadjoint representation of H on the Lie algebra h n +1 . It decomposes as h ⊕ std , where h implies the adjoint action of H on its Lie algebra and std is the stan-dard representation, plus a copy of the trivial representation in the unitarycase (but this will not contribute anything to the weights).We restrict this representation to a maximal split torus (inside of a chosenminimal parabolic subgroup), and use the recipe for constructing a fan andan exponent arrangement that was given in § m is the split rank of H , and we embed GL m in H into a Levi subgroup (let us assume that m ≥ in the even orthogonal case),and denote by (cid:15) , . . . , (cid:15) m the standard characters of the torus of diagonalelements in GL m , the anti-dominant cone with respect to the usual choiceof Borel is given by the inequalities (cid:15) ≤ · · · ≤ (cid:15) n − ≤ (cid:15) n ≤ , except in the even orthogonal case where it is given by (cid:15) ≤ · · · ≤ (cid:15) n − ≤ (cid:15) n ≤ − (cid:15) n − . The non-zero weights of the standard representation are ± (cid:15) , ± (cid:15) , . . . , ± (cid:15) n , and therefore they don’t introduce new walls to the anti-dominant chamber(they maintain their sign), except in the even orthogonal case where theyintroduce a wall at (cid:15) n = 0 .Thus, the fan F is the one corresponding to the anti-dominant cone a + ,except in the even orthogonal case where it corresponds to the partition ofthis cone into two subcones along the hyperplane (cid:15) n = 0 .In either case, the exponent arrangment is obtained by adding to themodular character of the minimal parabolic (coming from h ) the followingcharacter: • for the cone given by the inequalities (cid:15) ≤ · · · ≤ (cid:15) n − ≤ (cid:15) n ≤ , and for all its faces, the character (cid:15) + · · · + (cid:15) n − + (cid:15) n ; • in the even orthogonal case, for the other cone (with (cid:15) n ≥ ) and itsfaces, the character (cid:15) + · · · + (cid:15) n − − (cid:15) n . HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 71
In both cases, the character added is non-trivial , unless all (cid:15) i = 0 , i.e.,except at the point ∈ a . Therefore, the exponents are never critical. Thisfinishes the discussion of the Gross–Prasad quotients.In the linear case, the relevant stack is X = GL n,E \ GL n,E × GL n +1 ,E / GL n × GL n +1 , where GL n , GL n +1 are defined over our global field k , and with an in-dex E we denote the restriction of scalars of the base change to a qua-dratic extension E/k . However, there is a quadratic character enteringthe Jacquet-Rallis relative trace formula (on GL n or GL n +1 , whichever iseven-dimensional), which I have presently not included in the formalism.However, the linearization close to the most singular point of X → c (againrepresented by the element ) has been described in [46], and has to dowith H := H n := GL n -orbital integrals, against a quadratic character, onthe space of the representation V = { X ∈ h n +1 ,E | X + ¯ X = 0 } . That is, theonly difference from the setup of the present paper is that for the same pair ( H, V ) , one needs to put a quadratic character on H .We check again the asymptotic behavior of the functions of the form Σ V Φ , where Φ is a Schwartz function on V ( A k ) . The representation V isisomorphic to the adjoint action of H = H n on the Lie algebra h n +1 , andhence decomposes as h ⊕ std , plus a copy of the trivial representation. By the same arguments as above,the fan given by the recipe of § PPENDIX
A. F
ROM ALGEBRAIC TO N ASH STACKS : APRESENTATION - FREE APPROACH
A.1. Let X be a smooth algebraic stack of finite type over a local field F .In this appendix I will construct out of X a stack X over the (´etale) site N of Nash manifolds, that does not use a fixed presentation of X but, rather,a “limit” over all presentations. If there is an F -surjective presentation,this construction is equivalent to the construction by groupoids that waspresented in § X = X ( F ) over N as a “limit” of the Nashgroupoids [ R X ( F ) ⇒ X ( F )] , where X runs over all algebraic presentations X → X , and R X = X × X X .This “limit” will be obtained by localization of a category with respectto smooth surjective morphisms: X (cid:48) → X over X . More precisely, we firstdefine a pre-stack X pre-stack , as a localization of a category X naive . The cat-egory X naive is fibered not only over N , but also over Pres X , the categoryof smooth epimorphisms of algebraic stacks (“presentations”): X → X , where X is a (necessarily smooth) scheme of finite type over F , with mor-phisms between ( X (cid:48) → X ) and ( X → X ) being all smooth morphisms X (cid:48) → X over X . For X ∈ ob(Pres X ) , set R X = X × X X , with the “source”and “target” maps s and t : R X s ⇒ t X. The fiber category of X naive over ( X → X ) ∈ ob(Pres X ) is the Nash stackassociated to the Nash groupoid [ R X ( F ) ⇒ X ( F )] .Let X (cid:48) → X be a morphism in Pres X , and ( T → W, T → X ( F )) an objectin X naive over ( X → X ) . The fiber product T (cid:48) := T × X ( F ) X (cid:48) ( F ) is still smooth over W , and an epimorphism iff the image of X (cid:48) ( F ) in X ( F ) contains the image of T . The map T (cid:48) → W is naturally an R X (cid:48) -torsor.A morphism ( T (cid:48) → V, T (cid:48) → X (cid:48) ( F )) → ( T → W, T → X ( F )) in X naive isa Cartesian square of Nash manifolds: T (cid:48) (cid:47) (cid:47) (cid:15) (cid:15) T (cid:15) (cid:15) V × X (cid:48) ( F ) (cid:47) (cid:47) W × X ( F ) , (A.1)satisfying the following requirements: the bottom arrow is induced froma morphism V → W in N and a morphism X (cid:48) → X in Pres X , the image of X (cid:48) ( F ) → X ( F ) contains the image of T → X ( F ) , and the induced diagramof groupoids commutes: [ T (cid:48) × V T (cid:48) ⇒ T (cid:48) ] (cid:47) (cid:47) (cid:15) (cid:15) [ T × W T ⇒ T ] (cid:15) (cid:15) [ R X (cid:48) ( F ) ⇒ X (cid:48) ( F )] (cid:47) (cid:47) [ R X ( F ) ⇒ X ( F )] . For simplicity, we will usually represent such a morphism just by the arrow T (cid:48) → T , with all other arrows being implicit.Now we construct the pre-stack X pre-stack by localizing X naive over a class S of morphisms, i.e., X pre-stack = X naive [ S − ] . The class S is the class ofthose morphisms of the form (A.1) where V = W and the map V → W isthe identity, i.e., T (cid:48) is obtained by base change in Pres X only: T (cid:48) (cid:47) (cid:47) (cid:15) (cid:15) T (cid:15) (cid:15) X (cid:48) ( F ) (cid:47) (cid:47) X ( F ) (where, again, I remind the reader that the image of T has to lie, set-theoretically,in the image of X (cid:48) ( F ) ). The morphisms in S will be represented by doublearrows ⇒ . HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 73
A.1.1.
Proposition.
The class S allows a calculus of right fractions. This meansthat it verifies the right Ore conditions:(a) for any s ∈ S and any other morphism f in X naive with the same target, thereis a commutative diagram T f (cid:48) (cid:47) (cid:47) s (cid:48) (cid:11) (cid:19) T s (cid:11) (cid:19) T f (cid:47) (cid:47) T with s (cid:48) ∈ S .(b) If T f (cid:47) (cid:47) f (cid:47) (cid:47) T t (cid:43) (cid:51) T is such that t ◦ f = t ◦ f , with t ∈ S , then there is an s in S such that f ◦ s = f ◦ s : T s (cid:43) (cid:51) T f (cid:47) (cid:47) f (cid:47) (cid:47) T . Proof.
The first one is true, with the arrows induced by base change fromthe corresponding diagram: V × X ( F ) (cid:47) (cid:47) (cid:11) (cid:19) W × X ( F ) s (cid:11) (cid:19) V × X ( F ) f (cid:47) (cid:47) W × X ( F ) where X = X × X X .For the second one, consider the corresponding arrows: T (cid:15) (cid:15) f (cid:47) (cid:47) f (cid:47) (cid:47) T t (cid:43) (cid:51) (cid:15) (cid:15) T (cid:15) (cid:15) V × X ( F ) f (cid:47) (cid:47) f (cid:47) (cid:47) U × X ( F ) t (cid:43) (cid:51) U × X ( F ) , where we recall that corresponding squares are Cartesian. Since t is, bydefinition of the class S , the identity on U , and t ◦ f = t ◦ f by assumption,it follows that f | V = f | V . Therefore, the left squares can be completed to a Cartesian diagram: T (cid:15) (cid:15) s (cid:43) (cid:51) T (cid:15) (cid:15) f (cid:47) (cid:47) f (cid:47) (cid:47) T (cid:15) (cid:15) V × ( X × f ,X ,f X )( F ) (cid:43) (cid:51) V × X ( F ) f (cid:47) (cid:47) f (cid:47) (cid:47) U × X ( F ) . The square on the left corresponds to a morphism s in S , and by construc-tion we have f ◦ s = f ◦ s . (cid:3) The fact that S allows a calculus of right fractions means that morphisms T → T in X pre-stack = X naive [ S − ] can be described as equivalence classesof “roofs”: Z (cid:122) (cid:2) (cid:32) (cid:32) T T where two roofs with sources Z and Z (cid:48) are equivalent if they are dominatedby a third one, i.e., there is a commutative diagram: Z (cid:122) (cid:2) (cid:33) (cid:33) T Z (cid:48)(cid:48) (cid:107) (cid:115) (cid:79) (cid:79) (cid:15) (cid:15) (cid:47) (cid:47) T .Z (cid:48) (cid:92) (cid:100) (cid:61) (cid:61) This makes it easy to check that X pre-stack is indeed a pre-stack.A.1.2. Proposition. X pre-stack = X naive [ S − ] is a pre-stack in the ´etale topologyover N .Proof. We first check that it is a category fibered in groupoids over N .First, given an object ( T → U, T → X ( F )) in X pre-stack U and a morphism V → U in N , we get by base change an object ( T × U V → V, T × U V → X ( F )) in X pre-stack V as required by the first axiom of “category fibered ingroupoids”. Notice that both objects live over the same ( X → X ) ∈ ob(Pres X ) .For the second axiom, let W → V → U be a diagram in N and T W → T U , T V → T U morphisms in X pre-stack lying over W → U , V → U , respec-tively. If we denote by X W , X V and X U the smooth X -schemes over whichthey live, the morphisms between them mean correspond to equivalenceclasses of base changes: T (cid:48) W → X (cid:48) W ( F ) , T (cid:48) V → X (cid:48) V ( F ) (induced from X (cid:48) W → X W , X (cid:48) V → X V ) and morphisms: X (cid:48) W → X U , X (cid:48) V → X U , together HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 75 with Cartesian diagrams: T (cid:48) W (cid:15) (cid:15) (cid:47) (cid:47) T U (cid:15) (cid:15) X (cid:48) W ( F ) (cid:47) (cid:47) X U ( F ) (and similarly for T (cid:48) V ).Thus, T (cid:48) W (cid:39) T U × U × X U ( F ) ( W × X (cid:48) W ( F )) (and similarly for T (cid:48) V ) and themap W → V induces a unique map: T (cid:48)(cid:48) W → T (cid:48) V , where T (cid:48)(cid:48) W = T U × U × X U ( F ) ( W × ( X (cid:48) W × X U X (cid:48) V )( F )) . The natural map T (cid:48)(cid:48) W → T (cid:48) W corresponds to a morphism in S , hence the arrow W → V lifts toan arrow T W → T V in X pre-stack such that the composition: T W → T V → T U equals the given morphism T W → T U .It is easy to see that any two roofs representing such a morphism T W → T V are dominated by a third one (constructed again by a fiber product),hence the lift is unique and the category is indeed fibered in groupoidsover N .Now we verify the pre-stack axiom: Consider two isomorphisms be-tween two objects T , T ∈ ob( X pre-stack U ) . Suppose that these isomorphismsare identified over all elements of a covering family { U i → U } i , that is: allelements of a finite cover of U by ´etale maps. The isomorphisms are rep-resented by “roofs” as before, and the fact that they are identified locallymeans that, locally, these roofs are dominated by other roofs. The latter lieover objects ( X i → X ) ∈ Pres X ; notice that each of them is a presentation of X , since the cover only concerns the N -parameter. Therefore, a fiber prod-uct of the X i ’s (which are finite in number) over X will provide the requiredisomorphism between the original morphisms.Vice versa, assume that U = U ∪ U (for simplicity), and let T ,U i → T ,U i be isomorphisms which “agree” over U ∩ U . These morphisms arerepresented by morphisms in X naive : T (cid:48) ,U i → T ,U i , where T (cid:48) ,U i → T ,U i isin S (and we may by base change assume that for both i = 1 , the objects T (cid:48) ,U i live over the same object ( X (cid:48) → X ) ∈ Pres X . The condition of theiragreement on U ∩ U can be expressed with a further roof over U ∩ U ,and again by taking fiber products with the corresponding presentationwe may in fact assume that the actual morphisms: T (cid:48) ,U i → T ,U i agree over U ∩ U . But then they can be glued to a morphism of Nash stacks: T (cid:48) → T which together with the map T (cid:48) → T represents the desired isomorphism: T → T ∈ X pre-stack . (cid:3) A.1.3.
Definition.
Given a smooth algebraic stack X over F , we define astack over N as follows: X := X ( F ) := the stackification of X pre-stack .This completes the definition, which poses a number of interesting ques-tions, such as: Is the association X → X functorial? Is X a Nash stack? I willnot address these questions in this paper, except for observing that the def-inition coincides with the one given in § F -surjectivepresentations:A.1.4. Proposition.
Let X → X be an F -surjective presentation of the smoothalgebraic stacks X over F .Then X ( F ) → X := X ( F ) is a smooth epimorphism of stacks over N and, inparticular, X is a Nash stack.This Nash stack is equivalent to the stack defined by the groupoid object [( X × X X )( F ) ⇒ X ( F )] . Proof.
The point is that the stack [( X × X X )( F ) ⇒ X ( F )] is a final object, insome sense, in X naive . More precisely, let X (cid:48) → X be any other presentationof X , and let X (cid:48)(cid:48) = X × X X (cid:48) . Then we have obvious -morphisms of stacks: [ R X (cid:48)(cid:48) ( F ) ⇒ X (cid:48)(cid:48) ( F )] → [ R X ( F ) ⇒ X ( F )] , [ R X (cid:48)(cid:48) ( F ) ⇒ X (cid:48) ( F )] → [ R X (cid:48) ( F ) ⇒ X (cid:48) ( F )] . Since X → X is F -surjective, the smooth morphism of Nash manifolds X (cid:48)(cid:48) ( F ) → X (cid:48) ( F ) is an epimorphism, and hence by Lemma 2.2.3 the secondmorphism above is an equivalence. It follows that for any presentation X (cid:48) → X we have a canonical (up to 2-isomorphism) -morphism of stacks: [ R X (cid:48) ( F ) ⇒ X (cid:48) ( F )] → [ R X ( F ) ⇒ X ( F )] , and that the “limit” prestack X pre-stack (and hence its stackification X ) isequivalent to the stack associated to [ R X ( F ) ⇒ X ( F )] . Of course, the -morphism X ( F ) → X is automatically a smooth epimorphism in that case. (cid:3) A PPENDIX
B. H
OMOLOGY AND COSHSEAVES IN NON - ABELIANCATEGORIES
B.1.
Exact structures.
Our basic references are [18, 22].B.1.1.
Definition.
Let A be an additive category. A kernel-cokernel pair ( f, g ) in A is a pair of composable morphisms X f −→ Y g −→ Z in A , such that f is a kernel of g and g is a cokernel of f . If a class E ofkernel-cokernel pairs on A is fixed, an admissible monomorphism is a mor-phism f for which there exists a morphism g such that ( f, g ) ∈ E . Admissibleepimorphisms are defined dually.An exact structure on A is a class E of kernel-cokernel pairs which isclosed under isomorphisms and satisfies the following axioms: HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 77 E For all objects X in A , the identity morphism id X : X → X is anadmissible monomorphism.E op For all objects X in A , the identity morphism id X : X → X isan admissible epimorphism.E The class of admissible monomorphisms is closed under composi-tion.E op The class of admissible epimorphisms is closed under composi-tion.E The push-out of an admissible monomorphism along an arbitrarymorphism exists and yields an admissible monomorphism.E op The pull-back of an admissible epimorphism along an arbitrarymorphism exists and yields an admissible epimorphism.An additive category A is called quasi-abelian if:(1) every morphism has a kernel and a cokernel (such a category iscalled pre-abelian ), and(2) the class of kernels is stable under push-out along arbitrary mor-phisms and the class of cokernels is stable under pull-back alongarbitrary morphisms.In quasi-abelian categories, the class E max of all kernel-cokernel pairs isan exact structure. Vice versa, the property that all kernel-cokernel pairsform an exact structure characterizes quasi-abelian categories among pre-abelian ones.For example, the categories of Banach spaces and Fr´echet spaces (ornuclear Fr´echet spaces) are quasi-abelian categories. The kernel-cokernelpairs X f −→ Y g −→ Z in them are those pairs of mono- and epimorphismswhich are strict , i.e., with closed image. In other words, f should be injec-tive and g is surjective with ker( g ) = im( f ) (set-theoretically).Let A be an additive category equipped with an exact structure E . Achain complex → A n → A n +1 → . . . is called exact or acyclic if each differ-ential factors A n → B n → A n +1 in such a way that B n → A n +1 → B n +1 is in E . A chain map f : A → B is called a quasi-isomorphism if its mapping coneis homotopy equivalent to an acyclic complex; in pre-abelian categories onecan directly say that the mapping cone is acyclic.The derived category is then defined in the usual way, by localization ofthe homotopy category with respect to quasi-isomorphisms.B.2. Cosheaves.
Cosheaves are the notion that one obtains from sheavesby inverting the arrows in the source category. In other words, a cosheafon a site C valued in a suitable category A is a functor F : C → A (a pre-cosheaf ) that takes covers to colimits, that is, for every covering family ( U i → U ) i we have: F ( U ) = lim → (cid:71) i,j F ( U i × U U j ) ⇒ (cid:71) i F ( U i ) . (B.1)Moreover, if A is an additive category endowed with an exact structure, wemay impose a stricter condition on the cosheaf that is related to the exactstructure.In this paper, all cosheaves are valued either in complex vector spaceswithout topology (in the non-Archimedean case), or in Fr´echet spaces. More-over, the sites are such that any covering family can be replaced by a singleelement U (cid:48) → U . (We will tacitly assume this for all sites in the discussionthat follows.) Under these simplifying conditions, (B.1) becomes simply aco-equalizer diagram: F ( U (cid:48) × U U (cid:48) ) ⇒ F ( U (cid:48) ) → F ( U ) . The stricter condition implied above, when F is valued in Fr´echet spaces,is that the co-equalizer is strict , that is, if we denote by s and t the “source”and “target” arrows above, then the image of s ! − t ! is closed. From now on,when talking about a cosheaf valued in Fr´echet spaces, we will be assumingthis condition. (For a morphism s in C , the induced morphism in A isdenoted by s ! .)If C has a final object “ ∗ ”, the global sections functor is obtained by eval-uating at that object. If it does not have a final object, we can try to definethe global sections functor as a colimit: F ( ∗ ) := lim → U ∈ ob( C ) F ( U ) , but it is not, in general, clear that one can take the colimit valued in thesame target category (e.g., Fr´echet spaces).Here, I take a more ad-hoc point of view on global sections and homol-ogy, using only ˇCech homology for cosheaves. More precisely, for U ∈ ob( C ) and a covering V → U , we define ˇ H V • ( U, F ) : to be the (strict) quasi-isomorphism class of the complex: → F ([ V ] i +1 U ) → F ([ V ] iU ) → · · · → F ( V × U V ) → F ( V ) → , (B.2)where [ V ] iU denotes the i -fold fiber product of V over U , and the differen-tials are obtained by alternating sums of the morphisms that one gets from [ V ] i +1 U to [ V ] iU by forgetting the j -th copy, j = 0 , . . . , i .When C has a final object, we say that the cosheaf is acyclic if the complexabove is quasi-isomorphic to → F ( U ) → for every U ∈ ob( C ) andevery cover V → U . When C does not have a final object, but satisfies thesame property, I will call it locally acyclic . (A more careful definition wouldrequire this condition only after passing to covers, but this will be enough HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 79 for the purposes of this paper.) Here is the relation of this notion to thefunctor of global sections:B.2.1.
Lemma.
Assume that F f −→ G is a strict epimorphism of cosheaves of vec-tor spaces or Fr´echet spaces over a site C , that is: for every U ∈ ob( C ) the map: F ( U ) → G ( U ) is a strict epimorphism. Assume that G is locally acyclic. Then thekernel pre-cosheaf ker f ( U ) := ker( F ( U ) → G ( U )) is a cosheaf, the sequence ker f ( U ) → F ( U ) → G ( U ) is a kernel-cokernel pair in the category of vec-tor/Fr´echet spaces for every U and, moreover, for any cover V → U we have adistinguished triangle in the derived category of vector/Fr´echet spaces: → ˇ H V • ( U, G )[ − → ˇ H V • ( U, ker f ) → ˇ H V • ( U, F ) → ˇ H V • ( U, G ) → B.2.2.
Remark.
The statement about distinguished triangles holds in a muchmore general setting of kernel-cokernel pairs
H → F → G of cosheaves;however, since there may be issues with cosheafification (which is neededto construct kernels) in the category of Fr´echet spaces, I will avoid talkingabout exact structures of Fr´echet cosheaves in more generality.
Proof.
The proof is as in the abelian case, by chasing arrows along the fol-lowing diagram, and is left to the reader:... (cid:15) (cid:15) ... (cid:15) (cid:15) ... (cid:15) (cid:15) (cid:47) (cid:47) ker f ( V × U V ) (cid:15) (cid:15) (cid:47) (cid:47) F ( V × U V ) (cid:15) (cid:15) (cid:47) (cid:47) G ( V × U V ) (cid:15) (cid:15) (cid:47) (cid:47) (cid:47) (cid:47) ker f ( V ) (cid:15) (cid:15) (cid:47) (cid:47) F ( V ) (cid:15) (cid:15) (cid:47) (cid:47) G ( V ) (cid:15) (cid:15) (cid:47) (cid:47) (cid:47) (cid:47) ker f ( U ) (cid:15) (cid:15) (cid:47) (cid:47) F ( U ) (cid:15) (cid:15) (cid:47) (cid:47) G ( U ) (cid:15) (cid:15) (cid:47) (cid:47)
00 0 0
Notice that, without any assumptions on the acyclicity of G , the factthat f is a strict epimorphism implies that for any cover V → U , the map ker f ( V ) → ker f ( U ) is a strict epimorphism. The acyclicity assumption isused to say that the kernel of this map is precisely the image of ker f ( V × U V ) . (cid:3) Now let X be a restricted topological space, and let C = X Zar be the“Zariski site” generated by its topology, i.e., its objects are open subsetsof X , morphisms are inclusions, and covering families are finite families of morphisms which are jointly surjective. In fact, it is more convenientto enlarge C to include disjoint unions of open subsets of X as its objects,with the obvious morphisms, so that coverings can be represented by asingle morphism V → U .A cosheaf F on X Zar is called flabby if for any inclusion of open subsets U → V the induced map F ( U ) → F ( V ) is a closed inclusion.B.2.3. Lemma.
Let F be a cosheaf of vector spaces or of Fr´echet spaces on theZariski site of a restricted topological space. If F is flabby, then it is acyclic.Proof. If we were to form the complex (B.2) in the category of cosheaves val-ued in vector spaces without topology, then the statement about acyclicitywould follow from usual considerations (or from the even more commontheory of sheaves, applied to the linear duals). We know, moreover, thatthe maps in (B.2) are continuous, so in degree ≥ , where the complex asabstract vector spaces is exact, the images of the differentials are closed.On the other hand, in degree zero we have by the cosheaf axioms that F ( V × U V ) → F ( V ) → F ( U ) → is strictly exact, which shows that (B.2)is strictly quasi-isomorphic to the complex → F ( U ) → . (cid:3) B.3.
Homology for smooth F -representations. This subsection aims to ex-plain the homological meaning of the derived coinvariant functor of smooth F -representations of real algebraic (or Nash) groups, that I defined in an adhoc way in § M H of smooth F -representationsof such a group H is equivalent to the category of non-degenerate A -modules,where A := S ( H ) is the convolution algebra of Schwartz measures on H .The formalism of homological algebra that we need is due to Delignefrom SGA4 [19], and presented very nicely in [18]. To apply it to our setting,we will use the results of Taylor [43] which, however, only hold for unitalalgebras. Therefore, we set ˜ A = A ⊕ C , with the algebra structure extending that of A and making ∈ C the iden-tity element. It is a nuclear Fr´echet algebra, and hence we have uniquelydefined completed tensor products: ˜ A ˆ ⊗ V for any Fr´echet space V . We let A be the category of Fr´echet ˜ A -modules, and we endow it with the exactstructure of all strongly exact sequences as in [43, § ˜ A -modules · · · → V n +1 → V n → V n − → . . . is strongly exact if it is strictly exact, and in addition every kernel and cok-ernel admit complements as topological vector spaces (i.e., not necessarily as ˜ A -modules).On the other hand, the category F is quasi-abelian, and hence has acanonical (maximal) exact structure, that of strict exact sequences.We assume given a continuous, non-zero homomorphism of algebras: (cid:82) : A → C , or, equivalently a non-degenerate A -module structure on C HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 81 (where a ∈ A acts as multiplication by (cid:82) a ). For our Schwartz algebra thisfunctional is the total integral, as the notation suggests. We extend it to ˜ A as the identity on C . The rough coinvariant functor corresponds to theassociation V (cid:55)→ V / the closure of the subspace spanned by vectors of the form ( av − (cid:90) a · v ) , a ∈ ˜ A, v ∈ V. It is a functor C : A → F , and it coincides with our “rough” functor of H -coinvariants on non-degeneratemodules of S ( H ) .A free object in A is an object of the form V = ˜ A ˆ ⊗ W , where the action of ˜ A is on the first factor by left multiplication. Free objects have the followingproperties: • They are projective , i.e., if V is free, then the (vector space-valued)functor Hom A ( V, • ) is exact — it turns strongly exact sequences in A into exact sequences of vector spaces. • We have CV = C ˆ ⊗ A V = W , where C ˆ ⊗ A denotes the quotient of C ˆ ⊗ V by the image of the map: A ˆ ⊗ V → V , a ⊗ v (cid:55)→ av − (cid:82) a · v . Thequotient is taken without closure, and in particular the statementhere means that the image is closed.The first statement follows from the fact that Hom ˜ A ( V, Z ) = Hom(
W, Z ) ,cf. [43, Proposition 1.3]. (The morphisms on the left are in A , and thoseon the right in F .) The second is [43, Proposition 1.5]. Clearly, for bothstatements, it doesn’t matter whether we use A or ˜ A . The second statementeasily extends to all projective objects, since by [43, Proposition 1.4] thoseare precisely the direct summands of free ones. Thus, the functor C of coin-variants, restricted to the fully exact subcategory P of projective objects in A , is exact.As we will recall in a moment, every object in A admits a projective res-olution. By [18, Theorem 10.22] with arrows reversed (the second conditionof this theorem is trivially satisfied by projectivity), this implies that the de-rived category D − P is equivalent to D − A . By [18, Lemma 10.26] (againreversing arrows), the functor C , when applied to a complex of projectiveobjects representing a given object in D − A , yields the total derived functor L C of C : L C : D − A (cid:39) D − P → F , which is characterized by the following universal property:For every complex A • (of objects in A ) it represents, in the derived cat-egory D − F , the functor which assigns to each K • ∈ ob( D F ) , the set ofequivalence classes of pairs of diagrams: (cid:40) C ( A (cid:48)• ) f −→ K • ,A (cid:48)• s −→ A • , where s is a (strong) quasi-isomorphism of complexes in A . For more de-tails, and the notion of equivalence, cf. [18, § A admits a projective res-olution, thus establishing that D − P is equivalent to D − A . For notationalsimplicity, we set ˜ A n = the completed tensor product of n -copies of ˜ A , andanalogously for A . By [43, § V ∈ ob( A ) the sequence · · · → ˜ A n +1 ˆ ⊗ V d −→ ˜ A n ˆ ⊗ V → · · · → ˜ A ˆ ⊗ V → V → (B.3)is a strongly exact resolution by free modules. By definition, if we numberthe copies of ˜ A n +1 from to n , the action of ˜ A is only on the left copy, andthe boundary maps are d ( a ⊗· · ·⊗ a n ⊗ v ) = n − (cid:88) i =0 ( − i a ⊗· · ·⊗ a i a i +1 ⊗· · ·⊗ v +( − n a ⊗· · ·⊗ a n − ⊗ a n v. (B.4)Hence, by [18, Theorem 10.25], for every strongly exact sequence of ob-jects in A : → V → V → V → , we get a distinguished triangle in D − F : L C V [ − → L C V → L C V → L C V . With all this material obtained from the literature, we would now like toprove that, for the computation of L C , one may replace in (B.3) all copies of ˜ A by A , thus arriving at the description of derived coinvariants of § Proposition.
The natural inclusion from the complex C ˆ ⊗ A (cid:16) · · · → A n +1 ˆ ⊗ V d −→ A n ˆ ⊗ V → · · · → A ˆ ⊗ V → (cid:17) (B.5) into the complex C ˆ ⊗ A (cid:16) · · · → ˜ A n +1 ˆ ⊗ V d −→ ˜ A n ˆ ⊗ V → · · · → ˜ A ˆ ⊗ V → (cid:17) (B.6) is a (strict) quasi-isomorphism. (The application of C ˆ ⊗ A to the entire complex just means application toeach element or, by the above, applying the functor of coinvariants.) Proof.
Notice that C ˆ ⊗ A A n +1 ˆ ⊗ V (cid:39) A n ˆ ⊗ V . We explicate the boundary maps,which we will denote by δ in order to distinguish them from the boundarymaps d of (B.4): δ ( a ⊗ · · · ⊗ a n ⊗ v ) = ( (cid:90) a ) · a ⊗ · · · ⊗ a n ⊗ v + (B.7) + n − (cid:88) i =1 ( − i a ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ v + ( − n a ⊗ · · · ⊗ a n − ⊗ a n v. HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 83
Now, for every ≤ i ≤ n , let B in := ˜ A i − ˆ ⊗ C ˆ ⊗ A n − i ˆ ⊗ V. The convention is that B n = A n ˆ ⊗ V . It is then clear from the definitions: ˜ A n ˆ ⊗ V = n (cid:77) i =0 B in . Our goal is to show that the summands with i (cid:54) = 0 do not contributeanything to the homology of (B.6).We claim that, for i ≥ , δ ( B in ) ⊂ B i − n − ⊕ B in − , with the first summandnot appearing when i = 1 and the second not appearing when i = n . Moreprecisely, as can easily be seen from (B.7), we have δ ( a ⊗ . . . a i − ⊗ ⊗ a i +1 ⊗ · · · ⊗ a n ⊗ v ) = (B.8) = δ ( a ⊗ . . . a i − ) ⊗ ⊗ ( a i +1 ⊗ · · · ⊗ a n ⊗ v )++( − i +1 ( a ⊗ . . . a i − ) ⊗ ⊗ d ( a i +1 ⊗ · · · ⊗ a n ⊗ v ) . Here by δ we denote the morphism given by the same formula as (B.7)when V = ˜ A , and d denotes the morphism (B.4). When i = 1 , the con-vention is that δ = 0 , and when i = n , then d = 0 . Hence, omitting thesummands with i = 0 from (B.6) (which correspond to (B.5)), we still get acomplex n +1 (cid:77) i =1 B in +1 → n (cid:77) i =1 B in → · · · → C ⊗ V → . (B.9)Moreover, from the above formulas it is clear that (B.9) is the tensor productof the complexes ˜ A n +1 δ −→ ˜ A n → · · · → ˜ A → (with ˜ A in degree ) and A n +1 ˆ ⊗ V d −→ A n ˆ ⊗ V → · · · → A ⊗ V → V → (with V in degree ).Both complexes are acyclic (strictly exact): the first computes the homol-ogy of ˜ A as an ˜ A -module, which is free and hence acyclic, and the secondis the resolution (B.3), with ˜ A replaced by A (and easily seen to be a strictlyexact resolution, again, though not necessarily strongly exact). Thus, (B.9)is acyclic, and (B.6) is (strictly) quasi-isomorphic to (B.5). (cid:3) A PPENDIX
C. A
SYMPTOTICALLY FINITE FUNCTIONS
C.1.
Derivative arrangements and a criterion for asymptotic finiteness.
Let us consider a toric variety Y for a real torus T as in § Y is considered as a normal embedding for a torus quotient T (cid:48) of T .Let E be an exponent arrangement for Y ; it was used in § F E , consisting of functions on T ( R ) . We will describe an effi-cient way of checking whether a function on T ( R ) is a section of F E over Y ( R ) . The discussion here carries over, essentially verbatim, and will beapplied to the case of the equivariant toroidal compactifications of the auto-morphic quotient [ H ] = H ( k ) \ H ( A k ) . For notational simplicity we presentthe case of toric varieties.To describe such a criterion, recall that the toric variety Y is described bya fan F of strictly convex rational polyhedral cones on the vector space t (cid:48) = Hom( G m , T (cid:48) ) ⊗ R , and that the cones in the fan are in bijection C ↔ Z C with the geometric or-bits of T on Y , in such a way that the relative interior of C consists preciselyof those cocharacters λ into T (cid:48) such that lim λ → λ ( t ) ∈ Z C . The exponentarrangement Z (cid:55)→ E ( Z ) can be considered as an arrangement for the fan: C (cid:55)→ E ( C ) := E ( Z C ) . We recall that E ( Z C ) consists of characters of the R -points of the stabilizer T C of points on Z C .The sub-fan consisting of the cone C and all its faces corresponds to theaffine, open subvariety Y C ⊂ Y , previously denoted as Y Z C . To describethe behavior of sections locally, in a neighborhood of a point z ∈ Z C ( R ) ,we may replace Y by this affine open subvariety described by the fan ofthe cone C .Let D be a cone in the fan. (We use a different letter now, because thefollowing will be used for every face D of C to describe the behavior in aneighborhood of a point of Z C .)We define a derivative arrangement ( F (cid:48) D , E ) as follows: • The new fan F (cid:48) D will live on the vector space t (cid:48) D := t / t D , where t D is the subspace spanned by cocharacters into T D ; thus, thenew fan will correspond to an embedding of the torus T (cid:48) D := T /T D .To define the fan F (cid:48) D , consider the set of orbits which are con-tained in the closure of Z D ; they correspond to all cones D (cid:48) ∈ F which contain D . The fan F (cid:48) D will consist of the images of all thosecones D (cid:48) in t (cid:48) D , which are strictly convex, as required. • It inherits the multisets E ( D (cid:48) ) of exponents from the original ex-ponent arrangement. Thus, we do not need a new symbol for theexponents.The pair ( F (cid:48) D , E ) gives rise to a toric embedding Y (cid:48) D of T (cid:48) D , and a cosheafover it which we will denote by F D,E ; its restriction to the open stratum Z D is what in the notation of § F Z D ,E ( Z D ) .Now we describe how to detect if a function on T ( R ) coincides withan element of F E ( Y ) in a (semi-algebraic) neighborhood V of a point z ∈ Z C ( R ) . The neighborhood V will be taken in Y C , according to Remark 5.2.2.It will also be taken to be compact — and in particular with compact imagein Z C ( R ) under the contraction map.C.1.1. Lemma.
Let V = (cid:83) D U D be a decomposition into subsets, one for eachnon-trivial face D of C (i.e., each non-open orbit containing Z C in its closure), HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 85 such that U D is bounded away from every orbit not contained in the closure of Z D (i.e., from any Z E with D (cid:54)⊂ E ).Then a smooth function f on T ( R ) coincides with an element of F E ( Y ) on V ifand only if on U D it coincides, up to an element of F E ( T (cid:48) ) (i.e., a rapidly decayingsection), with a section of F D,E over the closure of Z D ( R ) . The picture here shows such a partition of a neighborhood:I will describe a way to get such a partition after the proof; the idea is, es-sentially, to start by removing from V neighborhoods of the maximal orbits(= one-dimensional edges of C ), then remove from the remainder neigh-borhoods of the orbits of codimension two, etc. The proof also has such aninductive structure. Proof.
Let us say that two orbits are not comparable if one is not containedin the closure of the other. In that case, they can be separated by (semi-algebraic) open neighborhoods.Let Z D be a non-open orbit of maximal dimension in Y C , that is: D is aface of C of dimension one. I claim:In a neighborhood U of Z D , which is bounded away fromany orbit not comparable to Z D , restrictions to U of elementsof F E ( Y ( R )) are precisely the restrictions of those functionson T ( R ) which differ by elements of F E ( T (cid:48) ( R )) (i.e., sectionsof rapid decay on T (cid:48) ( R ) ) from sections of F D,E over the clo-sure of Z D . (C.1)This is immediately true by definition for those sections whose germsat orbits in the closure of Z D are zero (i.e., which are sections of F E onan open set not containing any orbits smaller than Z D ). We now check itinductively on dim Z D − dim Z D (cid:48) , where Z D (cid:48) is an orbit in the closure of Z D ,of smallest dimension such that the germ of a given section f at Z D (cid:48) ( R ) isnon-zero. Using the notation of the definition, in a neighborhood of Z D (cid:48) ( R ) the section is equal to f σ plus a section whose germ at Z D (cid:48) ( R ) is zero. Since f σ is a section of F D (cid:48) ,E and hence coincides a fortiori with a section of F D,E in that neighborhood, by the inductive hypothesis we are done.We started with (C.1) because it is easier to state, but in fact it is a specialcase of the following, which is proved by exactly the same argument: By “bounded away” from an orbit Z we will mean that it is disjoint from a semi-algebraic neighborhood of Z ( R ) . If Z D is any orbit in Y , U is a neighborhood U of Z D which isbounded away from any orbit not comparable to Z D , and U (cid:48) is obtained from U by removing neighborhoods of all orbitsof larger dimension, then restrictions to U (cid:48) of elements of F E ( Y ( R )) are precisely the restrictions of those functions on T ( R ) which differ by elements of F E ( T (cid:48) ( R )) (i.e., sections ofrapid decay on T (cid:48) ( R ) ) from sections of F D,E over the closureof Z D . (C.2)And this implies the lemma. (cid:3) I explain how to get a partition into subsets U D as above. We can choosea semi-algebraic lift of the contraction map: s : Z C ( R ) → T (cid:48) ( R ) , and thenassume that the given neighborhood of z ∈ Z C ( R ) is a compact subset ofthe set { s ( z (cid:48) ) · t | z (cid:48) ∈ Z C ( R ) , t ∈ T C ( R ) , log( t ) ∈ C } . (Recall that under the log map: T ( R ) → t R → t (cid:48) R , elements of T C haveimage in the linear span of C .)We can then describe a partition by describing a partition of the cone C and pulling it back via the log map.We do this by “moving” the maximal faces of C into the strict interior;that is, if we take a cross-section of the cone with an affine hyperplane A inthe linear span of C , meeting all of its non-trivial faces, then the cone C isdefined by the “positive” sides of a set of hyperplanes H i ⊂ A , and we canconsider a small parallel translation H i → H (cid:48) i of each of those hyperplanestowards the interior of C . For a given face D , defined as the intersectionof the hyperplanes H i , i ∈ I D , the subset U D will be determined by theset D (cid:48) of those elements of the cone C which lie on the “negative” sideof the corresponding translated hyperplanes H (cid:48) i , i ∈ I D . We observe thatthe set D (cid:48) only meets the relative interiors of the cones containing D in theirclosure, and therefore the corresponding set U D will be bounded away fromorbits that are not in the closure of Z D .C.2. Completion of the proof of Theorem 5.7.1.
Recall that we have anequivariant toroidal embedding [ H ] F of [ H ] = H ( k ) \ H ( A k ) , determinedby the fan F on the anti-dominant Weyl chamber a + , which in turn is deter-mined by the weights of the representation V of H .Let us fix a minimal parabolic P with a Levi subgroup M as before,and consider all cocharacters into the universal maximal split torus A ascocharacters into H via the embedding A → M determined by P .For any cone C ∈ F we define f C ( h ) := (cid:88) γ ∈ V C, (cid:90) V C, + ( A k ) f (( γ + v ) h ) dv, (C.3)where the spaces V C, , V C, + , V C, − are as in (5.14). HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 87
It is easy to see:C.2.1.
Lemma.
The function f C is a smooth function on [ H ] P C , not dependingon the choice of ( P , M ) , and an A C ( A k ) -eigenfunction with eigencharacter χ C .Proof. Indeed, if P C , M C are the standard parabolic and Levi in the classassociated to C , induced from the chosen pair ( P , M ) , the subspaces V C, , V C, + are M C -stable, since A C lies in the center of M ; moreover, although V C, is not necessarily stable under the unipotent radical U ( P C ) of P C , anyaffine subspace of the form γ + V C, + is, because it can be characterized as thesubspace of those elements for which lim t → v · λ ( t ) = γ for λ a cocharacterin the relative interior of C , and lim t → λ ( t ) − uλ ( t ) = 1 for all such λ and u ∈ U ( P C ) . The action of U ( P C ) is by affine automorphisms on those affinesubspaces, and since U ( P C ) has trivial character group, it has to preserveHaar measure. (cid:3) Now we will use the derivative arrangements ( F (cid:48) C , E ) defined in § C.1.By decreasing induction on the dimension of C (s. the following remarks),we may assume that we have proven:The map f (cid:55)→ f C is a continuous map from F ( V ( A k )) to F C,E ( A C ( R ) \ [ H ] F (cid:48) C P C ) , the space of asymptotically finite func-tions on [ H ] P C which are ( A C ( R ) , χ C ) -eigenfunctions, withfan F (cid:48) C and exponent arrangement as in § C.1, that is: the fan F (cid:48) C on the vector space a / a C consists of the images of thecones of F containing C , and the arrangement is simply therestriction of E to these cones. (C.4)We remark the following: • This claim is obtained by inductively applying the Σ V -version (notthe Σ N -version) of Theorem 5.7.1 to the Schwartz function f (cid:48) : γ (cid:55)→ (cid:90) V C, + ( A k ) f (( γ + v )) dv on the M ( A k ) -stable vector space V C, ( A k ) . Notice that f C ( m ) = χ C ( m )Σ V C, f (cid:48) ( m ) (C.5)on the subspace [ M ] ⊂ [ H ] P C , where χ C is the (unique) charactercorresponding to C of the exponent arrangment E . (This embed-ding depends on the choice of parabolic P C , but so does the sub-space V C, .) • For any cone D which contains C as a face we have V C, ⊃ V D, , V C, ± ⊂ V D, ± . Thus, the exponent arrangement E restricted to thecones containing D in their closure is the same as the exponent ar-rangement on a obtained by the above recipe from the weights Φ( V C, ) of the representation V C, , multiplied by χ C . • The basis of the induction is the case when C is of full dimension, inwhich case A C = A , P C = P and the quotient A C ( R ) \ [ H ] P C is com-pact, and it is immediately clear that (C.5) represents a continuousmap to the space of smooth A C ( R ) -eigenfunctions with eigenchar-acter χ C on [ M ] .Now we will check the asymptotic finiteness of the function Σ V f inthe vicinity of a point z ∈ Z C by adapting the criterion of Lemma C.1.1.Namely, we restrict our attention to a sufficiently small neighborhood U ,namely a neighborhood satisfying the following conditions: • it belongs to a small neighborhood of the P C -cusp which is isomor-phic to a neighborhood of the cusp in P C ( k ) \ H ( A k ) under the map(5.8); • let U (cid:48) be the homeomorphic preimage of U in a neighborhood of thecusp in P C ( k ) \ H ( A k ) , then U (cid:48) is the preimage of a neighborhood U (cid:48)(cid:48) of z in [ H ] F P C as described before Lemma C.1.1, namely: it is compactand belongs to orbits which contain Z C in their closure (where Z C is, by definition, a stratum both in [ H ] and in [ H ] P C .We can then partition U into subsets U D , one for each non-zero face D of C , such that U D is bounded away from all orbits not in the closure of Z D .We may assume that the original neighborhood is sufficiently close to the P C -cusp, and hence also to the P C -cusp for every face D of C .Then I claim:C.2.2. Lemma.
The restriction of Σ V f to U D coincides, up to the restriction of arapidly decaying function on [ H ] (depending continuously on f ), with f D . By a straightforward adaptation of Lemma C.1.1 and by (C.4), this isenough to prove the theorem.This lemma is proven by a variant of the simplified argument that weused in § λ . By the assumptions on U (cid:48)(cid:48) , it iscontained in a set of the form M C ( k ) U ( P C )( A k ) A + C K ⊂ [ H ] P C , (C.6)where A + C is the preimage of cone C under the log map A C ( R ) (cid:55)→ a C ⊗ Q R , and K is a compact subset in H ( A k ) . We have implicitly chosen a parabolicin class of P C in order to write the set on the right-hand side of (C.6), andalthough its Levi subgroup M C is not needed in order to make sense ofthe above expression, let us now choose such a decomposition, in orderto consider elements of A C as elements in the center of M C , and hence aselements of H . Thus, we have decompositions of the vector space V as in(5.14), and similarly when C is replaced by any face D of it, by the choiceof Levi M D induced from M C .From (C.6) it follows that U ⊂ [ H ] is covered by the H ( k ) -coset of a set ofthe form Ω A + C K ⊂ [ H ] P C , where Ω is a compact subset of U ( P C )( A k ) . Since HE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 89 the subset U D is bounded away (by a semi-algebraic neighborhood) fromany stratum not in the closure of Z D , it is covered by a subset of the form Ω A DC K ⊂ [ H ] P C , where A DC is the preimage under the log map of a subcone D (cid:48) ⊂ C whichis bounded away from any face of C not containing D in its closure. (Thereader can compare with the suggested construction of the subsets U D atthe end of § C.1.)We can now attempt to estimate the restriction of Σ V f to a set of the form Ω A DC K where, we recall, the set A DC is considered as a subset of H ( A k ) bythe choice of specific Levi for the class of P C .Decomposing the sum over V ( k ) as in (5.17) we will get terms that decayrapidly in U D , except for the last one, which is equal to f D . More precisely,consider the term (cid:88) γ − ∈ V D, − ( k ) (cid:114) { } (cid:88) γ ∈ ( V D, + V D, + )( k ) f (( γ − + γ ) h ) , (C.7)and let h ∈ A DC . We can identify A DC modulo a compact subgroup with itsimage a DC in a C ⊗ R . The fact that the latter is bounded away from faceswhose closure does not contain D means that for every A C -weight χ of therepresentation V D, − , viewed as a functional on a C , the set a DC lies in a be-longing strictly in the negative half-space of χ . From this, it easily followsthat (C.7) is of rapid decay on A DC , in a way that depends continuously on f . To incorporate the whole set Ω A DC K we just need to translate f by k ∈ K and change our choice of Levi M C by ωM C ω − , for ω ∈ Ω (and, corre-spondingly, the decomposition (5.14) for the face D ). A way to encode thisuniformly is to think of the spaces V D, ± , V D, as abstract k -vector spaces,and let Ω parametrize their embeddings into V . These abstract spaces comewith an action of the abstract torus A C , and as ω varies in Ω and k varies in Ω K we apply the sum (C.7) to the ω -pullback of the k -translate of f to thesespaces, with h ∈ A DC . It is then clear that the above estimates are uniformin the parameters ( ω, k ) , and hence the expression (C.7) is of rapid decay in U D , in a way that depends continuously on f .The same arguments apply to the second line of (5.17), with f replacedby a Fourier transform, and thus the above lemma, and the theorem, havebeen proved. R EFERENCES [1] Avraham Aizenbud and Dmitry Gourevitch. Schwartz functions on Nash manifolds.
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REECE . RRATUM TO: THE SCHWARTZ SPACE OF A SMOOTHSEMI-ALGEBRAIC STACK
YIANNIS SAKELLARIDIS
The purpose of this note is to fix two gaps in the construction of Schwartzspaces of semi-algebraic stacks in [4], and to strenghen some statements,replacing quasi-isomorphisms by homotopy equivalences. I am grateful toAvraham Aizenbud, Shachar Carmeli, and Dmitry Gourevitch for pointingout the gaps, and suggesting the stronger statements.The first gap is in the proofs of Propositions 3.1.2 and 3.1.4, where I mis-quote [3, Theorem A.1.1] and write a Schwartz function as a product oftwo Schwartz functions. There is also an obvious typo in the statement ofProposition 3.1.4: the sequence appearing should end with ∂ −→ S ( Y ) → .Moreover, with this gap corrected, a stronger statement is actually provenin these two propositions than claimed. Namely, the sequence of Proposi-tion 3.1.4 (with the aforementioned typo corrected) is not just strictly exact,but homotopic to zero. I formulate this here as a proposition, which su-persedes both of Propositions 3.1.2 and 3.1.4 in the paper, and indicate thecorrections needed for a complete proof.D.2.3. Proposition.
Let π : X → Y be a smooth surjective morphism of Nashmanifolds. Let [ X ] nY = the fiber product of n copies of X over Y (whose projectionmap to Y is still denoted by π ), and consider the complex ( S ([ X ] nY )) n : · · · → S ([ X ] Y ) → S ([ X ] Y ) → S ( X ) → , with differentials ∂ n : S ([ X ] n +1 Y ) → S ([ X ] nY ) induced from the alternating sum ofpush-forwards when a copy of X is deleted, as in [4, Proposition 3.1.4] . Consider S ( Y ) as a complex in degree zero, and the morphism of complexes π ! : ( S ([ X ] nY )) n → S ( Y ) induced by the push-forward π ! : S ( X ) → S ( Y ) . This morphism is a homotopyequivalence.Proof. [3, Theorem A.1.1] states that any f ∈ S ([ X ] nY ) can be written as afinite sum f ( x ) = m (cid:88) i =1 φ i ( π ( x )) f i ( x ) , Mathematics Subject Classification.
Primary 58H05; Secondary 22A22, 11F70.
Key words and phrases. stacks, Schwartz spaces, orbital integrals.
934 YIANNIS SAKELLARIDIS AND YIANNIS SAKELLARIDIS where φ i is a Schwartz function on Y and f i ∈ S ([ X ] nY ) . Over an Archimedeanfield it is not true, in general, that it can be written as a product φ · f , asclaimed in [4].But now, assuming, as in the proofs of [4, Propositions 3.1.2 and 3.1.4],that the differential ∂ n − f ∈ S ([ X ] n − Y ) vanishes, an extra complicationarises, because only the sum (cid:80) mi =1 φ i ( π ( x )) ∂ n − f i ( x ) vanishes, not eachterm ∂ n − f i individually. The next step in the proofs is to disintegrate φ toan element h of some space of “relative Schwartz measures” S (cid:48) ( X ) (whosepush-forwards to Y are Schwartz functions — see the proof of [4, Proposi-tion 3.1.2] for details). For the argument to go through as stated, we needto do this compatibly for all φ i ’s. Namely, let us assume that the base fieldis F = R (because in the non-Archimedean case there is no issue, and inthe complex case we may work by restriction of scalars over R withoutchanging the final statement).Let us first discuss the special case where the morphism X → Y admitsa Nash section σ : Y → X , which, in addition, extends to a tubular neighbor-hood ι : Y × B r (cid:44) → X , where r is the relative dimension of the map π , and B r is the open unit ball in R r . Then, choosing a Schwartz measure µ ∈ S ( B r ) with total mass , we can set h i := ι ! ( φ i ⊗ µ ) ∈ S (cid:48) ( X ) . Then π ! h i = φ i , and (cid:80) i h i ( x ) ∂ n − f i ( x , . . . , x n − ) = 0 ; the proofs of the two Propositions nowgo through as stated. Moreover, the tubular neighborhood gives rise to anembedding, again to be denoted by the same letter: ι : [ X ] nY × B r (cid:44) → [ X ] n +1 Y (with B r determining the last coordinate), and a choice of µ as above allowsus to define linear maps H n : S ([ X ] nY ) → S ([ X ] n +1 Y ) by H n ( f ) = ι ! ( µ ⊗ f ) . This includes the case of H : S ( Y ) → S ( X ) , whichis a section for the push-forward map. One then easily checks that H n is ahomotopy between H ◦ π ! and the identity on the complex ( S ([ X ] nY )) n ; inother words, π ! : ( S ([ X ] nY )) n → S ( Y ) is a homotopy equivalence.We have up to now assumed that the morphism X → Y admitted asection with a tubular neighborhood. Such tubular neighborhoods existlocally over Y [2, 2.4.3], [1, Theorem 3.6.2]. The last step to correct theproof is to show that all statements are local over Y (in the semi-algebraictopology). For this, given a (finite) semi-algebraic upen cover Y = ∪ j Y j ,we use the “Schwartz partition of unity” of [1, Theorem 4.4.1], which isa collection of tempered functions u j , with u j supported on Y j , (cid:80) j u j =1 , and the property that multiplication by u j turns a Schwartz function(or measure) on Y to a Schwartz function (or measure) on Y j . Of course,multiplication by u j ◦ π will not change the property ∂ n − f = 0 (of f ∈S ([ X ] nY ) ), so we are reduced to the case where a section with a tubularneighborhood exists. (cid:3) RRATUM TO: THE SCHWARTZ SPACE OF A SMOOTH SEMI-ALGEBRAIC STACK 95
The second gap is in the proof of functoriality in Theorem 3.3.1. Again,once the gap is fixed a stronger statement is actually proven:D.2.4.
Theorem.
Let X be a Nash stack. For any two presentations X → X , X → X , the Schwartz complexes ( S ([ X i ] n X )) n : · · · → S ([ X i ] X ) → S ([ X i ] X ) → S ( X i ) → ( i = 1 , ) are canonically homotopy equivalent; and hence can be denoted by S • ( X ) .The association X (cid:55)→ S • ( X ) is functorial with respect to smooth 1-morphismsof Nash stacks, up to homotopy.Proof. For notational simplicity, let us in the proof denote X by X , X by Y , and X ( i ) Y ( j ) := [ X ] i X × X [ Y ] j X . We also denote X (1) Y (1) by R XY , X (2) by R X , and Y (2) by R Y ; hence, for i, j ≥ we have X ( i ) Y ( j ) = [ R X ] i − X × X R XY × Y [ R Y ] j − Y . (D.8)This is a unique Nash manifold up to unique isomorphism, once the Nashmanifolds R XY , R X , R Y (with their morphisms to X, Y ) have been fixed.The “presentations” X → X , Y → X implicitly include the groupoids R X ⇒ X , R Y ⇒ Y , but the gap in the proof of Theorem 3.3.1 is that it isnot taken into account that R XY is only defined up to automorphisms over X × Y . Thus, we need to make sure that, in the proof of Theorem 3.3.1,composition with automorphisms τ : R XY → R XY over X × Y does notchange the homotopy class of the equivalence ( S ( X ( n ) )) n ∼ −→ ( S ( Y ( n ) )) n .We revisit the proof, in order also to explain that it can be strengthenedto a homotopy equivalence. The essential statement is that, if we considerthe total complex T XY associated to the bicomplex ( S ( X ( i ) Y ( j ) )) i,j ≥ , itsnatural push-forward maps to the complexes ( S ( X ( i ) )) i ≥ , ( S ( Y ( j ) )) j ≥ arehomotopy equivalences.Let us briefly see why: Applying Proposition D.2.3 above, for any i , thenatural push-forward ( S ( X ( i ) Y ( j ) )) j −→ (0 → S ( X ( i ) ) → is a homotopy equivalence. The construction of a homotopy inverse reliedon choosing, locally, a section with a tubular neighborhood: X ( i ) × B r (cid:44) → X ( i ) Y (1) . In our setting, we can choose once and for all a section with a tubularneighborhood ι : X × B r (cid:44) → R XY , (D.9)at least locally on X . (It is clearly enough to work locally over X here, in or-der to prove the homotopy equivalence of the total complex with ( S ( X ( i ) )) i .)This induces sections X ( i ) × B r = [ R X ] ( i − X × X X × B r (cid:44) → X ( i ) Y (1) = [ R X ] ( i − X × X R XY for all i ≥ ; the resulting homotopy inverses constructed in the proof ofProposition D.2.3 (denoted by H there) will now be, by construction, chainmaps of complexes : H : ( S ( X ( i ) )) i ≥ → ( S ( X ( i ) Y (1) )) i ≥ (cid:44) → T XY . Thus, the statement of Proposition D.2.3 extends to the morphism ofcomplexes T XY → ( S ( X ( i ) ) i ≥ , and shows that it is a homotopy equiva-lence.Let us now assume that τ : R XY → R XY is an automorphism over X × Y ;by (D.8), it induces automorphisms τ ! of all Schwartz spaces S ( X ( i ) Y ( j ) ) , i, j ≥ . I claim that the composition τ ! ◦ H is still a homotopy inverse tothe push-forward map p X : T XY → ( S ( X ( i ) )) i ≥ . Indeed, in the construc-tion of this homotopy inverse, τ just modifies the tubular neighborhood(D.9), that is, τ ! H is obtained by the same construction, using the tubularneighborhood τ ◦ ι . Thus, it is homotopy inverse to the canonical push-forward map p X .But τ ! ◦ H is also homotopy inverse to the composition p X ◦ τ − . Wededuce that the morphisms p X and p X ◦ τ − are homotopic. The sameholds for the analogous morphisms p Y and p Y ◦ τ − , thus the homotopyclass of the equivalence of Schwartz complexes ( S ( X ( i ) )) i ≥ ∼ −→ ( S ( Y ( j ) )) j ≥ is fixed under automorphisms of X × X Y . (cid:3) R EFERENCES [1] Avraham Aizenbud and Dmitry Gourevitch. Schwartz functions on Nash manifolds.
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Selecta Math.(N.S.) , 22(4):2401–2490, 2016. doi:10.1007/s00029-016-0285-3 . E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS AND C OMPUTER S CIENCE , R
UTGERS U NIVERSITY –N EWARK , 101 W
ARREN S TREET , S
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