aa r X i v : . [ m a t h . N T ] F e b H OF IGUSA VARIETIES VIA AUTOMORPHIC FORMS
ARNO KRET, SUG WOO SHIN
Abstract.
Our main theorem describes the degree 0 cohomology of Igusa varieties in terms ofone-dimensional automorphic representations in the setup of mod p Hodge-type Shimura varietieswith hyperspecial level at p , mirroring the well known analogue for complex Shimura varieties. Asan application, we obtain a completely new approach to two geometric questions. (See § p -adic groups due to Howe–Moore and Casselman. Contents
1. Introduction 12. Preliminaries in representation theory and endoscopy 93. Jacquet modules, regular functions, and endoscopy 244. Asymptotic analysis of the trace formula 345. Shimura varieties of Hodge type 466. Igusa varieties 547. Cohomology of Igusa varieties 588. Applications to geometry 65References 681.
Introduction
Igusa varieties were studied by Igusa [Igu68] and Katz–Mazur [KM85] in the case of modularcurves. Harris–Taylor and Mantovan [HT01,Man05] have generalized the construction to PEL-typeShimura varieties. Recently Caraiani–Scholze [CS17] gave a slightly different definition in the PELcase which gives the same cohomology. Hamacher, Zhang, and Hamacher–Kim went further todefine Igusa varieties for Hodge-type Shimura varieties [Ham17, Zha, HK19]. In the ( µ -)ordinarysetting, Igusa varieties are also referred to as Igusa towers. (Often the definitions differ in a minorway.) There are versions of Igusa varieties as p -adic formal schemes or adic spaces over p -adicfields, but we concentrate on the characteristic p varieties in this paper. We also mention thatfunction-field analogues of Igusa varieties are studied in a forthcoming paper by Sempliner.The ℓ -adic cohomology of Igusa varieties (with ℓ = p ) has several arithmetic applications. In[HT01, Man05, HK19], the authors prove a formula computing the cohomology of Shimura varietiesin terms of that of Igusa varieties and Rapoport–Zink spaces. This means that, if we understand thecohomology of Igusa varieties well enough, then our knowledge of cohomology can be propagatedfrom Rapoport–Zink spaces to Shimura varieties or the other way around. This is the basic principle Date : February 23, 2021. underlying [Shi11,Shi12] on the global Langlands correspondence and the Kottwitz conjecture. Foranother application, a description of ℓ -adic cohomology of Igusa varieties was one of the mainingredients in [CS17,CS] to prove vanishing of cohomology of certain Shimura varieties with ℓ -torsioncoefficients, which in turn supplied a critical input for a recent breakthrough on the Ramanujanand Sato–Tate conjecture for cuspidal automorphic representations of GL of “weight 2” over CMfields [ACC + ].Thus an important long-term goal is to compute the ℓ -adic cohomology of Igusa varieties witha natural group action. A major first step would be a Langlands–Kottwitz-style trace formula forIgusa varieties, which has been obtained for Shimura varieties of Hodge type at hyperspecial levelin [Shi09,Shi10,MC] building upon [HT01, Ch. 5] in analogy with [LR87,Kot92b,KSZ]. One wishesto turn that into an expression of the cohomology in terms of automorphic forms, but this requiresa solution of various complicated problems; some should be tractable but others are out of reach ingeneral, most notably an endoscopic classification and Arthur’s multiplicity formula for the relevantgroups.The main objective of this paper is twofold. Firstly, we describe the H of Igusa varieties viaone-dimensional automorphic representations over non-basic Newton strata of Hodge-type Shimuravarieties at hyperspecial level. This mirrors the well-known fact that H of complex Shimuravarieties is governed by one-dimensional automorphic representations. Secondly, to achieve this, wedevelop a method and obtain various technical results with a view towards the entire cohomologyof Igusa varieties (as an alternating sum over all degrees). Our method, partly inspired by Laumon[Lau05] and also by Flicker–Kazhdan [FK88], should prove useful for studying ℓ -adic cohomologyof Shimura varieties as well.Our result on H not only sets a milestone in its own right, but also reveals deep geometricinformation. Namely, our theorem readily implies the discrete Hecke orbit conjecture for Shimuravarieties and the irreducibility of Igusa varieties in the same generality as above. (The irreducibilitymeans that Igusa varieties are no more reducible than the underlying Shimura varieties in someprecise sense.) Our work provides a completely new approach and perspective to these two problemsby means of automorphic forms and representation theory.One of our main novelties consists in a careful asymptotic argument via the trace formula tosingle out H (or compactly supported cohomology in the top degree) without reliance on anyclassification, a key to obtain an unconditional result. Since the “variable” for asymptotics isencoded in a test function at p , a good amount of local harmonic analysis naturally enters thepicture. Another feature of our approach is to allow induction on the semisimple rank of the group;this would make little sense in a purely geometric argument (as endoscopy is hard to realize in thegeometry of Shimura varieties).Roughly speaking, cohomology of Igusa varieties is closely related to that of Shimura varieties viathe Jacquet module operation at p , relative to a proper parabolic subgroup in the non-basic case.To show that only one-dimensional automorphic representations contribute to H of Igusa varieties,the key representation-theoretic input is an estimate for the central action on Jacquet modules dueto Casselman and Howe–Moore. Though there is no direct link, it would be interesting to remarkthat a similar situation occurs in the context of beyond endoscopy (e.g., [FLN10, § § The main theorem.
Let (
G, X ) be a Shimura datum of Hodge type with reflex field E ⊂ C .Assume that the reductive group G over Q admits a reductive model over Z p , and take K p := G ( Z p ). In the basic case, Igusa varieties are 0-dimensional, and their H is expressed as the space of algebraic automorphicforms on an inner form of G , through a description of points with suitable group actions as in [MC]. This idea goesback to Serre [Ser96] for modular curves, and Fargues [Far04, Ch. 5] in the PEL case. OF IGUSA VARIETIES 3 (Namely G is unramified at p , and K p is a hyperspecial subgroup.) For simplicity, the adjoint groupof G is assumed to be simple over Q throughout the introduction. (Otherwise the notion of basicelements needs to be modified. See Definition 5.3.2 below.) We do not assume p > p = 2 will be included in [KSZ, MC].Fix field maps Q ֒ → Q p , Q p ≃ C , and Q ℓ ≃ C (which will be mostly implicit). The resultingembedding E ֒ → Q p induces a place p of E above p . Let k ( p ) denote the residue field of E at p ,which embeds into the residue field F p of Q p . Thereby we identify k ( p ) ≃ F p . Let S K p denote theintegral canonical model over O E p with a G ( A ∞ ,p )-action. In the main text, we work with Shimuraand Igusa varieties at finite level and then take limits over open compact subgroups K p ⊂ G ( A ∞ ,p ).(For instance, the fixed-point formula should be applied at finite level.) However, we will ignorethis point and pretend that we are always at infinite level away from p to simplify exposition.A fixed symplectic embedding of ( G, X ) into a Siegel Shimura datum yields a G ( A ∞ ,p )-equivariantmap from S K p to a suitable Siegel moduli scheme (over O E p after a base change from Z p ). Viapullback, we obtain a universal abelian scheme A over S K p , which can be equipped with a familyof ´etale and crystalline tensors over geometric points x → S K p ,k ( p ) . This assigns to x the p -divisiblegroup A x [ p ∞ ] (with G -structure) up to isomorphism.Let µ p : G m → G Q p denote the cocharacter arising from ( G, X ) (and via Q p ≃ C ). This cutsout a finite subset B ( G Q p , µ − p ) in the Kottwitz’s set B ( G Q p ) of G -isocrystals. Fix b ∈ G ( ˘ Q p )whose image [ b ] lies in B ( G Q p , µ − p ). (Without the latter condition, C b and N b below are knownto be empty.) Then b gives rise to an isomorphism class of p -divisible group Σ b with G -structureover F p via the symplectic embedding above and the Dieudonn´e theory. We also obtain a Newtoncocharacter ν b from b . We may arrange that ν b is dominant with respect to a suitable Borel subgroupof G Q p defined over Q p . For simplicity, assume that Σ b is defined over k ( p ) and that [ k ( p ) : F p ] ν b is a cocharacter, not just a fractional cocharacter. (In practice, these assumptions are unnecessarysince it is sufficient to have a finite extension of k ( p ) in the last sentence.)Write ρ for the half sum of all B -positive roots. Write J b for the Q p -group of self-quasi-isogeniesof Σ b (preserving G -structure) over F p , and J int b for the subgroup of J b ( Q p ) consisting of automor-phisms. Then J int b is an open compact subgroup of J b . As a general fact, J b is an inner form of a Q p -rational Levi subgroup M b of G Q p . We say that b is basic if ν b is a central in G Q p , or equivalentlyif M b = G Q p (namely if J b is an inner form of G Q p ).The central leaf C b (resp. Newton stratum N b ) is the locus of x ∈ S K p on which the geometricfibers of A x [ p ∞ ] are isomorphic (resp. isogenous) to Σ b . By construction, C b and N b are stable underthe G ( A ∞ ,p )-action on S K p . We also define the Igusa variety Ig b over S K p ,k ( p ) to be the parameterspace of isomorphisms between Σ b and A [ p ∞ ]. The obvious action of J int b on Ig b naturally extendsto a J b ( Q p )-action. Below are some basic facts ( § § § q := k ( p ).Fact 1. C b is (formally) smooth over k ( p ) and closed in N b ,Fact 2. C b is equidimensional of dimension h ρ, ν b i ,Fact 3. Ig b is a J int b -torsor over the perfection of C b ,Fact 4. the q -th power Frobenius on Ig b coincides with the action of ν b ( q ) ∈ Z J b ( Q p ).In particular dim Ig b = h ρ, ν b i , and every connected component of Ig b, F p (resp. C b, F p ) is irreducible.(Fact 4 is actually considered only in the completely slope divisible case, cf. part (2) of Lemma6.2.1.)Our main theorem describes the connected components (= irreducible components) of Igusavarieties over F p with a natural group action. In the statement, writing J b ( Q p ) ab and G ( Q p ) ab for the abelianizations as topological groups, the one-dimensional representation π p of G ( Q p ) isnaturally viewed as a one-dimensional representation of J b ( Q p ) via the canonical map J b ( Q p ) ։ J b ( Q p ) ab → G ( Q p ) ab , cf. § ARNO KRET, SUG WOO SHIN
Theorem A (Theorem 6.1.3) . Assume that b is non-basic with [ b ] ∈ B ( G Q p , µ − p ) . Then there is a G ( A ∞ ,p ) × J b ( Q p ) -module isomorphism H ( Ig b , Q ℓ ) ≃ M π π ∞ ,p ⊗ π p , where the sum runs over one-dimensional automorphic representations π of G ( A ) whose real com-ponent is trivial on the preimage of the neutral component G ad ( R ) in G ( R ) . Before we sketch the idea of proof, let us discuss two geometric applications in some detail.1.2.
Application to the discrete Hecke orbit conjecture.
Oort introduced a central leaf in the specialfiber of PEL-type Shimura varieties modulo a prime p as a locus on which the universal p -divisiblegroup is fiberwise constant. In 1995 [Oor19, §
15] (also see [EMO01, Problem 18]), he proposed theHecke Orbit (HO) conjecture that the prime-to- p Hecke orbit of a point in a central leaf should beZariski dense in the leaf, if the point is outside the basic Newton stratum. The reader is referredto [CO19] for an excellent survey of the HO conjecture with updates. Oort drew analogy withthe Andr´e–Oort conjecture for a Shimura variety in characteristic zero, which asserts that theirreducible components of the Zariski closure of a set of special points are special subvarieties.(See [Tsi18] and references therein for recent results on the Andr´e–Oort conjecture.) A commonfeature is that a set of points with an extraordinary structure (being a prime-to- p Hecke orbit orspecial points) is Zariski dense in a distinguished class of subvarieties. We can also compare the HOconjecture with the Hecke equidistribution theorems for locally symmetric spaces in characteristiczero [COU01, EO06], stating roughly that the Hecke orbit of an arbitrary point is equidistributedin the locally symmetric space a suitable sense. (In particular the Hecke orbit is dense in the entirespace even for the analytic topology, to be contrasted with the phenomenon in characteristic p .) Itis also worth noting works to investigate Hecke orbits for the p -adic topology [GK, HMRL].Chai and Oort verified the HO conjecture for Siegel modular varieties [Cha06, Thm. 3.4] (detailsto appear in a monograph), in particular the irreducibility of leaves [Cha05,CO11]. The conjecture isalso known for Hilbert modular varieties [Cha06, Thm. 3.5] due to Chai and Yu. (Also see [YCO20].)The HO conjecture has seen several new results in recent years. Shankar proved the conjecturefor Deligne’s “strange models” (in the sense of [Del71, § disc ) and the continuous HO conjecture (HO cont ), corresponding toglobal and local geometry, respectively. In a nutshell (HO disc ) asserts that the prime-to- p Heckeorbit of a point x in a non-basic stratum meets every irreducible component of the central leafthrough x . Then (HO cont ) is designed to tell us that the closure of the prime-to- p Hecke orbit hasthe same dimension as the central leaf in each irreducible component, so that (HO disc ) and (HO cont )together imply the HO conjecture. To our knowledge, apart from some special cases mentionedabove, no theorems of general type have been obtained on either (HO disc ) or (HO cont ) until theresults by us and independently by van Hoften and Xiao ( § § Theorem B.
Conjecture (HO disc ) holds for Hodge-type Shimura varieties with hyperspecial level at p . Let C b ⊂ Sh K p be a non-basic stratum. It is not difficult to observe that the theorem boils downto showing that the G ( A ∞ ,p )-equivariant immersion C b → Sh K p induces a bijection on the sets ofgeometric connected components. (Informally speaking, C b is no more irreducible than Sh K p .) Since OF IGUSA VARIETIES 5 G ( A ∞ ,p ) acts transitively on π (Sh K p , F p ), it is enough to show that H ( C b, F p , Q ℓ ) ≃ H (Sh K p , F p , Q ℓ )as G ( A ∞ ,p )-modules (abstractly, not necessarily via the map induced by C b ⊂ Sh K p ). It is standardto compute H (Sh K p , F p , Q ℓ ) from H (Sh K p , C , Q ℓ ) as an application of compactification for integralmodels [MP19], so the problem is to compute H ( C b, F p , Q ℓ ). In light of Fact 3, this is done bytaking the J int b -invariants of the formula in Theorem A.1.3. Application to irreducibility of Igusa towers and a generalization.
In Hida theory of p -adicautomorphic forms, an important role is played by Igusa varieties over the ordinary Newton stratum,namely when the underlying p -divisible group is ordinary. In this case, Igusa varieties (and theirnatural extension to p -adic formal schemes) are usually referred to as Igusa towers. Recently Eischenand Mantovan [EM] developed Hida theory in the more general µ -ordinary PEL-type setup, whereHowe [How20] (and its sequel) also shed new light on the role of Igusa varieties (`a la Caraiani–Scholze). Igusa towers are also featured in Andreatta–Iovita–Pilloni’s work [AIP16, AIP18] onoverconvergent automorphic forms.A key property of Igusa towers is irreducibility, roughly meaning that they are as irreducibleas possible under the given constraints. The irreducibility has an application to the q -expansionprinciple for p -adic automorphic forms, which is a basic ingredient for the construction of p -adic L -functions. See p.96 in [Hid04] for the relevant remark, and also refer to Thm. 3.3 (Igusa), 4.21(Ribet), 6.4.3 (Faltings–Chai), and Cor. 8.17 (Hida) therein for the known cases (elliptic modular,Hilbert, Siegel, and PEL type A/C cases, respectively, all over the ordinary stratum) and furtherreferences. Irreducibility in the µ -ordinary case of PEL type A was proven in [EM]. Such aresult was obtained for Igusa varieties of a specific PEL type A by Boyer [Boy] without assuming µ -ordinariness.There are various methods to show the irreducibility as explained in [Cha08] and the introductionof [Hid11], for instance by using the automorphism group of the function fields of Shimura varietiesin characteristic 0 or by showing that the monodromy of the family of abelian varieties is large. Asan application of Theorem A, we obtain an entirely different representation-theoretic proof and alsoa natural generalization from the µ -ordinary case to the general non-basic case (and from the PELcase to the Hodge-type case). In the non- µ -ordinary case, Igusa varieties do not live over the entireNewton stratum (that is, the Newton stratum is not a central leaf) but our method is insensitiveto such a distinction.Write J b ( Q p ) ′ for the kernel of the map J b ( Q p ) → G ( Q p ) ab in § Theorem C.
Assume that b is non-basic. The stabilizer subgroup in J b ( Q p ) of each connectedcomponent of Ig b is equal to J b ( Q p ) ′ . The preimage of each connected component of C b along Ig b → C b is the disjoint union of J int b ∩ J b ( Q p ) ′ -torsors. Roughly speaking, the stabilizer subgroup cannot be larger than J b ( Q p ) ′ , and this should bethought of as saying that Igusa varieties are at least as reducible as Shimura varieties. The pointof the theorem is that, conversely, the stabilizer is as large as possible under the given constraint.The proof is almost immediate from the J b ( Q p )-action on H described in Theorem A.The above theorem is easily translated into an irreducibility statement about Igusa varieties;identifying the stabilizer is the key here. The details are worked out in § Some details on the proof of Theorem A.
Changing Σ b by a quasi-isogeny, as this does notaffect Ig b up to isomorphism, we may assume that Σ b is completely slope divisible and definedover a finite field. Then the main advantage is that Ig b can be written, up to taking perfection, asthe projective limit of smooth varieties of finite type defined over F p r for some sufficiently divisible r ∈ Z > . (The latter is denoted by Ig b in the main text, but we do not distinguish Ig b from Ig b inthe introduction.) This allows us to apply a trace formula technique to compute the cohomologyof Ig b , noting that the trace of a Hecke algebra element can be computed at a finite level. Via ARNO KRET, SUG WOO SHIN
Poincar´e duality, Theorem A may be rephrased as a description of the top degree compact-supportcohomology H h ρ,ν b i c ( Ig b , Q ℓ ), which we may hope to access by the Lang–Weil estimate.Adapting the Langlands–Kottwitz method to Igusa varieties, as worked out in [Shi09] and [MC],one obtains a formula of the formTr ( φ ∞ ,p φ p × Frob jp r | H c ( Ig b , Q ℓ )) = (geometric expansion) , j ∈ Z ≫ , where φ ∞ ,p φ p ∈ H ( G ( A ∞ ,p ) × J b ( Q p )) and j ∈ Z ≫ . In fact, one can show that the Frob p r -actionon Ig b is represented by the action of a central element of J b ( Q p ). Thereby φ p × Frob jp r in (1.4.1)may be replaced with a translate φ ( j ) p ∈ H ( J b ( Q p )) of φ p by a central element. The right hand sideof (1.4.1) is a linear combination of orbital integrals of φ ∞ ,p φ ( j ) p on G ( A ∞ ,p ) × J b ( Q p ) over a certainset of conjugacy classes. After stabilization (adapting [Shi10] to the Hodge-type case), we have aformula of the formTr ( φ ∞ ,p φ ( j ) p | H c ( Ig b , Q ℓ )) = X e (constant) · ST e ell ( f e ,p f e , ( j ) p ) , j ∈ Z ≫ , (1.4.1)where the sum runs over endoscopic data e for G ( § f e ,p f e , ( j ) p is a suitable function on thecorresponding endoscopic group G e . By ST e ell we mean the stable elliptic terms in the trace formulafor G e . The most nontrivial point in the stabilization is the “transfer” at p . Indeed, as G e is not anendoscopic group of J b , this requires a special construction as detailed in § H h ρ,ν b i c ( Ig b , Q ℓ ) but H c ( Ig b , Q ℓ ) in the Grothendieck group of G ( A ∞ ,p ) × J b ( Q p )-representations.This is the long-term goal stated earlier. On the analogous problem for Shimura varieties, a roadmap has been laid out in [Kot90], which can be mimicked for Igusa varieties to some extent. How-ever there are serious obstacles: (1) An endoscopic classification for most reductive groups is outof reach; exactly the same issue occurs for Shimura varieties as well. (2) The geometric side (stableelliptic terms) is very difficult to compare with the spectral side. One could imagine making thecomparison more tractable by passing from H c to intersection cohomology, following the strategyfor Shimura varieties to “fill in” the stable non-elliptic terms, but no theory of compactification isavailable for Igusa varieties to allow it. (Franke’s formula for H c of locally symmetric spaces [Fra98]suggests that one should expect a similarly complicated answer for H c of Igusa varieties.)Our goal is to extract spectral information on H h ρ,ν b i c ( Ig b , Q ℓ ) from the leading terms in (1.4.1)in the variable j via the Lang–Weil estimate. Thus we can get away with less by proving equalitiesup to error terms of lower order. To bypass (1) and (2), a key is to show that (stable) non-ellipticterms as well as endoscopic (a.k.a. unstable) terms have slower growth in j than the (stable) ellipticterms. This is the technical heart of our paper taking up §
4. Let us provide more details.The basic strategy is an induction on the semisimple rank, based on our observation that somekey property of the function f e , ( j ) p is replicated after taking an endoscopic transfer or a constantterm. (For instance, we need to pass along the Newton cocharacter through the inductive steps.)So we want to prove a bound on the trace formula for a quasi-split group over Q , with a testfunction f p f ( j ) p satisfying such a property. The desired bound partly comes from a root-theoreticcomputation, involving a curious interaction between p and ∞ such as “evaluating” the Newtoncocharacter (coming from p ) at the infinite place (Lemma 4.1.1). The most interesting part in thispart of the argument isa spectral expansion of T ell , the elliptic part of the trace formula.(In (1.4.1) we can replace ST e ell with T ell once the difference is known to be of lower order.) Inour setup, where the archimedean test function is stable cuspidal, we have Arthur’s simple traceformula [Art89] of the following shape: T disc ( f p f ( j ) p ) = T ell ( f p f ( j ) p ) + (geometric terms on proper Levi subgroups) . (1.4.2) OF IGUSA VARIETIES 7
The proper Levi terms at finite places look similar to the elliptic part of the trace formula forproper Levi subgroups, but a complicated behavior is seen at the infinite place as this comes fromstable discrete series characters along non-elliptic maximal tori of the ambient group. On each openWeyl Chamber, we have a formula as a finite linear combination of finite dimensional charactersof the Levi subgroup, but this quickly spirals out of control in the induction. Adapting an idea ofLaumon [Lau97] from the non-invariant setup to the invariant setup, we overcome the difficulty byimposing a regularity condition on the test function at an auxiliary prime q ( = p ) and show thatthe Q -conjugacy classes with nonzero contributions land in a single Weyl Chamber. Then a finitedimensional character of a Levi subgroup is itself a stable discrete series character of the same Levisubgroup along elliptic maximal tori, and the inductive argument can continue. (The auxiliaryhypothesis at q is harmless in that no information is lost, as shown in § φ ∞ ,p φ ( j ) p | H c ( Ig b , Q ℓ )) = X π ∗ m ( π ∗ )Tr ( f ∗ ,p f ∗ , ( j ) p ) + (error terms) , where f ∗ ,p f ∗ , ( j ) p is the test function on the quasi-split inner form G ∗ of G , and the sum runs overdiscrete automorphic representations of G ∗ ( A ). At this point, we apply a trace identity. Let φ ∗ , ( j ) p denote a transfer of φ ( j ) p from J b to its quasi-split inner form M b . For each irreducible smoothrepresentation π ∗ of G ∗ ( Q p ), we have (Lemma 3.1.2)Tr π ∗ p ( f ∗ , ( j ) p ) = Tr J ( π ∗ p )( φ ∗ , ( j ) p ) , where J is the normalized Jacquet module relative to the parabolic subgroup determined by ν b whose Levi component is M b . Since b is non-basic, M b is a proper Levi subgroup. Moreover thetranslation ( j ) is given by a central element satisfying a positivity condition with respect to ν b . Inthese circumstances, we make a crucial use of an estimate due to Casselman and Howe–Moore ( § J ( π ∗ p )( φ ∗ , ( j ) p ) has the highest growth if and only if dim π ∗ p = 1. A strong approximationargument ( § π ∗ = 1, under a group-theoretic conditionguaranteed in our setting. Moreover, it is not hard to transfer one-dimensional representationsfrom M b ( Q p ) to J b ( Q p ) compatibly with the transfer of functions ( § Work of van Hoften and Xiao.
While this paper was being completed, Pol van Hoften an-nounced [vH] as well as [vHX] with Luciena Xiao Xiao, which prove the discrete HO conjecture andthe irreducibility of Igusa varieties. (Either this work or our work, combined with Xiao’s earlier pa-per [Xia], implies the full HO conjecture in certain cases, cf. Corollary 5.4.6 below.) Their methodis more geometric in nature and totally different from ours in that no use is made of automorphicforms. Further goals in their work and ours are disparate. For instance [vH] proves new results onstratification of Shimura varieties and the Langlands–Rapoport conjecture in the parahoric case,whereas our work is a stepping stone for computing the cohomology of Igusa varieties in all de-grees (up to alternating sums). The two threads could have a future intersection though, as theLanglands–Rapoport conjecture in the parahoric case would be an important ingredient for derivingthe analogue for Igusa varieties, extending Mack-Crane’s thesis [MC] from the hyperspecial case tothe case when G ( Q p ) has no hyperspecial subgroup. This would in turn allow the current paper’smethod to extend beyond the unramified case.1.6. A guide for the reader.
On a first reading, we suggest that all complexities arising fromcentral characters and z -extensions should be skipped, by assuming that all central character dataare trivial and that z -extensions are unnecessary. In fact this should be the case in most examples. ARNO KRET, SUG WOO SHIN
The central character datum is always trivial on the level of G appearing in the Hodge-type datum,but we allow it to be nontrivial mainly because we do not know whether H in the endoscopic datum( § L -group. Another good idea is to start reading in §
5, especiallyif one’s main interests lie in geometry, referring to the earlier sections only as needed and takingthe results there for granted.Sections 2 and 3 consist of mostly background materials in local harmonic analysis and rep-resentation theory. Though we claim little originality, there may be some novelty in the way weorganize and present them. Some statements would be of independent interest. Section 4 is perhapsthe most technical as this is where the main trace formula estimates are obtained. As such, mostreaders may want to take the results in § § p . Our main theorem on Igusa varieties is stated in § § § § § Jacquet moduleestimate ( § § § H c ( Ig b , Q ℓ ) ( § / / Thm. 6.1.3on H ( Ig b , Q ℓ )via autom. forms(main theorem) § / / § / / discrete HO conjecture+irreducibility of Ig b Notation. • The trivial character (of the group that is clear from the context) is denoted by . • If T is a torus over a field k with algebraic closure k , X ∗ ( T ) := Hom k ( T, G m ) and X ∗ ( T ) :=Hom k ( G m , T ). When R is a Z -algebra, we write X ∗ ( T ) R := X ∗ ( T ) ⊗ Z R and X ∗ ( T ) R := X ∗ ( T ) ⊗ Z R . • D := lim ←− G m is the protorus (over an arbitrary base), where the transition maps are the n -th power maps. • ˘ Z p := W ( F p ), ˘ Q p := Frac ˘ Z p , and σ ∈ Aut( ˘ Q p ) is the arithmetic Frobenius. • P ( S ) is the power set of a set S . • If H is an algebraic group over a field k , we write H ⊂ H for its neutral component.Let G be a connected reductive group over a field k of characteristic 0. • If k is a finite extension of k , then Res k/k G denotes the restriction of scalars group. • If k ′ is an extension field of k then G k ′ := G × Spec k Spec k ′ . • G der is the derived subgroup, ̺ : G sc → G der ⊂ G the simply connected cover, Z G the center(we also write Z ( G )), G ad := G/Z G the adjoint group, and G ab := G/G der the maximalcommutative quotient. Write A G ⊂ Z G for the maximal split subtorus over k . • G ( k ) ? is the set of semisimple (resp. regular semisimple, resp. strongly regular) elementsin G ( k ) for ? = ss (resp. reg, resp. sr). We put T ( k ) ? := T ( k ) ∩ G ( k ) ? for ? ∈ { reg , sr } . • If k is a local field and G a reductive group over k , we then write I ( G ( k )) and S ( G ( k )) forthe space of invariant and stable distributions on G ( k ). (For more details, see § G ( k )) we mean the set of isomorphism classes of irreducible admissible representationsof G ( k ). • When k = Q p , two elements δ, δ ′ ∈ G ( ˘ Q p ) are ( G ( ˘ Q p ) , σ ) -conjugate (resp. ( G (˘ Z p ) , σ ) -conjugate ) if there exists a g ∈ G ( ˘ Q p ) (resp. g ∈ G (˘ Z p )) such that δ ′ = σ ( g ) δg − . OF IGUSA VARIETIES 9
Let T (resp. S ) be a maximal torus (resp. maximal split torus) of G over k with T ⊃ S . Let M be a minimal k -rational Levi subgroup containing T . • Φ( T, G ) is the set of absolute roots, Φ(
S, G ) = Φ k ( S, G ) the set of k -rational roots. • Ω G = Ω( T, G ) for the Weyl group over k , and Ω Gk = Ω( S, G ) for the k -rational Weyl group.We often omit k from Φ k ( S, G ) and Ω Gk when it is clear from the context. • L ( G ) or L k ( G ) is the set of all k -rational Levi subgroups of G containing M . Write L < ( G ) := L ( G ) \{ G } . Lemma 1.7.1. If G der is simply connected then every k -rational Levi subgroup of G has simplyconnected derived subgroup.Proof. This can be checked after base change to k , so assume k = k . For every maximal torus T ⊂ G , the cocharacter lattice X ∗ ( T ) modulo the coroot lattice is torsion free by hypothesis. Thus X ∗ ( T ) modulo the lattice generated by an arbitrary subset of simple coroots is torsion free, implyingthat every Levi subgroup of G has simply connected derived subgroup. (cid:3) Acknowledgments.
AK was partially supported by an NWO VENI grant. SWS is partially sup-ported by NSF grant DMS-1802039 and NSF RTG grant DMS-1646385. AK and SWS are gratefulto Erez Lapid, Gordan Savin, and Maarten Solleveld for pointing them in the right direction re-garding § Preliminaries in representation theory and endoscopy
Estimates for Jacquet modules of unitary representations.
Here we recall some facts fromwork of Howe–Moore [HM79] and Casselman [Cas95] in order to bound the absolute value ofcentral characters in the Jacquet modules of unitary representations of p -adic reductive groups.We consider the following setup and notation. • Let F be a non-archimedean local field of characteristic 0. We write val F , O F , k , q , ̟ F respectively for the normalized valuation of F , the ring of integers of F , the residue field of F , the cardinality of k , and an uniformizer of F so that val F ( ̟ F ) = 1, • G is a connected reductive group over F with center Z = Z G , • Rep( G ) is the category of smooth representations of G ( F ), • P = M N is a Levi decomposition of an F -rational proper parabolic subgroup of G , • A M is the maximal F -split torus in the center of M , • ∆ is the set of roots of A M in N , • A − P := { x ∈ A M ( F ) : | α ( x ) | ≤ , ∀ α ∈ ∆ } , • A −− P := { x ∈ A M ( F ) : | α ( x ) | < , ∀ α ∈ ∆ } , • δ P : M ( F ) → R × > is the modulus character given by δ P ( m ) := | det(Ad( m ) | Lie N ( F )) | . • J P : Rep( G ) → Rep( M ) is the normalized Jacquet module functor, so J P ( π ) = π N ⊗ δ − / P with π N denoting the N ( F )-coinvariants of π , • I GP : Rep( M ) → Rep( G ) is the normalized parabolic induction functor, sending π M to thesmooth induction of π M ⊗ δ / P from P ( F ) to G ( F ). • When R ∈ Rep( M ) has finite length, write Exp( R ) for the set of A M ( F )-characters appear-ing as central characters of irreducible subquotients of R . Lemma 2.1.1. If G is simply connected, F -simple, and F -isotropic, then every normal subgroup of G ( F ) is either G ( F ) itself or contained in Z ( F ) .Proof. A normal subgroup N of G ( F ) not contained in Z ( F ) is open of finite index in G ( F )by [PR94, Prop. 3.17] since G is F -simple. Since G ( F ) is F -isotropic and simply connected, G ( F )is generated by the F -points of the unipotent radicals of F -rational parabolic subgroups. By Tit’stheorem proven in [Pra82], every open proper subgroup of G ( F ) is compact. On the other hand, N is easily seen to be non-compact by considering the adjoint action of a maximal F -split torus ona root subgroup. Therefore N = G ( F ). (cid:3) Proposition 2.1.2 (Howe–Moore) . Assume that G sc is F -simple. Let π be an infinite dimensionalirreducible unitary representation of G ( F ) . Then there exists an integer ≤ k < ∞ such that everymatrix coefficient of π belongs to L k ( G ( F ) /Z ( F )) .Proof. This follows from the explanation on pp.74–75 of [HM79] below Theorem 6.1, once we verifythe following claim: if π ( g ) is a scalar operator for g ∈ G ( F ) then g ∈ Z ( F ). Taking a z -extension of G , we reduce to the case when G der is simply connected. Pulling back π via the multiplication map Z ( F ) × G der ( F ) → G ( F ) and passing to one of the finitely many constituents (cf. [Xu16, Lem. 6.2])which is infinite-dimensional, we may assume that G is itself F -simple and simply connected. Now Z ′ be the group of g ∈ G ( F ) such that π ( g ) is a scalar. Then Z ′ is a normal subgroup of G ( F ). If Z ′ = G ( F ) then dim π = 1, contradicting the initial assumption. Therefore Z ′ ⊂ Z ( F ) by Lemma2.1.1, proving the claim. (cid:3) Proposition 2.1.3 (Casselman) . Let π be an irreducible unitary representation of G ( F ) . For every ω ∈ Exp( π N ) and every a ∈ A − P , we have the inequality | ω ( a ) | ≤ . (2.1.1) Now suppose that G sc is F -simple and that a ∈ A −− P . Then the equality holds if and only if π isfinite dimensional.Proof. This follows from [Cas95, § p < ∞ is assumed), with the obvious extension tocover the case p = ∞ . Let us explain more details, freely using the notation and definition of thatpaper.The analogue of [Cas95, Lem. 4.4.1] for p = ∞ is the following. Let F : Z n → C be a Z n -finitefunction. Then sup x ∈ Z n F ( x ) < ∞ if and only if | χ ( x ) | ≤ x ∈ N n and χ associatedwith F . The proof is elementary. Similarly [Cas95, Lem. 4.4.3, Prop. 4.4.4] have the analogues for p = ∞ , with “bounded” in place of “summable” and “ | χ ( x ) | ≤
1” in place of “ | χ ( x ) | < p = ∞ , the conclusion of [Cas95, Cor. 4.4.5] should read “ |F | is bounded if and only if for everycharacter χ associated to Φ and a ∈ A − Θ , | χ ( a ) | ≤ a ∈ A − Θ ” is replaced with“ a ∈ A − Θ \ A ∅ ( O ) A ∆ ”.) Let us refer to the p = ∞ analogue of [Cas95, Cor. 4.4.5] as Cor. 4.4.5 ∞ .Now we prove (2.5.2) by imitating the proof for the “only if” direction of [Cas95, Thm. 4.4.6].Consider K, Γ as in that proof as well as the decomposition G = ` i,j ∈ I Kγ i A −∅ γ j K , where { γ i } i ∈ I isa set of representatives for Γ /KA ∆ . Let v ∈ V , ˜ v ∈ ˜ V be given, and consider the matrix coefficient c v, ˜ v ( g ) = h π ( g ) v, ˜ v i of π . We apply Cor. 4.4.5 ∞ to F = c v, ˜ v . Then every ω ∈ Exp( π N ) is associatedto the restriction Φ of F to Θ A −∅ ( ǫ ) with 0 < ǫ ≤
1, and Cor. 4.4.5 ∞ tells us that | ω ( a ) | ≤ a ∈ A − Θ . (His A − Θ corresponds to our A − P .)It remains to check the last assertion of the theorem. Suppose that dim π = ∞ . Let K, Γ , v, ˜ v, c v, ˜ v be as above. By Proposition 2.1.2, | c v, ˜ v | k is integrable modulo center for some 2 ≤ k < ∞ .Applying [Cas95, Cor. 4.4.5] in the same way as above to p = k and a ∈ A −− P , we obtain that | ω ( a ) δ − /kP ( a ) | <
1. Therefore | ω ( a ) | <
1. Now suppose that dim π < ∞ . Then ker π is an opensubgroup of G ( F ). As the open subgroup N ( F ) ∩ ker π of the unipotent subgroup N ( F ) actstrivially on π , we see that N ( F ) itself acts trivially on π . (Use conjugation by A M ( F ).) ThereforeExp( π N ) consists of the central character ω of π (restricted to M ( F )) only, which is unitary. Inparticular | ω ( a ) | = 1 for all a ∈ A −− P . (cid:3) For instance, see the proof of Proposition 3.9 in http://virtualmath1.stanford.edu/~conrad/JLseminar/Notes/L2.pdf for details. The roman F is used in [Cas95]. We change it to F to avoid conflict with the field F . OF IGUSA VARIETIES 11
Remark . Proposition 2.1.3 is sharp in general (though some strengthening is available undera hypothesis, e.g., see [Oh02]). For example, consider G = GL ( F ) with P (resp. N ) consistingof upper triangular (resp. upper triangular unipotent) matrices. The complementary series rep-resentations π ǫ = I GP ( | · | ǫ , | · | − ǫ ) with ǫ ∈ R with 0 < ǫ < / π ǫ ) N = J N ( π ǫ ) ⊗ δ / P = δ / P ⊗ (cid:0) ( | · | ǫ , | · | − ǫ ) ⊕ ( | · | − ǫ , | · | ǫ ) (cid:1) = ( | · | ǫ + , | · | − ǫ − ) ⊕ ( | · | − ǫ + , | · | ǫ − ) . So in this case, Exp(( π ǫ ) N ) contains the character ω = ( | · | − ǫ + , | · | ǫ − ) of Q × p × Q × p . Then a = (cid:0) p
00 1 (cid:1) ∈ A −− P . We get ω ( a ) = p ǫ − which gets arbitrarily close to 1 as ǫ tends to 1 / Lemma 2.1.5.
Assume that G ad has no F -anisotropic factor. Then every irreducible smooth repre-sentation of G ( F ) is either one-dimensional or infinite-dimensional.Proof. We may assume that G der is simply connected via z -extensions. Suppose that π is a finite-dimensional irreducible smooth representation of G ( F ). Then the normal subgroup ker π ∩ G der ( F )of G der ( F ) is open. Lemma 2.1.1 implies that ker π ∩ G der ( F ) = G der ( F ), thus π factors throughthe abelian quotient G ( F ) /G der ( F ). Therefore dim π = 1, completing the proof. (cid:3) Local Hecke algebras and their variants.
We retain the notation from the preceding sectionbut allow the local field F to be either non-archimedean or archimedean. A basic setup of localHecke algebras will be introduced, partly following [Art96, § G ( F ) and a maximal compact subgroup K ⊂ G ( F ). Let G ( F ) sr denote thesubset of strongly regular elements g ∈ G ( F ), namely the semisimple elements whose centralizers in G are (maximal) tori. By [Ste65, 2.15], G ( F ) sr is open and dense in G ( F ) (for both the Zariski andnon-archimedean topologies). Write R ( G ) for the space of finite C -linear combinations of irreduciblecharacters of G ( F ), which is a subspace in the space of functions on G ( F ) sr . We also identify R ( G )with the Grothendieck group of smooth finite-length representations of G ( F ) with C -coefficients.Let H ( G ) = H ( G ( F )) denote the space of smooth compactly supported bi- K -finite functions on G ( F ). Let I ( G ) denote the invariant space of functions on G ( F ), namely the quotient of H ( G ) bythe ideal generated by functions of the form g f ( g ) − f ( hgh − ) with h ∈ G ( F ) and f ∈ H ( G ).From [Kaz86, Thm. 0], we see that f ∈ H ( G ) has trivial image in I ( G ) if and only if its orbitalintegral vanishes on G ( F ) sr if and only if Tr π ( f ) = 0 for all irreducible tempered representations of G ( F ); moreover, the same is true if G ( F ) sr is replaced with G ( F ) and if the temperedness conditionis dropped. By abuse of notation, we frequently write f ∈ I ( G ) to mean a representative f ∈ H ( G )of an element in I ( G ). The trace Paley–Wiener theorem [BDK86] describes I ( G ) as a subspace of C -linear functionals on R ( G ) via f (cid:18) Θ Z G ( F ) sr f ( g )Θ( g ) dg (cid:19) . (2.2.1)If R ( G ) is thought of as a Grothendieck group, the above map is simply f ( π Tr π ( f )).Denote by S ( G ) the quotient of H ( G ) by the ideal generated by functions each of which has van-ishing stable orbital integrals on G ( F ) sr . Thus we have natural surjections H ( G ) ։ I ( G ) ։ S ( G ).By R ( G ) st we mean the subspace of R ( G ) consisting of stable linear combinations (i.e., constanton each stable conjugacy class in G ( F ) sr ). Then S ( G ) is identified with a subspace of functionson R ( G ) st via (2.2.1) (since Θ ∈ R ( G ) st now, the image depends only on the image of f in S ( G ));the subspace is characterized by [Art96, Thm. 6.1, 6.2] in the p -adic case, cf. last paragraph onp.491 of [Xu17]. Via the obvious quotient map I ( G ) → S ( G ) and the restriction map from R ( G ) to R ( G ) st , we have a commutative diagram I ( G ) / / Tr (cid:15) (cid:15) S ( G ) Tr (cid:15) (cid:15) Hom C -linear ( R ( G ) , C ) / / Hom C -linear ( R ( G ) st , C ) . Let us extend the setup so far to allow a fixed central character. By a local central characterdatum for G , we mean a pair ( X , χ ), where • X is a closed subgroup of Z ( F ) with a Haar measure µ X on X , • χ : X → C × is a smooth character.Let H ( G, χ − ) = H ( G ( F ) , χ − ) denote the local Hecke algebra consisting of smooth bi- K -finitefunctions f on G ( F ) which have compact support modulo X and satisfy f ( xg ) = χ − ( x ) f ( g ) for x ∈ X and g ∈ G ( F ). The χ -averaging map H ( G ) → H ( G, χ − ) , f (cid:18) g Z X f ( gz ) χ ( z ) dµ X (cid:19) , is a surjection. We have the obvious definitions of I ( G, χ − ) and S ( G, χ − ), the χ -averaging maps I ( G ) → I ( G, χ − ) and S ( G ) → S ( G, χ − ), as well as the quotient maps H ( G, χ − ) ։ I ( G, χ − ) ։ S ( G, χ − ) . We can think of I ( G, χ − ) as a subspace of functions on R ( G, χ ), the subspace of R ( G ) generated byirreducible characters with central character χ . Analogously S ( G, χ − ) is the subspace of functionson R ( G, χ ) st defined similarly.2.3. Transfer of one-dimensional representations.
Let G and G ∗ be connected reductive groups overa non-archimedean local field F of characteristic zero, related by an F -isomorphism ξ : G F ∼ → G ∗ F .We assume that G ∗ is quasi-split over F and that ξ is an inner twisting, namely that ξ − σ ( ξ ) is aninner automorphism of G F for every σ ∈ Gal(
F /F ). As in § F -morphisms ̺ : G sc → G and ̺ ∗ : G ∗ sc → G ∗ . Define an F -torus and two groups G ♭ := G/G der , G ( F ) ♭ := cok( G sc ( F ) ̺ → G ( F )) , G ( F ) ab := G ( F ) /G ( F ) der , where G ( F ) der is the commutator subgroup (as an abstract group). Then G ( F ) der is contained inim( G ( F ) sc → G ( F )) [Del79, 2.0.2], so there are natural morphisms G ( F ) ։ G ( F ) ab ։ G ( F ) ♭ ։ G ( F ) /G der ( F ) ֒ → G ♭ ( F ) . (2.3.1)In particular, G ( F ) ♭ is an abelian group. We view G ( F ) ♭ and G ( F ) ab as topological groups usingthe quotient topology as G sc ( F ) and G ( F ) der have closed images in G ( F ). Only the latter caserequires explanation: if G is a torus then G der = { } , and if not, G ( F ) der is not contained in Z G der ( F ) so an open subgroup of finite index in G der ( F ) by [PR94, Prop. 3.17] (reduce to thesimply connected and F -simple case via z -extensions).The last two maps in (2.3.1) are isomorphisms if G der = G sc by Kneser’s vanishing theoremfor H of simply connected groups (applicable since F is non-archimedean). The definition anddiscussion above applies to G ∗ in the same way.Let 1 → Z → G α → G → z -extension of G over F . Since G → G induces G ad1 ∼ → G ad ,the classifying data for inner twists of G and those of G are identified (up to isomorphism). Thuswe may assume that there is a z -extension 1 → Z → G ∗ α ∗ → G ∗ → ξ : G ,F ∼ → G ∗ ,F such that ξ and ξ form a commutative square together with the maps α and α ∗ . The map G , der → G der induced by α is a simply connected cover, allowing an identification G , der = G sc . Likewise we have G ∗ , der = G ∗ sc . OF IGUSA VARIETIES 13
Lemma 2.3.1.
There is a commutative diagram in which rows are exact and vertical maps areisomorphisms: / / Z ( F ) /Z ( F ) ∩ G , der ( F ) / / ∼ (cid:15) (cid:15) G ( F ) ♭ = G ♭ ( F ) ∼ (cid:15) (cid:15) / / G ( F ) ♭ / / ∼ (cid:15) (cid:15) / / Z ( F ) /Z ( F ) ∩ G ∗ , der ( F ) / / G ′ ( F ) ♭ = G ∗ ,♭ ( F ) / / G ∗ ( F ) ♭ / / . Here the second vertical map is given by the isomorphism G ♭ ∼ → G ∗ ,♭ induced by ξ , and the firstand third vertical maps are induced by the second. Moreover the isomorphism G ( F ) ♭ ∼ → G ∗ ( F ) ♭ iscanonical, i.e., independent of the choice of z -extensions. We will write ξ ♭ : G ( F ) ♭ ∼ → G ∗ ( F ) ♭ for the canonical isomorphism. Proof.
Let us verify that the first row is exact in the lemma; the exactness of the second row followsin the same way. Consider the following commutative diagram where all maps are the natural ones:1 / / G , sc ( F ) / / (cid:15) (cid:15) G ( F ) / / (cid:15) (cid:15) G ♭ ( F ) / / (cid:15) (cid:15) / / im( G , sc ( F ) → G , der ( F )) / / G ( F ) / / G ( F ) ♭ / / . Applying the snake lemma, we obtain the exact sequence in the first row of the lemma.The map ξ induces an F -isomorphism G ♭ ∼ → G ∗ ,♭ and restricts to an F -isomorphism from Z onto Z . Thus the first two vertical maps are isomorphisms, which implies that the last verticalmap is also.As for the last assertion, if 1 → Z → G α → G → → Z → G α → G → z -extensions then there is a third z -extension 1 → Z × Z → G → G → G = { ( g , g ) ∈ G × G : α ( g ) = α ( g ) } , equipped with projections G ։ G and G ։ G . Thus weare reduced to showing that G and G (and likewise G and G ) induce the same isomorphism G ( F ) ♭ ∼ → G ∗ ( F ) ♭ . This is an elementary compatibility check. (cid:3) Lemma 2.3.2. If G sc has no F -anisotropic factor, then G ( F ) ♭ = G ( F ) ab .Proof. We may assume that G is not a torus. Via a z -extension, we reduce to the case when G sc = G der . Then G ( F ) der is a noncentral normal subgroup of G der ( F ). Applying Lemma 2.1.1 toeach F -simple factor of G der , we deduce that G ( F ) der = G der ( F ), hence G ( F ) ♭ = G ( F ) ab . (cid:3) Corollary 2.3.3. If G sc has no F -anisotropic factor, then the following four groups (under multipli-cation) are in canonical isomorphisms with each other:(1) the group of smooth characters G ( F ) → C × ,(2) the group of smooth characters G ( F ) ♭ → C × ,(3) the group of smooth characters G ∗ ( F ) ♭ → C × ,(4) the group of smooth characters G ∗ ( F ) → C × ,where the maps from (2) to (1) and from (3) to (4) are given by pullbacks, and the map between(2) and (3) is via the isomorphism of Lemma 2.3.1. With no assumption on G sc , we still havecanonical isomorphisms between (2), (3), and (4), and a canonical embedding from (2) to (1).Proof. We have seen that G ( F ) ab is the maximal abelian topological quotient of G ( F ), so G ( F )may be replaced with G ( F ) ab in (1). The analogue holds for (4). From (2.3.1) and Lemma 2.3.1,we have G ( F ) ab ։ G ( F ) ♭ ≃ G ∗ ( F ) ♭ և G ∗ ( F ) ab . Lemma 2.3.2 tells us that the last map is always an isomorphism (since G ∗ has no F -anisotropicfactor); so is the first map if G has no F -anisotropic factor. The corollary follows. (cid:3) Remark . The only nontrivial F -anisotropic simply connected simple group over F is of type A,more precisely of the form Res F ′ /F SL ( D ) for a central division algebra D over a finite extension F ′ of F with [ D : F ] = n and n ≥
2. So the condition in the corollary is that G sc has no such factor.Two exemplary cases are (i) G = GL ( D ), G ∗ = GL n ( F ) and (ii) G = SL ( D ), G ∗ = SL n ( F ). In (i),it is standard (e.g., [Rie70, Intro.]) that G ( F ) der = G der ( F ), and the four sets are still isomorphic.However, in (ii), G ( F ) der is the group of 1-units in the maximal order of D by [Rie70, §
5, Cor.]. Inparticular (1) is a nontrivial group, whereas (2) and (3) are evidently trivial, thus (4) is trivial bythe corollary.
Remark . One can also construct a natural map from (4) to (1) through the continuous coho-mology H ( W F , Z ( b G )) = H ( W F , Z ( b G ∗ )) following Langlands. (This works for archimedean localfields F as well.) Indeed, [Xu16, App. A] explains the isomorphism between H ( W F , Z ( b G ∗ )) and(4), and a map from H ( W F , Z ( b G ∗ )) to (1). Let us define the notion that semisimple elements g ∈ G ( F ) and g ∗ ∈ G ∗ ( F ) are matching ,depending only on the G ( F )-conjugacy class of the inner twisting ξ : G F ∼ → G ∗ F . (In practice, onlythe G ( F )-conjugacy class of ξ is well-defined.) When G der = G sc , stable conjugacy classes are thesame as F -conjugacy classes, and g and g ∗ are considered matching if their stable conjugacy classesare identified via ξ . In general, matching can be defined by lifting ξ to an inner twisting between z -extensions of G and G ∗ as in [Kot82, pp.799–800] (specializing to the case E = F ), which alsoshows that the definition is independent of the choice of z -extensions and the choice of ξ in its G ( F )-conjugacy class. Since G ∗ is quasi-split, every g admits a matching element in G ∗ ( F ).When g ∈ G ( F ) and g ∗ ∈ G ∗ ( F ) are matching, we have an inner twisting between the connectedcentralizers I g , I g ∗ in G, G ∗ by [Kot82, Lem. 5.8]. Fix Haar measures on the pairs of inner forms( G ( F ) , G ∗ ( F )) and ( I g ( F ) , I g ∗ ( F )) compatibly in the sense of [Kot88, p.631] to define orbital inte-grals at g and g ∗ as well as stable orbital integrals [Kot88, pp.637–638]. Write e ( G ) ∈ {± } for theKottwitz sign. Now f ∈ H ( G ( F )) and f ∗ ∈ H ( G ∗ ( F )) are said to be matching if for every stronglyregular g ∗ ∈ G ∗ ( F ), we have the identity of stable orbital integrals SO g ∗ ( f ∗ ) = ( SO g ( f ) , if there exists a matching g ∈ G ( F ) , , if there is no such g ∈ G ( F ) . (2.3.2) Remark . The sign convention in (2.3.2) is chosen in favor of simplicity. (See also Remark 7.4.1below.) One could require SO g ∗ ( f ∗ ) = e ( G ) SO g ( f ) instead, so that the Kottwitz sign e ( G ) playsthe role of transfer factor, but that would introduce e ( G ) in the trace identity of Lemma 2.3.7.A standard fact (cf. § f , there always exists a matching f ∗ as above, called a transfer of f (between inner forms). If the Harish-Chandra character Θ π ∗ of π ∗ ∈ Irr( G ∗ ( F )) is stable, i.e., Θ π ∗ ( g ∗ ) = Θ π ∗ ( g ∗ ) whenever two strongly regular semisimpleelements g ∗ , g ∗ ∈ G ∗ ( F ) are stably conjugate, then the value Tr π ∗ ( f ∗ ) = R G ∗ ( F ) sr f ∗ ( g ∗ )Θ π ∗ ( g ∗ ) dg ∗ is determined by the stable orbital integrals of f ∗ on strongly regular semisimple elements. Thisfollows from the stable version of the Weyl integration formula, cf. (2.3.3) below. This discussionapplies to π ∗ with dim π ∗ = 1 for example, since the characters of such π ∗ are clearly stable. Theanalogue holds true with G and f in place of G ∗ and f ∗ . The latter map is asserted to be also an isomorphism in [Xu16, App. A], but this is false for G = SL ( D ) (inwhich case Z ( b G ) = { } ) as explained in Remark 2.3.4. In loc. cit. , for a simply connected group G ′ over F , it is saidthat all continuous characters G ′ ( F ) → C × are trivial, but this is not guaranteed unless G sc has no F -anisotropicfactor (e.g., this is okay for G ∗ ). OF IGUSA VARIETIES 15
Lemma 2.3.7.
Let f ∈ H ( G ( F )) and f ∗ ∈ H ( G ∗ ( F )) be matching functions. Let π (resp. π ∗ ) bea one-dimensional smooth representation of G ( F ) (resp. G ∗ ( F ) ). If π ∗ corresponds to π via thepreceding corollary then Tr π ( f ) = Tr π ∗ ( f ∗ ) . Proof.
As dim π = dim π ∗ = 1, the characters Θ π and Θ π ∗ are stable, and Θ π ( g ) = π ( g ), Θ π ∗ ( g ∗ ) = π ∗ ( g ∗ ) for g ∈ G ( F ) , g ∗ ∈ G ∗ ( F ). For a maximal torus T of G over F , write W ( T ) for theassociated Weyl group. By the stable Weyl integration formula,Tr π ( f ) = Z G ( F ) sr f ( g )Θ π ( g ) dg = X T | W ( T ) | Z T ( F ) sr SO t ( f )Θ π ( t ) dt, (2.3.3)where the summation runs over a set of representatives for stable conjugacy classes of maximal toriof G over F . The analogous formula holds for G ∗ ( F ). Since f and f ∗ are matching, if a maximaltorus T ∗ of G ∗ does not transfer to G over F (in the sense of [Kot84b, 9.5]) then SO t ∗ ( f ∗ ) = 0on t ∗ ∈ T ∗ ( F ) sr . If T ∗ transfers to a maximal torus T of G (since G ∗ is quasi-split over F , everymaximal torus of G over F arises in this way), and if t ∈ T ( F ) sr and t ∗ ∈ T ∗ ( F ) sr have matchingconjugacy classes, then the image of t in G ( F ) ab and that of t ∗ in G ∗ ( F ) ab correspond via thesurjection G ( F ) ab ։ G ∗ ( F ) ab in the proof of Corollary 2.3.3. Indeed, this is straightforward when G der is simply connected, and reduced to the latter case in general via z -extensions. Hence Θ π ( t ) =Θ π ∗ ( t ∗ ) in view of Corollary 2.3.3. It follows from (2.3.2) and (2.3.3) that Tr π ( f ) = Tr π ∗ ( f ∗ ). (cid:3) Remark . In Lemma 2.3.7, π and π ∗ are not related by the Jacquet–Langlands correspondencewhen G ∗ = GL n and G is not quasi-split. For instance, consider the case G = GL ( D ) for a centraldivision algebra D over a p -adic field F with n >
1. The trivial representation of D × corresponds tothe Steinberg representation of GL n ( F ) under Jacquet–Langlands, but to the trivial representationof GL n ( F ) in the lemma.2.4. Lefschetz functions on real reductive groups.
Let G be a connected reductive group over R containing an elliptic maximal torus. Fix a maximal compact subgroup K ∞ ⊂ G ( R ). Denote by G ( R ) + the preimage of the neutral component G ad ( R ) (for the real topology) under the naturalmap G ( R ) → G ad ( R ). Lemma 2.4.1.
We have G ( R ) + = Z ( R ) · ̺ ( G sc ( R )) .Proof. Since G sc ( R ) is connected, clearly ̺ ( G sc ( R )) maps into G ad ( R ) . Therefore G ( R ) + ⊃ Z ( R ) · ̺ ( G sc ( R )). We have surjections G sc ( R ) × Z ( R ) ։ G ( R ) ։ G ad ( R ) by [Mil05, Prop. 5.1]. Thisimplies that G ( R ) + ⊂ Z ( R ) G ( R ) = Z ( R ) · ̺ ( G sc ( R )). (cid:3) Let ξ be an irreducible algebraic representation of G C , and ζ : G ( R ) → C × be a continuouscharacter. By restriction ξ yields a continuous representation of G ( R ) on a complex vector space,which we still denote by ξ . Write ω ξ : Z ( R ) → C × for the central character of ξ . By Π ∞ ( ξ, ζ ) wemean the set of isomorphism classes of irreducible discrete series representations whose central andinfinitesimal characters are equal to those of the contragredient of ξ ⊗ ζ . This is a discrete series L -packet by the construction of [Lan89], which assigns to Π ∞ ( ξ, ζ ) an L -parameter ϕ ξ,ζ : W R → L G. Thus we also write Π ∞ ( ϕ ξ,ζ ) for Π ∞ ( ξ, ζ ). We have ξ ⊗ ζ ≃ ξ ′ ⊗ ζ ′ as representations of G ( R ) ifand only if there exists an algebraic character χ of G C such that ξ ′ = ξ ⊗ χ and ζ ′ = ζ ⊗ χ − . Inthis case Π ∞ ( ξ, ζ ) = Π ∞ ( ξ ′ , ζ ′ ), and ϕ ξ,ζ ≃ ϕ ξ ′ ,ζ ′ .Write A G for the maximal split torus in the center of G . Let χ : A G ( R ) → C × be a continuouscharacter. Let Irr temp ( G ( R ) , χ ) be the set of (isomorphism classes of) irreducible tempered repre-sentations of G ( R ) whose central character equals χ on A G ( R ) . Following [Art89, § f ∈ H ( G ( R ) , χ − ) is said to be stable cuspidal if Tr π ( f ) = 0 for every π ∈ Irr temp ( G ( R ) , χ ) that is not a discrete series representation, and if Tr π ( f ) is constant as π varies over each discrete series L -packet.Fix a Haar measure on G ( R ) and the Lebesgue measure on A G ( R ) , so as to determine a Haarmeasure on G ( R ) /A G ( R ) . Choose a pseudo-coefficient f π for each π ∈ Π ∞ ( ξ, ζ ) `a la [CD85], sothat f π ∈ H ( G ( R ) , ω ξ ζ ). The characterizing property of f π is that Tr π ( f π ) = 1 and Tr π ′ ( f π ) = 0for π ′ ∈ Irr temp ( G ( R ) , ( ω ξ ζ ) − ). Although f π is not unique, its orbital integrals are uniquelydetermined. It depends on the Haar measure on G ( R ): if the measure is multiplied by c ∈ C × then f π is to be replaced with c − f π . An averaged Lefschetz function associated with ( ξ, ζ ), to bedenoted by either f ξ,ζ or f ϕ ξ,ζ , is defined as f ξ,ζ := | Π ∞ ( ξ, ζ ) | − X π ∈ Π ∞ ( ξ,ζ ) f π ∈ H ( G ( R ) , ω ξ ζ ) . By construction, f ξ,ζ is stable cuspidal in the above sense.In fact | Π ∞ ( ξ, ζ ) | is a constant depending only on G , namely the ratio of the Weyl groups for G and a maximal compact subgroup. Write d ( G ) ∈ Z ≥ for this constant. When ξ = (the trivialrepresentation), we also write Π ∞ ( ζ ) and f ζ for Π ∞ ( ξ, ζ ) and f ξ,ζ .For elliptic γ ∈ G ( R ), let I γ denote its connected centralizer in G ( R ), and e ( I γ ) ∈ {± } itsKottwitz sign. Let I cpt γ denote an inner form of I γ over R that is anisotropic modulo Z G ( R ).From [Kot92a, p.659] (as our O γ ( f ξ,ζ ) equals d ( G ) − SO γ ∞ ( f ∞ ) there), we see that O γ ( f ξ,ζ ) = ( d ( G ) − vol( A G ( R ) \ I cpt γ ( R )) − ζ ( γ ) e ( I γ )Tr ξ ( γ ) , γ : elliptic , , γ : non-elliptic . (2.4.1)In (2.4.1), the Haar measure on I cpt γ ( R ) is chosen to be compatible (in the sense of [Kot88, p.631])with the measure on I γ ( R ) used in the orbital integral, to compute vol( A G ( R ) \ I cpt γ ( R )) withrespect to the Lebesgue measure on A G ( R ) . Again by loc. cit. we have SO γ ( f ξ,ζ ) = ( vol( A G ( R ) \ I cpt γ ( R )) − ζ ( γ )Tr ξ ( γ ) , γ : elliptic , , γ : non-elliptic . (2.4.2)Let G ∗ be a quasi-split group over R with inner twisting G C ∼ → G ∗ C , through which ξ, ζ aboveare transported to G ∗ . Thereby we obtain an averaged Lefschetz function f ∗ ξ,ζ on G ∗ ( R ). Lemma 2.4.2.
The function f ∗ ξ,ζ is a transfer of f ξ,ζ (in the sense of (2.3.2) ).Proof. This is immediate from (2.4.2). (cid:3)
Lemma 2.4.3.
Assume that ξ = . Let π : G ( R ) → C × be a continuous character whose centralcharacter equals ζ − when restricted to A G ( R ) . Then π | G ( R ) + = ζ − | G ( R ) + if and only if π | Z ( R ) = ζ − | Z ( R ) . If the equivalent conditions hold then Tr ( f ζ | π ) = 1 if π = ζ − ; otherwise Tr ( f ζ | π ) = 0 .Proof. The first assertion is clear from Lemma 2.4.1. For the second assertion, it follows from (2.4.1)via the Weyl integration formula that Tr ( f ζ | π ) = vol( K ) − R K ζ ( k ) π ( k ) dk , where K is a maximalcompact-modulo- A G ( R ) subgroup of G ( R ). The integral vanishes unless π = ζ − on K , in whichcase π = ζ − on the entire G ( R ) (since K meets every component of G ( R )) and Tr ( f ζ | π ) = 1. (cid:3) One-dimensional automorphic representations.
Now let G be a connected reductive group overa number field F . Let v be a place of F and set G v := G F v . We have a finite decomposition of G sc into F -simple factors G sc = Y i ∈ I G i , with G i = Res F i /F H i , (2.5.1)for a finite extension F i /F and an absolutely F -simple simply connected group H i over each F i .Accordingly G ad = Q i ∈ I G ad i . Note that we have a natural composite map G → G ad → G ad i for OF IGUSA VARIETIES 17 each i ∈ I , where the last arrow is the projection onto the i -component. Let P v = M v N v be a Levidecomposition of a parabolic subgroup of G v . We consider the following assumption, where “nb”stands for non-basic (cf. Definition 5.3.2 below). ( Q -nb( P v )) The image of P v in ( G ad i ) v is a proper parabolic subgroup for every i ∈ I .The assumption implies that G ad has no (nontrivial) F -simple factor that is anisotropic over F v . This leads to a useful fact that the embedding G sc ( F ) ֒ → G sc ( A vF ) has dense image by strongapproximation. When G ad is itself F -simple, ( Q -nb( P v )) is simply saying that P v is a proper parabolic subgroup of G v . Lemma 2.5.1.
Assume that G der = G sc and that G i is isotropic over F v for every i ∈ I . Let π bea discrete automorphic representation of G ( A F ) , and π ′ an irreducible G der ( A F ) -subrepresentationof π . Decompose π ′ = ⊗ i π ′ i according to G der ( A F ) = Q i ∈ I G i ( A F ) . Write G i ( F v ) = H i ( F i ⊗ F F v ) = Y w | v H i ( F i,w ) , where w runs over the set of places of F i above v , and decompose π ′ i,v = ⊗ w | v π ′ i,w accordingly. Iffor every i ∈ I , there exists w | v such that π ′ i,w is trivial, then dim π = 1 .Proof. Thanks to the isotropicity assumption, we apply the strong approximation to see that theembedding H i ( F i ) ֒ → H i ( A wF i ) has dense image. Since the underlying space of π ′ i consists of auto-morphic forms which are left-invariant under H i ( F i ), and since π ′ i,w is trivial, we argue as in theproof of [KST, Lem. 6.2] to see that π ′ i is trivial on the entire H i ( A F i ). The same argument appliesto every i ∈ I to imply that π ′ is trivial. Since G ( A F ) /G der ( A F ) is abelian, we deduce dim π = 1noting that π is generated by π ′ as a G ( A F )-module. (cid:3) Corollary 2.5.2.
Let π be an irreducible G ( A F ) -subrepresentation of L ( G ( F ) \ G ( A F ) /A G, ∞ ) andlet ω v ∈ Exp( J P v ( π v )) . Then δ / P v ( a ) ≤ | ω v ( a ) | ≤ δ − / P v ( a ) , a ∈ A −− P v . (2.5.2) Now assume ( Q -nb( P v )) . Then the left (resp. right) equality holds for some a ∈ A −− P v if and onlyif the left (resp. right) equality holds for all a ∈ A −− P v if and only if π is one-dimensional.Proof. The assertion about the (in)equality on the left follows from the right counterpart by con-sidering the contragredient of π . From now on, we concentrate on the right (in)equality.The right inequality in (2.5.2) is immediate from Proposition 2.1.3 and the normalization J P v ( π v ) =( π v ) N v ⊗ δ − / P v . It remains to check the three conditions for the equality are equivalent. The onlynontriviality is to show that dim π = 1, assuming that | ω v ( a ) | = δ − / P v ( a ) for some a ∈ A −− P v .We may assume G der = G sc via z -extensions. We decompose P ′ v := P v ∩ G der = Y i ∈ I,w | v P v,i,w according as ( G der ) v = Y i ∈ I,w | v ( H i ) w , where w runs over places of F i above v . Similarly A P ′ v = Q i,w A P v,i,w . Assumption ( Q -nb( P v ))tells us that for every i , there exists w | v such that P v,i,w is a proper parabolic subgroup of ( H i ) w .In particular, G i,v is isotropic for every i . So the assumptions of Lemma 2.5.1 are satisfied, andwe adopt the setup and notation from there. By the lemma, it suffices to show that for every i ,there exists a place w of F i above v such that π ′ i,w = . In fact we only need to find w | v suchthat dim π ′ i,w < ∞ by Lemma 2.1.5 and Corollary 2.3.3. (The corollary implies that the only1-dimensional representation of a simply connected isotropic group is the trivial one.)The central isogeny Z × G der → G induces a map A G v × A P ′ v → A P v , which has finite kerneland cokernel on the level of F v -points. Replacing a with a finite power, we may assume that a is the image of ( a , a ′ ) ∈ A G v ( F v ) × A P ′ v ( F v ), so that | ω ( a ) | = | ω ( a ′ ) | . (The central character of π ′ v is unitary on A G v ( F v ), so | ω ( a ) | = 1.) Write a ′ = ( a i,w ) i,w and ω v | A P ′ v ( F v ) = ( ω v,i,w ) i,w accordingto the decomposition of P ′ v above. We have | ω v,i,w ( a i,w ) | ≤ δ − / P v,i,w ( a i ) by Proposition 2.1.3, while Q i,w | ω v,i,w ( a i,w ) | = Q i,w δ − / P v,i,w ( a i,w ) from our running assumption. Therefore | ω v,i,w ( a i,w ) | = δ − / P v,i,w ( a i,w ) , ∀ i ∈ I. Since J P ′ v ( π ′ v ) ⊂ J P v ( π v ), we see that ω v | A P ′ v ( F v ) ∈ Exp( J P ′ v ( π ′ v )). Thus we have ω v,i,w ∈ Exp( J P v,i,w ( π ′ v,i,w )) . Finally for each i , we apply the equality criterion of Proposition 2.1.3 at a place w where P v,i,w isproper in ( H i ) w . Thereby we deduce that dim π ′ i,w < ∞ as desired. (cid:3) Let ξ : G F ∼ → G ∗ F be an inner twisting, with G ∗ a connected reductive group over F . Lemma 2.5.3.
There is a canonical bijection between one-dimensional automorphic representationsof G ( A F ) and those of G ∗ ( A F ) , compatible with the bijection of Corollary 2.3.3 at every place of F .Proof. Define G ( A F ) ♭ := cok( G sc ( A F ) ̺ → G ( A F )). Similarly we have G ∗ ( A F ) ♭ , G ( F ) ♭ , and G ∗ ( F ) ♭ .The arguments of § z -extensions are easily adapted to show that G ( A F ) ♭ is an abelian groupand that there exists a canonical isomorphism G ( A F ) ♭ ≃ G ∗ ( A F ) ♭ compatible with the isomorphismof Lemma 2.3.1 at every place of v and that the above isomorphism carries G ( F ) ♭ onto G ∗ ( F ) ♭ .With this input, the proof is analogous to that of Corollary 2.3.3. In fact we can assume that G sc = G der again by taking a z -extension, and it suffices to show that the inclusion G der ( F ) G ( A F ) der ⊂ G der ( A F ) is an equality so that every one-dimensional automorphic representations of G ( A F ) fac-tors through G ( A F ) ♭ (and likewise for G ∗ ). Since G ( A F ) der contains G ( F v ) der = G der ( F v ) whenever G is quasi-split over F v (Lemma 2.3.2), the desired equality follows from the strong approximationfor G der . (cid:3) To state the next lemma, define a (global) central character datum to be a pair ( X , χ ) as follows,where Q ′ v means the restricted product over all places of F . • X = Q ′ v X v is a closed subgroup of Z ( A F ) such that Z ( F ) X is closed in Z ( A F ), and • χ = Q v χ v : X ∩ Z ( F ) \ X → C × , with χ v : X v → C × a continuous character. (It is implicitthat for each x = ( x v ) ∈ X , we have χ v ( x v ) = 1 for almost all v , so that χ is well definedon X .) .) Lemma 2.5.4.
Let ( X , χ ) be a central character datum for G . Let v be a finite place of F , and g v ∈ G ( F v ) such that the image of g v in G ( F v ) ab is not contained in the image of X v . Then thereexists a one-dimensional automorphic representation π of G ( A F ) with π | X = χ such that π v ( g v ) = 1 .Proof. Replacing G with a z -extension and ( X , χ ) with its pullback to the z -extension, we mayassume that G der = G sc . Then we may replace G with G ab as ( X , χ ) factors through a centralcharacter datum for G ab .Thus we assume that G = T is a torus. By assumption g v ∈ T ( F v ) lies outside X v and thus g v / ∈ T ( F ) X in T ( A F ), in which g v is an element via the obvious embedding T ( F v ) ֒ → T ( A F ). Thusthe proof is complete by the fact (part of Pontryagin duality) that, for every non-identity element x in a locally compact Hausdorff abelian group X , there exists a unitary character of X whose valueis nontrivial at x . (Take X = T ( A F ) /T ( F ) X → C × and x = g v .) (cid:3) OF IGUSA VARIETIES 19
Endoscopy with fixed central character.
Let F be a local or global field of characteristic 0. Let G be a connected reductive group over F with an inner twisting G F ∼ → G ∗ F with G ∗ quasi-split over F . Let E ( G ) (resp. E ell ( G )) denote a set of representatives for isomorphism classes of endoscopic(resp. elliptic endoscopic) data for G as defined in [LS87, KS99]. A member of E ( G ) is representedby a quadruple e = ( G e , G e , s e , η e ) consisting of a quasi-split group G e , a split extension G e of W F by b G e , s e ∈ Z G e , and η e : G e ֒ → L G satisfying the conditions detailed in loc. cit. In particular e ∗ := ( G ∗ , L G ∗ , , id) ∈ E ell ( G ). Write E < ell ( G ) := E ell ( G ) \{ e ∗ } .From now on, let e ∈ E ( G ). Set ι ( G, G e ) := τ ( G ) τ ( G e ) − ζ ( e ) − ∈ Q . Throughout § z -extensions in thenext subsection. (The assumption is known to be true if e = e ∗ , when it is evident, or if G der issimply connected, by [Lan79, Prop. 1].) • (assumption) G e = L G e .For now we restrict to the case when F is local. Let e be as above. Consider a local centralcharacter datum ( X , χ ) for G as in § X e ⊂ Z G e ( F ) denote the image of X under the canonicalembedding Z G ֒ → Z G e . Thus we can identify X = X e . We say a semisimple element γ e ∈ G e ( F )is strongly G -regular if γ e corresponds to (the G ( F )-conjugacy class of) an element of G ( F ) sr viathe correspondence between semisimple conjugacy classes in G e ( F ) and those in G ( F ) [LS87, 1.3].Write G e ( F ) G -sr ⊂ G e ( F ) for the subset of strongly G -regular elements.Thanks to the proof of the transfer conjecture and the fundamental lemma [Wal06,CL10,Ngˆo10],we know that each f ∈ H ( G ( F )) admits a transfer f e ∈ H ( G e ( F )) whose stable orbital integralson strongly G -regular semisimple elements are determined by the following formula, where thesum runs over strongly regular G ( F )-conjugacy classes, and ∆( · , · ) denotes the transfer factor asin [LS87] (see the remark below on normalization). SO γ e ( f e ) = X γ ∈ G ( F ) sr / ∼ ∆( γ e , γ ) O γ ( f ) , γ e ∈ G e ( F ) G -sr . (2.6.1)The assignment of f e to f is not unique on the level of Hecke algebras, but (2.6.1) determines awell-defined map LS e : I ( G ) → S ( G e ).The transfer satisfies an equivariance property. For each z ∈ Z G ( F ) ⊂ Z G e ( F ), define thetranslates f z , f e z of f, f e by f z ( g ) = f ( zg ) and f e z ( h ) = f e ( zh ). The equivariance of transfer factorsunder translation by central elements (see [LS87, Lem. 4.4.A]) implies that f e z is a transfer of λ e ( z ) f z for a smooth character λ e : Z G ( F ) → C × . The character λ e is independent of f e and f , and itsrestriction λ e | Z G ( F ) can be described as follows. Consider the composite map W F → L G e η e → L G → L Z G , (2.6.2)where the last map is dual to the embedding Z G ֒ → G . Then λ e | Z G ( F ) is the character of Z G ( F )corresponding to the composite map above. See [KSZ, § χ e : X e → C × by the relation χ e ( z ) = λ e ( z ) − χ ( z ) , z ∈ X = X e . (2.6.3)In light of the equivariance property above, the transfer map LS e : I ( G ) → S ( G e ) descends toLS e : I ( G, χ − ) → S ( G e , χ e , − ) (2.6.4)via averaging, still denoted by LS e for simplicity. The identity (2.6.1) still holds if f e = LS e ( f )under (2.6.4). In the special case of e = e ∗ (so that χ e = χ ), we write f ∗ ∈ H ( G ∗ ( F ) , χ − ) for atransfer of f ∈ H ( G ( F ) , χ − ). If X = { } then f ∗ here coincides with the one in § e ( G ) in (2.3.2) plays the role of transfer factor. The fundamental lemma tells us the following. Assume that G and e are unramified; the lattermeans that G e is an unramified group and that the L -morphism η e is inflated from a morphismof L -groups with respect to an unramified extension of F . We also assume that χ is unramified,i.e., χ is trivial on X ∩ K for some (thus every) hyperspecial subgroup K of G ( F ). We normalizethe transfer factors canonically as in [LS87] (which is possible as G is quasi-split). Then LS e canbe realized by an algebra morphism on the unramified Hecke algebras ξ e , ∗ : H ur ( G ( F ) , χ − ) → H ur ( G e ( F ) , χ e , − ) . We turn to the case of global field F . Recall that Z is the center of G . Let ( X , χ ) be a globalcentral character datum ( § X e = Q v X e v to be the image of X under the canonical embedding Z G ( A F ) ֒ → Z G e ( A F ). We have χ e := Q v χ e v : X e → C × , where χ e v was given by the local consideration above, so that functions in H ( G ( A F ) , χ − ) transfer tothose in H ( G e ( A F ) , ( χ e ) − ). Denote by λ e = Q v λ e v : Z G ( F ) \ Z G ( A F ) → C × the character with λ e v as in the local context above. (The Z G ( F )-invariance of λ e follows from the equivariance oftransfer factors [LS87, Lem. 4.4.A] and the product formula [LS87, Cor. 6.4.B].) The restriction of λ to Z G ( A F ) corresponds to the composite map (2.6.2) (with F now global). There is an equality χ e = λ e , − χ as characters on X = X e as in (2.6.3) since this holds at every place of F . In particular χ e is trivial on Z G ( F ) ∩ X e , and ( X e , χ e ) is a central character datum for G e . Remark . The local transfer factors are well defined only up to a nonzero scalar (there is nocanonical choice unless G is quasi-split or G e = G ∗ , without further rigidification), so we alwayschoose a normalization implicitly, for instance throughout §
3. Observe that scaling the transferfactor results in scaling the transfer map (2.6.4). However the Langlands–Shelstad product formula[LS87, § is the canonical global transfer factor. We willalways make such a consistent choice across all places without further comments. This will notintroduce ambiguity in our main argument as it takes place in the global context.It simplifies some later arguments if e is chosen to enjoy a boundedness property. We say that asubgroup of L G = b G ⋊ W F is bounded if its projection to b G ⋊ Gal(
E/F ) is contained in a compactsubgroup for some (thus every) finite Galois extension
E/F containing the splitting field of G . Lemma 2.6.2.
In either local or global case, we can choose the representative e = ( G e , G e , s e , η e ) inits isomorphism class to satisfy the following condition: η e ( W F ) is a bounded subgroup of L G . (Werestrict η e to W F via the splitting W F → G e built into the data.)Proof. Since η e | b G e will be fixed throughout, we use it to identify b G e with a subgroup of b G . We takethe convention that all cocycles/cohomology below are continuous cocycles/cohomology.It suffices to show that there exists an L -morphism η e : G e → L G extending η e | b G e such that η e ( W F ) is bounded. Indeed, given such an η e , write η e ( w ) = a ( w ) ⋊ w and η e ( w ) = a ( w ) ⋊ w for each w ∈ W F . Since η e ( wgw − ) = η e ( wgw − ) for every g ∈ b G e , one deduces that c ( w ) := a ( w ) − a ( w )centralizes η e ( b G e ), thus c ( w ) is a 1-cocycle representing a class in H ( W F , Z ( b G e )). By defining η e by g ⋊ w ∈ G e c ( w ) η e ( g ⋊ w ), we see that e = ( G e , G e , s e , η e ) belongs to the isomorphism classof e and satisfies the boundedness condition by hypothesis.To prove the existence of η e as above, we assume for the moment that G der = G sc and that G e = L G e . There may be a more direct proof, but we will show this by slightly refining the proofof [Lan79, Prop. 1], where Langlands shows that η e | b G e extends to an L -morphism η e : L G e → L G under the hypothesis but without guaranteeing boundedness of image. To construct η e (denoted ξ therein), Langlands reduces to the elliptic endoscopic case, chooses a sufficiently large finiteextension K/F , and then constructs ξ ′ : W K/F → L G such that η e ( g ⋊ w ) := η e ( g ) ξ ′ ( w ) gives thedesired L -morphism. (From here on, in the current proof, we follow Langlands to use the Weilgroup W K/F to form the L -group, i.e., L G = b G ⋊ W K/F .) OF IGUSA VARIETIES 21
It is enough to arrange that ξ ′ has bounded image in Langlands’s construction. Write b N forthe normalizer of b T (which is L T in loc. cit. ) in b G . Let b N c (resp. Z ( b G e ) c ) denote the maximalcompact subgroup of b N (resp. Z ( b G e )). The starting point is a choice of ξ ′ : W K/F → L G as aset-theoretic map satisfying the second displayed formula on p.709 therein. Such a ξ ′ can be chosenusing the Langlands–Shelstad representatives of each Weyl group element ω , denoted by n ( ω ) ∈ b N in [LS87, § σ -action ω T/G ( σ ) on L T and the action ω ( σ ) differ by theWeyl action ω ( σ ) in his notation. See the seventh displayed formula on p.703.) In fact n ( ω ) ∈ b N c since it is a product of finite-order elements in b N . Thereby ξ ′ has image in b N c ⋊ W K/F (thusbounded). It follows that the 2-cocycle of W K/F given by a w ,w = ξ ′ ( w ) ξ ′ ( w ) ξ ′ ( w w ) − has values in Z ( b G e ) c (not just Z ( b G e ) as in [Lan79, p.709]). We need to verify the claim that the2-cocycle is trivial in H ( W K/F , Z ( b G e ) c ); then ξ ′ can be made a homomorphism after multiplyinga Z ( b G e ) c -valued 1-cocycle, without affecting boundedness of image. Moreover, thanks to Lemma 2therein (stated for Z ( b G e ) but also applicable for Z ( b G e ) c since both groups have the same group ofconnected components), we may assume that a w ,w ∈ ( Z ( b G e ) c ) . Then the claim follows from avariant of Lemma 4 therein, with S replaced by the maximal compact subtorus in the statementand proof. (In particular, the map (1) on p.719 is still surjective if S and S are replaced withtheir maximal compact subtori, by considering unitary characters.) This completes the proof thatthere exists a choice of ξ ′ under which the Weil group has bounded image, thus proving the lemmain the case of simply connected derived subgroup.Going back to the general case, let us remove the preceding hypothesis by replacing G with a z -extension 1 → Z → G → G →
1, which gives rise to a Γ F -equivariant exact sequence1 → b G → b G → b Z → . (2.6.5)From e , we obtain e = ( G e , G e , s e , η e ) with η e ( G e ) = η e ( G e ) b G and G e ≃ L G e as split extensionsof W F by b G . Our preceding proof tells us that there exists a 1-cocycle c : W F → Z ( b G e ) such thatthe image of W F under c · η e is bounded in L G . It is enough to prove the claim that there exists achoice of c such that ( c · η e )( W F ) is contained and bounded in L G (viewed as a subgroup of L G ).Indeed, we can then take η e to be the restriction of η e via G e ⊂ G e .To prove the claim, we may work with respect to a sufficiently finite extension K/F as above. Letus designate maximal compact subtori by the subscript c as before. Fix a 1-cocycle c : W K/F → Z ( b G ) such that ( c · η e )( W K/F ) is a bounded subgroup of L G . Composing ( c · η e ) | W K/F withthe natural projection L G → L Z , we obtain an L -morphism ǫ : W K/F → L Z with boundedimage, which is also viewed as a 1-cocycle valued in b Z ,c . From the surjection Z ( b G ) c ։ b Z ,c , weobtain a surjection H ( W K/F , Z ( b G ) c ) → H ( W K/F , b Z ,c ) in the same way as on p.719 of [Lan79](as we remarked above). Therefore, after multiplying a Z ( b G ) c -valued cocycle whose image in H ( W K/F , b Z ,c ) is the same as ǫ − , we may assume that ( c · η e ) | W K/F maps to the trivial cocycle W K/F → b Z ,c ) via the projection L G → L Z . This means that ( c · η e ) | W K/F has image containedin L G in light of (2.6.5), and the image remains bounded. This completes the proof the claim, andwe are done. (cid:3) Endoscopy and z -extensions. Here we explain a general endoscopic transfer with fixed centralcharacter by removing the assumption that G e = L G e in § z -extensions. For the time being,let the base field F of G be either local or global. Fix a z -extension over F (defined in [Kot82, § → Z → G → G → . Let e = ( G e , G e , s e , η e ) ∈ E < ell ( G ). As explained in [LS87, § z -extension1 → Z → G e → G e → , and e can be promoted to an endoscopic datum e = ( G e , L G e , s e , η e ) for G such that η e : L G e ֒ → L G extends η e : G e ֒ → L G . Moreover, changing e and e in their isomorphism classes if necessary, wemay ensure that η e ( W F ) and η e ( W F ) are bounded subgroups in L G and L G , respectively. Indeed,this is done in the course of proof of Lemma 2.6.2 in the general case. Write X (resp. X e ) for thepreimage of X in G (resp. G e ), and χ : X → C × for the character pulled back from χ .To describe endoscopic transfers, it is enough to work locally, so let F be a local field. Applying § G and e in place of G and e , we obtain an identification X e = X under the canonicalembedding Z G ֒ → Z G e as well as characters λ e : Z G ( F ) → C × and χ e : X e = X → C × suchthat χ e = λ e , − χ as characters on X e = X . Again λ e | Z G ( F ) corresponds to the parameter (2.6.2)(with G e , G replacing G e , G ). We also have a transferLS e : I ( G, χ − ) = I ( G , χ − ) (2.6.4) → S ( G e , χ e , − ) , where the equality is induced by G ( F ) ։ G ( F ).2.8. The trace formula with fixed central character.
In this subsection, G is a connected reductivegroup over Q . Let A G denote the maximal Q -split torus in Z G , and A G R denote the maximal R -splittorus in Z G R . Put A G, ∞ := A G ( R ) , A G R , ∞ := A G R ( R ) . Let χ : A G, ∞ → C × denote a continuous character. By L ,χ ( G ( Q ) \ G ( A )) we mean the dis-crete spectrum in the space of square-integrable functions (modulo A G, ∞ ) on G ( Q ) \ G ( A ) whichtransforms under A G, ∞ by χ .Let ( X = Q v X v , χ = Q v χ v ) be a central character datum as in § A G, ∞ ⊂ X ∞ . We can define L ,χ ( G ( Q ) \ G ( A )) in the same way as L ,χ ( G ( Q ) \ G ( A )). Let A disc ,χ ( G ) standfor the set of isomorphism classes of irreducible G ( A )-subrepresentations in L ,χ ( G ( Q ) \ G ( A )).The multiplicity of π ∈ A disc ,χ ( G ) in L ,χ ( G ( Q ) \ G ( A )) is denoted m ( π ).Define H ( G ( A ) , χ − ) := ⊗ ′ v H ( G ( Q v ) , χ − v ) as a restricted tensor product. Each f ∈ H ( G ( A ) , χ − )defines a trace class operator, yielding the discrete part of the trace formula: T G disc ,χ ( f ) := Tr (cid:0) f | L ,χ ( G ( Q ) \ G ( A )) (cid:1) = X π ∈A disc ,χ ( G ) m ( π )Tr ( f | π ) . (2.8.1)Fix a minimal Q -rational Levi subgroup M ⊂ G . Write L for the set of Q -rational Levi sub-groups of G containing M . Define the subset L cusp ⊂ L of relatively cuspidal Levi subgroups; bydefinition, M ∈ L belongs to L cusp exactly when the natural map A M, ∞ /A G, ∞ → A M R , ∞ /A G R , ∞ isan isomorphism. Let M ∈ L and γ ∈ M ( Q ) be a semisimple element. Write M γ for the centralizerof γ in M , and I Mγ := ( M γ ) for the identity component. Write ι M ( γ ) ∈ Z ≥ for the number ofconnected components of M γ containing Q -points. Write | Ω M | for the order of the Weyl group of M . For γ ∈ M ( Q ), let Stab M X ( γ ) denote the set of x ∈ X such that γ and xγ are M ( Q )-conjugate.Note that Stab M X ( γ ) is necessarily finite, cf. [KSZ, § M = G , we often omit M from thenotation, e.g., I γ = I Gγ and ι ( γ ) = ι G ( γ ).Fix Tamagawa measures on M ( A ) and I Mγ ( A ) for M ∈ L cusp and fix their decomposition intoHaar measures on M ( A ∞ ) and M ( R ) (resp. I Mγ ( A ∞ ) and I Mγ ( R )). This determines a measureon the quotient I Mγ ( A ) \ M ( A ), which is used to define the ad`elic orbital integral at γ in M , andsimilarly over finite-ad`elic groups. We also fix Haar measures on X and X ∞ . We equip I Mγ ( Q ) and OF IGUSA VARIETIES 23 X Q := X ∩ Z ( Q ) with the counting measures and A G ( R ) with the multiplicative Lebesgue measure.Thereby we have quotient measures on I Mγ ( Q ) \ I Mγ ( A ) / X , X Q \ X /A G ( R ) , and X ∞ /A G ( R ) .We define the elliptic part of the trace formula as T G ell ,χ ( f ) := X γ ∈ Γ ell , X ( G ) | Stab G X ( γ ) | − ι ( γ ) − vol( I γ ( Q ) \ I γ ( A ) / X ) O γ ( f ) , f ∈ H ( G ( A ) , χ − ) , (2.8.2)where Γ ell , X ( G ) is the set of X -orbits of elliptic conjugacy classes of G .Now we assume that G R contains an elliptic maximal torus. Let ξ be an irreducible algebraicrepresentation of G C , and ζ : G ( R ) → C × a continuous character. Let M ∈ L cusp and T ∞ an R -elliptic torus in M . Arthur introduced the function Φ M ( γ, ξ ) in γ ∈ T ∞ ( R ) in [Art89, (4.4),Lem. 4.2]. (We will see a concrete description of Φ M ( γ, ξ ) later.) Now let γ ∈ M ( Q ) and supposethat γ is elliptic in M ( R ). Let I M, cpt γ denote a compact-mod-center inner form of ( I Mγ ) R . Wechoose a Haar measure on I M, cpt γ ( R ) compatibly with that on I Mγ ( R ). Write q ( I γ ) ∈ Z ≥ for thereal dimension of the symmetric space associated with the adjoint group of ( I Mγ ) R . Following [Art89,(6.3)], define χ ( I Mγ ) := ( − q ( I γ ) τ ( I Mγ )vol( A I Mγ , ∞ \ I M, cpt γ ( R )) − d ( I Mγ ) . (2.8.3)For f ∞ ∈ H ( G ( A ∞ ) , ( χ ∞ ) − ), let f ∞ M ∈ H ( M ( A ∞ ) , ( χ ∞ ) − ) denote the constant term, cf. § § X is imposed below because it is also in [Dal19]. It is aharmless condition that is satisfied in our setup, but we expect it to be superfluous. Proposition 2.8.1.
Assume that X = Y ( A ) A G, ∞ for a central torus Y ⊂ Z G over Q . Let ξ, ζ be asabove. Then for each f ∞ ∈ H ( G ( A ∞ ) , ( χ ∞ ) − ) , T G disc ,χ ( f ξ,ζ f ∞ ) = 1vol( X Q \ X /A G, ∞ X M ∈L cusp ( − dim( A M /A G ) | Ω M || Ω G | X γ χ ( I Mγ ) ζ ( γ )Φ M ( γ, ξ ) O Mγ ( f ∞ M ) ι M ( γ ) · | Stab M X ( γ ) | , where the second sum runs over X -orbits on the set of R -elliptic conjugacy classes of M ( Q ) .Proof. This is [Dal19, Cor. 6.5.1]. (cid:3)
The stable trace formula.
Let H be a quasi-split group over Q . Let ( X H , χ H ) be a centralcharacter datum for H . Write Σ ell ,χ H ( H ) denote the set of stable elliptic conjugacy classes in H ( Q )up to X H -equivalence, where two stable conjugacy classes are considered X H -equivalent if thereexists representative γ H and γ ′ H such that γ ′ H = xγ H for some x ∈ X H . Following [KSZ, § ST H ell ,χ H ( h ) := τ X H ( H ) X γ H ∈ Σ ell ,χH ( H ) | Stab X H ( γ H ) | − SO H ( A ) γ H ( h ) , h ∈ H ( H ( A ) , χ − H ) . Consider a central character datum ( X , χ ) for G as well as f = ⊗ ′ v f v ∈ H ( G ( A ) , χ − ). For each e ∈ E < ell ( G ), we have e and a central character datum ( X e , χ e ) (whose v -components are given asin the preceding subsection). Write f e ,v ∈ H ( G e ( A ) , ( χ e ) − ) for a transfer of f v at each v . Put f e := ⊗ ′ v f e ,v . For e = e ∗ , we transfer f to f ∗ ∈ H ( G ∗ ( A ) , χ − ) as in § Proposition 2.9.1.
Let f = ⊗ ′ v f v ∈ H ( G ( A ) , χ − ) . Assume that there exists a finite place q suchthat O g ( f q ) = 0 for every non-regular semisimple g ∈ G ( Q q ) . If f ∗ and f e are associated with f asabove, then T G ell ,χ ( f ) = ST G ∗ ell ,χ ( f ∗ ) + X e ∈E < ell ( G ) ι ( G, G e ) ST G e ( A )ell ,χ e ( f e ) . Proof.
By hypothesis, the stable orbital integral of f e ,q vanishes outside G -regular semisimple con-jugacy classes. Suppose that the central character datum is trivial. In this case, the stabilization ofregular elliptic terms is due to Langlands [Lan83], cf. [Kot86, Thm. 9.6] (which assumes G der = G sc but stabilizes all elliptic terms) and [KS99, § G -regular terms in thetwisted trace formula). The latter can be specialized to the untwisted case, but extended to all G -regular terms (or all elliptic terms, in fact) following [Lab99].For general central character data, the argument is essentially the same by using the Langlands–Shelstad transfer with fixed central character as in § §
11] in the setting of Shimura varieties (without the regularityassumption). The method carries over to the current case (if we perform the usual transfer at p and ∞ rather than what is done in loc. cit. specific to Shimura varieties). (cid:3) The following finiteness result is going to tell us that the sum in Theorem 7.5.1 (and a similarsum in Theorem 7.1.1 below) is finite for each choice of φ ∞ ,p . Lemma 2.9.2.
The following are true.(1) Let v be a rational prime such that G Q v and χ v are unramified. Let f v ∈ H ur ( G ( Q v ) , χ − v ) .Then f v transfers to the zero function on G e ( Q v ) for each e = ( G e , G e , s e , η e ) ∈ E < ell ( G ) if G e is ramified over Q v .(2) Let S be a finite set of rational primes. The set of e ∈ E < ell ( G ) such that G e Q v is unramifiedat every rational prime v / ∈ S is finite.Proof. The first point follows from [Kot86, Prop. 7.5]. The second point is well known; see [Lan83,Lem. 8.12]. (cid:3) Jacquet modules, regular functions, and endoscopy
Throughout this section, let F be a finite extension of Q p with a uniformizer ̟ and residuefield cardinality q . The valuation on F is normalized such that | ̟ | = q − . Let G be a connectedreductive group over F . We study how certain maps of invariant or stable distributions between G and its Levi subgroups interact with Jacquet modules and endoscopy. The material here is largelybased on [Shi10, Xu17].3.1. ν -ascent and Jacquet modules. Let ν : G m → G be a cocharacter defined over F . Let M ν denote the centralizer of ν in G , which is an F -rational Levi subgroup. The maximal F -split torusin the center of M ν is denoted by A M ν .Write P ν (resp. P op ν ) for the F -rational parabolic subgroup of G which contains M ν as a Levicomponent and such that h α, ν i < h α, ν i >
0) for every root α of A M ν in P ν (resp. P op ν ). Theset of α as such is denoted by Φ + ( P ν ) (resp. Φ + ( P op ν )). Let N ν , N op ν denote the unipotent radicalof P ν , P op ν . For every α ∈ Φ + ( P op ν ), we have | α ( ν ( ̟ )) | = q −h α,ν i <
1. Therefore ν ( ̟ ) ∈ A −− P op ν . Thefollowing definition is a rephrase of [Shi10, Def. 3.1]. Definition 3.1.1.
An element γ ∈ M ν ( F ) is considered acceptable (with respect to ν ) if the actionof Ad( γ ) on (Lie N op ν ) F is contracting, i.e., all its eigenvalues λ ∈ F have the property that | λ | < a ∈ A M ν ( F ) is acceptable if and only if a ∈ A −− P op ν . Evidently the subset ofacceptable elements is nonempty, open, and stable under M ν ( F )-conjugacy. Define H acc ( M ν ) ⊂H ( M ν ) as the subspace of functions supported on acceptable elements. We also write H ν -acc ( M ν )to emphasize the dependence on ν . As in § F for simplicity. Lemma 3.1.2.
Let φ ∈ H acc ( M ν ) . There exists f ∈ H ( G ) with the following properties.(1) For every g ∈ G ( F ) ss , O Gg ( f ) = δ P ν ( m ) − / O M ν m ( φ ) OF IGUSA VARIETIES 25 if there exists an acceptable m ∈ M ν ( F ) which is conjugate to g in G ( F ) (in which case m is unique up to M ν ( F ) -conjugacy, and the Haar measures are chosen compatibly on theconnected centralizers of m and g ), and O Gg ( f ) = 0 otherwise.(2) Tr ( f | π ) = Tr (cid:0) φ | J P op ν ( π ) (cid:1) for π ∈ Irr( G ( F )) .Proof. This is [Shi10, Lem. 3.9] except that we corrected typos in the statement. The same proofstill works with two remarks. Firstly, we removed the assumption in loc. cit. that orbital integralsof φ vanish on semisimple elements with disconnected centralizers. This is possible by reducing tothe case of G with simply connected derived subgroup (then M ν, der is also simply connected byLemma 1.7.1) so that the centralizers of semisimple elements are connected in both M ν and G .Secondly, the mistake in loc. cit. occurs in line 1, p.806, where it should read φ := φ · δ − / P ν . (cid:3) Corollary 3.1.3.
Let φ and f be as in Lemma 3.1.2. For every g ∈ G ( F ) ss , SO Gg ( f ) = δ P ν ( m ) − / SO M ν m ( φ ) if there exists an acceptable m ∈ M ν ( F ) which is conjugate to g in G ( F ) , and SO Gg ( f ) = 0 otherwise.Proof. The proof is immediate from the preceding lemma, using the following fact [Shi10, Lem. 3.5]:if m ∈ M ν ( F ) is acceptable then the natural map from the set of M ν ( F )-conjugacy classes in thestable conjugacy class of m in M ν to the set of G ( F )-conjugacy classes in the stable conjugacy classof m in G is a bijection. (cid:3) Definition 3.1.4.
In the setup of Lemma 3.1.2, we say that f is a ν -ascent of φ .Recall the definition of I ( · ) and the trace Paley–Wiener theorem from § J ν : I ( M ν ) → I ( G ) , F 7→ (cid:0) π
7→ F ( J P op ν ( π )) (cid:1) . (3.1.1)Write I acc ( M ν ) for the image of H acc ( M ν ) in I ( M ν ). Then Lemma 3.1.2 means that, when φ ∈I acc ( M ν ), a ν -ascent of φ is well defined as an element of I ( G ), which is nothing but J ν ( φ ).The lemma yields extra information on orbital integrals. Xu [Xu17, Prop. C.4] shows that (3.1.1)induces a similar map for the stable analogues, which we denote by the same symbol: J ν : S ( M ν ) → S ( G ) . (3.1.2)Write X ∗ F ( G ) for the group of F -rational characters of G . Put X ∗ F ( G ) Q := X ∗ F ( G ) ⊗ Z Q and a G := Hom( X ∗ F ( G ) Q , R ). We have the map H G : G ( F ) → a G , g ( χ log | χ ( g ) | ) . It is easy to see that H G is invariant under G ( F )-conjugacy, i.e., if g , g ∈ G ( F ) are conjugate in G ( F ) then H G ( g ) = H G ( g ). Indeed, if g , g become conjugate in G ( F ′ ) for a finite extension F ′ /F , then since the map H G is functorial with respect to G ֒ → G ′ := Res F ′ /F G , and a G → a Res F ′ /F G is injective, it boils down to checking that H G ′ ( g ) = H G ′ ( g ), which is obvious.For f ∈ H ( G ), define the following subsets of a G :supp a G ( f ) := { H G ( x ) : x ∈ G ( F ) ss s.t. f ( x ) = 0 } , supp O a G ( f ) := { H G ( x ) : x ∈ G ( F ) ss s.t. O x ( f ) = 0 } , (3.1.3)supp SO a G ( f ) := { H G ( x ) : x ∈ G ( F ) ss s.t. SO x ( f ) = 0 } . Evidently supp SO a G ( f ) ⊂ supp O a G ( f ) ⊂ supp a G ( f ) . Writing P ( ∗ ) := collection of subsets of ∗ , we obtain a map supp a G (resp. supp O a G , supp SO a G ) from H ( G ) (resp. I ( G ), S ( G )) to P ( a G ). We define analogous objects for M ν in place of G . The injective restriction map X ∗ F ( G ) Q → X ∗ F ( M ν ) Q induces a canonical surjection pr G : a M ν → a G . (3.1.4)In fact we write a P ν := a M ν and identify X ∗ ( A M ν ) R ≃ a P ν induced by µ ∈ X ∗ ( A M ν ) ( χ
7→ h χ, µ i ).Via this identification, it is an easy exercise to describe pr G as the average map along Weyl orbits.Namely if T is a maximal torus of M ν (thus also of G ) over F , and if the Weyl group is takenrelative to T , thenpr G ( µ ) = | Ω G | − X ω ∈ Ω G ω ( µ ) = | Ω G | − X ω ∈ Ω G ω ( µ ) , µ ∈ X ∗ ( A M ν ) R . (3.1.5) Lemma 3.1.5.
The sets supp a G ( f ) , supp O a G ( f ) , and supp SO a G ( f ) remain unchanged if we restrict x inthe definition (3.1.3) to a subset D ⊂ G ( F ) reg that is open dense in G ( F ) .Proof. Since the map H G is continuous with discrete image, for each y in supp a G ( f ) (resp. supp O a G ( f ),supp SO a G ( f )), the preimage ( H G ) − ( y ) is open and closed. If y ∈ supp a G ( f ) then supp( f ) ∩ ( H G ) − ( y )is nonempty open in G ( F ) thus intersects D . This proves the assertion for supp a G ( f ).Next let y ∈ supp O a G ( f ). Then ( H G ) − ( y ) ∩ D is open dense in ( H G ) − ( y ). If O x ( f ) = 0 forevery x ∈ ( H G ) − ( y ) ∩ D , we claim that O x ( f ) = 0 , x ∈ ( H G ) − ( y ) ∩ G ( F ) ss . If x is regular, this follows from local constancy of O x ( f ) on regular elements. A Shalika germargument then proves O x ( f ) = 0 for non-regular semisimple x . (Compare with the proof of Lemma3.4.5 (1) below.) However, the claim contradicts y ∈ supp O a G ( f ). The lemma for supp O a G ( f ) follows.Finally, the stable analogue is proved likewise. (cid:3) Lemma 3.1.6.
The following diagrams commute. I acc ( M ν ) J ν / / supp O a Mν (cid:15) (cid:15) I ( G ) supp O a G (cid:15) (cid:15) P ( a P ν ) pr G / / P ( a G ) S acc ( M ν ) J ν / / supp SO a Mν (cid:15) (cid:15) S ( G ) supp SO a G (cid:15) (cid:15) P ( a P ν ) pr G / / P ( a G ) Proof.
This follows from Lemma 3.1.2 and Corollary 3.1.3 since, for each m ∈ M ν ( F ), the canonicalmap a M ν → a G sends H M ν ( m ) to H G ( m ). (cid:3) Let k ∈ Z and φ ∈ H ( M ν ). Define φ ( k ) ( l ) := φ ( ν ( ̟ ) − k l ) for l ∈ M ν ( F ) so that φ ( k ) ∈ H ( M ν ).Since ν is central in M ν , this induces a map( · ) ( k ) : I ( M ν ) → I ( M ν ) . (3.1.6) Lemma 3.1.7. If φ ∈ I acc ( M ν ) then φ ( k ) ∈ I acc ( M ν ) for all k ≥ . Given φ ∈ I ( M ν ) , there exists k = k ( φ ) such that φ ( k ) ∈ I acc ( M ν ) for all k ≥ k . The analogue holds true with H in place of I .Moreover, letting f ( k ) denote the ν -ascent of φ ( k ) for k ≥ k , we have supp ⋆ a G (cid:16) f ( k ) (cid:17) = pr G (supp ⋆ a Mν ( φ ( k ) )) = k · H G ( ν ( ̟ )) + pr G (supp ⋆ a Mν ( φ )) , ⋆ ∈ { O, SO } , where pr G : a M ν → a G is the canonical surjection.Proof. The second equality in the displayed formula is obvious, so we check the first equality. ByLemma 3.1.5 it is enough to verify firstly that if O g ( f ( k ) ) = 0 for g ∈ G ( F ) reg then H G ( g ) ∈ pr G (supp O a Mν ( φ ( k ) )), and secondly that if O m ( φ ( k ) ) = 0 for m ∈ M ( F ) reg then pr G ( H M ( m )) ∈ supp O a G ( f ( k ) ). This follows from Lemma 3.1.2 (1) and Lemma 3.1.6. The case of stable orbitalintegrals is analogous. (cid:3) OF IGUSA VARIETIES 27
Let Groth( M ν ( F )) denote the Grothendieck group of admissible representations of M ν ( F ). Lemma 3.1.8.
Let π , π ∈ Groth( M ν ( F )) . Assume that for each φ ∈ I ( M ν ) , there exists aninteger k ( φ ) such that Tr π ( φ ( k ) ) = Tr π ( φ ( k ) ) for all k ≥ k ( φ ) . Then we have π = π in Groth( M ν ( F )) .Proof. This is proved by the argument of [Shi09, p.536]. (cid:3) ν -ascent and constant terms. Fix an F -rational minimal parabolic subgroup P ⊂ P op ν of G with a Levi factor M ⊂ M ν . Let P be another F -rational parabolic subgroup of G containing P ,with a Levi factor M containing M . Henceforth we will often write L := M ν .We have the constant term map (compare with (3.1.1)) C GM : I ( G ) → I ( M ) , F 7→ (cid:0) ( π M
7→ F (n-ind GM ( π M )) (cid:1) , (3.2.1)where n-ind GM : Groth( M ( F )) → Groth( G ( F )) is the normalized parabolic induction (which doesnot change if P is replaced with a different parabolic with Levi factor M ). On the level of functions,when f ∈ H ( G ), we can define f M ∈ H ( M ) by an integral formula (e.g., [Shi11, (3.5)]) so that O Gg ( f ) = 0 if g ∈ G ( F ) reg is not conjugate to any element of M ( F ), and O Gm ( f ) = D G/M ( m ) / O Mm ( f M ) , ∀ G -regular m ∈ M ( F ) , (3.2.2)where D G/M : M ( F ) → R × > denotes the Weyl discriminant of G relative to M . This identity andparts (i) and (ii) of [Shi11, Lem. 3.3] tell us that f f M descends to the map C GM above. (Eventhough G is a general linear group in loc. cit. , everything applies to general reductive groups sincethat lemma is based on the general results of [vD72].)Since n-ind GM induces a map R ( M ) st → R ( G ) st [KV16, Cor. 6.13], the map C GM descends to amap on the stable spaces, still denoted by the same symbol: C GM : S ( G ) → S ( M ) . Define Ω
GM,L := { ω ∈ Ω G : ω ( M ∩ P ) ⊂ P , ω − ( L ∩ P ) ⊂ P } , which is a set of representativesfor Ω L \ Ω G / Ω M . For ω ∈ Ω GM,L , write M ω := M ∩ ω − ( L ), P ω := M ∩ ω − ( P ν ), and L ω := ω ( M ) ∩ L .Note that M ω (resp. L ω ) is an F -rational Levi subgroup of M (resp. L ) and that ω induces anisomorphism M ω ∼ → L ω , which in turn induces ω : I ( M ω ) ∼ → I ( L ω ) by φ ( g φ ( ω − g )). Since ν is central in L , its image lies in L ω . Applying ω − we obtain a cocharacter ν ω := ω − ( ν ) of M ω .Thus we have a chain of maps I ( L ) C LLω −→ I ( L ω ) ω − ≃ I ( M ω ) J νω −→ I ( M ) . Lemma 3.2.1. If φ ∈ I ν -acc ( L ) then C LL ω ( φ ) is contained in I ν -acc ( L ω ) .Proof. The proof of Lemma 3.3.5 below works verbatim: just replace stable orbital integrals therewith ordinary orbital integrals and use the identity (3.2.2) as well as the vanishing statement inthe same sentence. (Since Lemma 3.3.5 is more general, we supply a detailed argument only forthe latter.) (cid:3)
Lemma 3.2.2.
We have the following commutative diagram. I ( L ) J ν / / ⊕ C LLω (cid:15) (cid:15) I ( G ) C GM / / I ( M ) L ω ∈ Ω GM,L I ( L ω ) ⊕ ω − ∼ / / L ω ∈ Ω GM,L I ( M ω ) P ω J νω O O Finally, all this holds true with S ( · ) in place of I ( · ) . Proof.
Let φ ∈ I ( L ). We check that the images of φ in I ( M ) given in the two different ways havethe same trace against every π M ∈ Irr( M ( F )):Tr (cid:0) C LL ω ( J ν ( φ )) | π M (cid:1) = Tr (cid:0) J ν ( φ ) | n-ind GL ( π M ) (cid:1) = Tr (cid:0) φ | J P ν (n-ind GL ( π M )) (cid:1) = X ω ∈ Ω GM,L Tr (cid:0) φ | n-ind LL ω (cid:0) ω ( J P νω ( π M )) (cid:1)(cid:1) = X ω ∈ Ω GM,L Tr (cid:0) J ν ω ( ω − ( C LL ω ( φ ))) | π M (cid:1) , where the second last equality comes from Bernstein–Zelevinsky’s geometric lemma [BZ77, 2.12].The S ( · )-version is immediate from the I ( · )-version proven just now, since each map in the bigdiagram descends to a map between the stable analogues. (cid:3) ν -ascent and endoscopic transfer. In this subsection we assume that G is quasi-split over F .Let e = ( G e , G e , s e , η e ) be an endoscopic datum for G such that G e = L G e . (The last condition willbe removed via z -extensions in § F -pinnings ( B e , T e , {X α e } ) and ( B , T , {X α } ) for b G e and b G , respectively. (These choices are implicit in the discussion of § η e wemay and will assume that η e ( T e ) = T and η e ( B e ) ⊂ B .We have a standard embedding L P ν ֒ → L G and a Levi subgroup L M ν ⊂ L P ν as in [Bor79, 3.3, 3.4].Choose a subtorus S ⊂ T such that Cent( S, L G ) = L M ν . (This is possible by [Bor79, Lem. 3.5].)Following [Xu17, § G ( e , ν ) := { ω ∈ Ω G | Cent( ω ( S ) , L G e ) → W F is surjective } and Ω e ,ν := Ω G e \ Ω G ( e , M ν ) / Ω M ν . For each ω ∈ Ω e ,ν , we obtain an endoscopic datum e ω =( G e ω , L G e ω , s e ω , η e ω ) for L = M ν as follows. (Henceforth we view L G e as a subgroup of L G via η e .) Pick g ∈ b G such that Int( g ) induces ω on S . Then g L P ν g − ∩ L G e is a parabolic sub-group of L G e with Levi subgroup g L M ν g − , so there is a corresponding standard parabolic sub-group P e ν = M e ν N e ν such that the standard embedding L P e ν ֒ → L G e (resp. L M e ν ֒ → L G e ) becomes g L P ν g − ∩ L G e (resp. g L M ν g − ∩ L G e ) after composing with Int( g e ) for some g e ∈ b G e . Then thereis a unique L -embedding η e ω : L M e ν ֒ → L M ν such that Int( g ) ◦ η e ω = η e ◦ Int( g e ). Set G e ω := M e ν , and s e ω := g − sg ∈ c M ν . Then it is a routine exercise to check that ( G e ω , L G e ω , s e ω , η e ω ) is an endoscopicdatum for M ν .There is a canonical embedding A M ν ֒ → A M e ν = A G e ω (just like Z H ֒ → Z in § ν : G m → A M ν , we obtain ν ω : G m → A G e ω . By the above construction, G e ω = M e ν is a Levi subgroup of G e that is the centralizer of ν ω . Inparticular we have a map J ν ω : S ( G e ω ) → S ( G e ) as in (3.1.2). Consider the following commutativediagram W F / / L G e ω η e ω / / (cid:127) _ Int( g e ) (cid:15) (cid:15) L M ν / / (cid:127) _ Int( g ) (cid:15) (cid:15) L Z M ν (cid:15) (cid:15) W F / / L G e η e / / L G / / L Z G , (3.3.1)where the maps out of W F come from canonical splittings for the L -groups, the two horizontal mapson the right are induced by Z M ν ⊂ M ν and Z G ⊂ G , the first two vertical maps correspond to theLevi embeddings (coming from G e ω ⊂ G e and M ν ⊂ G ) followed by Int( g e ) and Int( g ) respectively,and finally the rightmost vertical map is induced by Z G ⊂ Z M ν . The left square in (3.3.1) commutesby Int( g ) ◦ η e ω = η e ◦ Int( g e ) above. The commutativity of the right square is obvious since Int( g )acts trivially on L Z G . Denote by λ e ω : Z M ν ( F ) → C × (resp . λ e : Z G ( F ) → C × ) OF IGUSA VARIETIES 29 the smooth character corresponding to the composite morphism from W F to L Z M ν (resp. L Z G ) inthe first (resp. second) row. The character λ e is the same as in § λ e ω | Z G ( F ) = λ e . The canonical splittings from W F to L G e ω and L G e commute with theLevi embedding L G e ω ֒ → L G e without Int( g e ), but the point is that Int( g e ) on L G e is equivariantwith the trivial action on L Z G via the horizontal maps in (3.3.1). Lemma 3.3.1.
Assume that η e ( W F ) is a bounded subgroup of L G in the sense above Lemma 2.6.2.(This condition can always be ensured by that lemma.) Then λ e ω is a unitary character.Proof. By assumption and commutativity of (3.3.1), η e ω ( W F ) is a bounded subgroup of L M ν , whoseimage in L Z M ν is a bounded subgroup accordingly. Therefore λ e ω is a unitary character via theLanglands correspondence for tori. (cid:3) Proposition 3.3.2.
The following diagram commutes. I ( M ν ) J ν / / ⊕ LS e ,ω % % ❑❑❑❑❑❑❑❑❑❑❑❑ I ( G ) LS e / / S ( G e ) L ω ∈ Ω e ,ν S ( G e ω ) P ω J νω ssssssssssss Let φ ∈ C ∞ c ( M ν ( F )) . If f ( k ) = J ν ( φ ( k ) ) then writing φ ( k ) ω := LS e ,ω ( φ ) ( k ) , we have φ ( k ) ω = λ e ω ( ν ( ̟ )) − k LS e ,ω ( φ ( k ) ) , LS e ( f ( k ) ) = X ω ∈ Ω e ,ν λ e ω ( ν ( ̟ )) k J ν ω (cid:16) φ ( k ) ω (cid:17) . Remark . When e is given by a Levi subgroup M as in § G e = M ), we haveLS e = C GM , LS e ,ω = C LL ω , and the meaning of ν ω is consistent between § § M and M ω . Proof.
The first equality follows from the equivariance property of transfer as discussed in theparagraph containing (2.6.3) (applied to z = ν ( ̟ ) − k , G = M ν , G e = G e ω , and f = φ ). Thecommutative diagram comes from (C.4) in [Xu17] (when θ is trivial). This, together with the firstequality, implies the last equality. (cid:3) Corollary 3.3.4.
Let φ ( k ) , φ ( k ) ω , and f ( k ) be as in Proposition 3.3.2. Then supp SO a G e (cid:16) J ν ω ( φ ( k ) ω ) (cid:17) = k · H G e ( ν ω ( ̟ )) + pr G e (cid:16) supp SO a Lω (LS e ,ω ( φ )) (cid:17) , ω ∈ Ω e ,ν , where pr G e : a G e ω ։ a G e is the natural projection.Proof. By Lemma 3.1.7 and Proposition 3.3.2,supp SO a G e (cid:16) J ν ω ( φ ( k ) ω ) (cid:17) = pr G e (cid:16) supp SO a G e ω ( φ ( k ) ω ) (cid:17) = pr G e (cid:16) supp SO a G e ω (LS e ,ω ( φ ( k ) )) . (cid:17) = pr G e (cid:16) k · H G e ω ( ν ω ( ̟ )) + supp SO a G e ω (LS e ,ω ( φ )) (cid:17) . We finish by observing that pr G e ( H G e ω ( ν ω ( ̟ ))) = H G e ( ν ω ( ̟ )). (cid:3) It is useful to know preservation of acceptability in the setting of Proposition 3.3.2 as this willallow an inductive argument in the proof of Corollary 4.2.3 below.
Lemma 3.3.5. If φ ∈ I acc ( M ν ) then φ ω := LS e ,ω ( φ ) is contained in S acc ( G e ω ) . Proof.
Suppose that SO γ ω ( φ ω ) = 0 for some strongly M ν -regular element γ ω ∈ G e ω ( F ). We needto check that γ ω is ν ω -acceptable. (It is enough to consider strongly M ν -regular elements thanksto Lemma 3.1.5.)From the orbital integral identity for SO γ ω ( φ ω ) (cf. (2.6.1)), we see the existence of γ ∈ M ν ( F ) sr whose stable conjugacy class matches that of γ ω such that O γ ( φ ) = 0. The latter implies that γ is ν -acceptable. Write T , T ω for the centralizers of γ , γ ω in M ν , G e ω , respectively. The matchingof conjugacy classes tells us that there is a canonical F -isomorphism i : T ≃ T ω which carries γ to γ ω , cf. [Kot86, § i sends the stable conjugacy class of γ to that of γ ω and is canonicalup to a Weyl group orbit. But i is determined if required to send γ to γ ω .) Since ν is centralin M ν , the map i necessarily carries ν to ν ω . Regarding T and T ω as maximal tori of G and G e ,respectively, we have an injection i ∗ : R ( G e ω , T ω ) ֒ → R ( G, T ) between the sets of roots induced by i (again [Kot86, § h α, ν ω i = h i ∗ ( α ) , ν i , α ∈ R ( G e ω , T ω ) . (3.3.2)We are ready to show that γ ω is ν ω -acceptable. Let α ∈ R ( G e ω , T ω ) such that h α, ν ω i >
0. We needto verify that | α ( γ ω ) | <
1, cf. Definition 3.1.1. But h i ∗ ( α ) , ν i > ν -acceptabilityof γ implies that | i ∗ ( α )( γ ) | <
1. Since i ∗ ( α )( γ ) = α ( γ ω ), the proof is finished. (cid:3) C -regular functions and constant terms. Assume that G is split over F and fix a reductivemodel over O F , still denoted by G . Let T be a split maximal torus of G over O F . Let C ∈ R > . Definition 3.4.1.
A cocharacter µ : G m → T is C - regular if the following two conditions hold.(1) |h α, µ i| > C for every α ∈ Φ( T, G ),(2) |h α | A M , pr M ( ωµ ) i| > C for every proper Levi subgroup M of G containing T , every ω ∈ Ω G ,and every α ∈ Φ( T, G ) \ Φ( T, M ).Write X ∗ ( T ) C -reg for the set of C -regular cocharacters. Lemma 3.4.2.
The following are true.(1) The subset X ∗ ( T ) C -reg of X ∗ ( T ) is nonempty, and stable under both Z -multiples and the Ω G -action.(2) Let µ, µ ∈ X ∗ ( T ) . If µ is C -regular, then there exists k ∈ Z > such that µ + kµ is C -regular for all k ≥ k .Proof. (1) Let X ∗ ( T ) R ,C -reg denote the subset of X ∗ ( T ) R defined by the same inequalities as inDefinition 3.4.1. We choose an inner product on X ∗ ( T ) R invariant under the Weyl group action.Clearly X ∗ ( T ) C -reg and X ∗ ( T ) R ,C -reg are stable under Z -multiples and the Weyl group action, andthe latter is open. It suffices to verify the claim that X ∗ ( T ) R ,C -reg is nonempty. Indeed, if the claimis true, we choose an open ball U ⊂ X ∗ ( T ) R ,C -reg . For k ∈ Z > large enough, k · U contains a pointof X ∗ ( T ), which then also lies in X ∗ ( T ) C -reg .Let us prove the claim. Identify X ∗ ( T ) R with the standard inner product space R n via a linearisomorphism. Say that a measurable subset A ⊂ R n has density 0 if vol( A ∩ B (0 , r )) / vol( B (0 , r )) → r → ∞ , where B (0 , r ) denotes the ball of radius r centered at 0. We will show that thecomplement of X ∗ ( T ) R ,C -reg in X ∗ ( T ) R is a density 0 set. Since a finite union of density 0 sets still hasdensity 0, it is enough to check that each of the conditions |h α, µ i| ≤ C and |h α | A M , pr M ( ωµ ) i| ≤ C defines a density 0 subset in X ∗ ( T ) R . Either condition defines a subset of the form | a x + · · · + a n x n | ≤ C (3.4.1)in the standard coordinates of R n , with coefficients a , ..., a n ∈ R . Notice that neither h α, µ i nor h α | A M , pr M ( ωµ ) i is trivially zero on all µ ∈ X ∗ ( T ) R . (In the latter case, the reason is thatpr M : X ∗ ( T ) R → X ∗ ( A M ) is surjective, and that α | A M ∈ X ∗ ( A M ) is nontrivial since α / ∈ Φ( T, M ).)Therefore not all a i ’s are zero. Now it is elementary to see that (3.4.1) determines a density 0subset. This proves the claim. OF IGUSA VARIETIES 31 (2) Since the pairings in Definition 3.4.1 are linear in µ , it is enough to choose k such that( k − C is greater than |h α, µ i| and |h α | A M , pr M ( ωµ ) i| for all α, M, ω as in that definition. (cid:3) Define T ( F ) C -reg to be the union of µ ( ̟ F ) T ( O × F ) as µ runs over the set of C -regular cocharacters.For each M ∈ L ( G ) containing T , set a M,C -reg := { a ∈ a M : |h α, a i| > C | log | ̟ || , ∀ α ∈ Φ + ( T, G ) \ Φ + ( T, M ) } . (3.4.2)Here we use the pairing X ∗ ( T ) R × X ∗ ( T ) R → R to compute h α, a i , viewing a in X ∗ ( T ) R via a M = X ∗ ( A M ) R ⊂ X ∗ ( T ) R . Recall that X ∗ ( A M ) R ≃ Hom( X ∗ ( M ) R , R ) via a ( χ
7→ h χ, a i ).Analogously X ∗ ( A T ) R ≃ Hom( X ∗ ( T ) R , R ). Write pr M : X ∗ ( A T ) R ։ X ∗ ( A M ) R for the map inducedby the restriction X ∗ F ( M ) → X ∗ F ( T ). (This is the analogue of pr G in § Lemma 3.4.3.
Let M ( G be a Levi subgroup containing T over F . Then H M ( T ( F ) C -reg ) ⊂ a M,C -reg .Proof.
Consider t := µ ( ̟ ) with µ ∈ X ∗ ( T ) C -reg . Then H M ( t ) ∈ Hom( X ∗ ( M ) R , R ) is identifiedwith the unique element a ∈ X ∗ ( A M ) R such that h χ, a i = log | χ ( µ ( ̟ )) | = h χ, µ i log | ̟ | , χ ∈ X ∗ ( M ) R . Let α ∈ Φ( T, G ) \ Φ( T, M ). Since the composite of the restriction maps X ∗ F ( M ) R → X ∗ F ( T ) R → X ∗ ( A M ) R is an isomorphism, we can find χ ∈ X ∗ F ( M ) R such that χ | A M = α | A M . Hence h α, a i = h α | A M , a i = h χ | A M , a i = h χ, a i = h χ, µ i log | ̟ | = h χ | A T , µ i log | ̟ | = h χ, pr M ( µ ) i log | ̟ | = h α | A M , pr M ( µ ) i log | ̟ | . Since µ is C -regular, |h α | A M , pr M ( µ ) i| > C . Hence |h α, a i| > C | log | ̟ || . (cid:3) The following definition is motivated by [FK88, p.195].
Definition 3.4.4.
Let
C >
0. We say f ∈ H ( G ) is C -regular if supp( f ) is contained in the G ( F )-conjugacy orbit of T ( F ) C -reg . Write H ( G ) C -reg for the space of C -regular functions. Lemma 3.4.5.
Let f ′ ∈ H ( G ) . Assume that every g ∈ G ( F ) reg such that O g ( f ′ ) = 0 (resp. SO g ( f ′ ) =0 ) is G ( F ) -conjugate (resp. stably conjugate) to an element of T ( F ) C -reg . Then(1) O g ( f ′ ) = 0 (resp. SO g ( f ′ ) = 0 ) if g ∈ G ( F ) ss is not regular, and(2) there exists f ∈ H ( G ) C -reg such that f and f ′ have the same image in I ( G ) (resp. S ( G ) ).Proof. (1) If g ∈ G ( F ) ss is not regular then no regular element in a sufficiently small neighborhoodof g intersects the G ( F )-orbit of T ( F ) C -reg . (Since every t ∈ T ( F ) C -reg satisfies | − α ( t ) | = 1for α ∈ Φ( T, G ), no α ( t ) approaches 1.) Thus f ′ has vanishing regular orbital integrals in aneighborhood of g . This implies that O g ( f ′ ) = 0, by an argument as in the proof of [Rog83, Lem. 2.6]via the Shalika germ expansion around g . The case of stable orbital integrals is analogous.(2) The point is that the G ( F )-conjugacy orbit of T ( F ) C -reg is open and closed in G ( F ). (Since T ( F ) C -reg is open and closed in T ( F ), and the map G ( F ) /T ( F ) × T ( F ) → G ( F ) induced by( g, t ) gtg − is a local isomorphism.) Thus the product of f ′ and the characteristic function onthe latter orbit is smooth and compactly supported, and thus belongs to H ( G ) C -reg . Denoting theproduct by f , we see that f and f ′ have equal orbital integrals (resp. stable orbital integrals) onregular semisimple elements. Therefore have the same image in I ( G ) (resp. S ( G )). (cid:3) Corollary 3.4.6.
Fix a C -regular cocharacter µ : G m → T . Let φ ∈ H ( T ) . Then there exists aninteger k = k ( φ ) such that for every integer k ≥ k , the µ -ascent of φ ( k ) is represented by a C -regular function on G ( F ) . In practice it seems enough to impose the condition on supp O ( f ). However when producing examples of C -regular f , often we have this condition satisfied. Proof.
There is a finite subset µ ∈ X ∗ ( T ) such that supp( φ ) ⊂ ∪ µ ∈ µ µ ( ̟ F ) T ( O F ). ApplyingLemma 3.4.2 to each member of µ and also Lemma 3.1.7, we can find k = k ( φ ) ∈ Z ≥ such that φ ( k ) is µ -acceptable and supp( φ ( k ) ) ⊂ T ( F ) C -reg for all k ≥ k . Write f ( k ) for a µ -ascent of φ ( k ) .By Lemma 3.4.5 it suffices to check for each k ≥ k and g ∈ G ( F ) reg that if O g ( f ( k ) ) = 0 then g is in the G ( F )-orbit of T ( F ) C -reg . This follows from the observed properties of φ ( k ) by Lemma3.1.2. (cid:3) Lemma 3.4.7.
Let f ∈ H ( G ) C -reg , M ∈ L < ( G ) , and e ∈ E < ( G ) . The following are true.(1) C GM ( f ) ∈ I ( M ) is represented by a function f M ∈ H ( M ) whose support is contained in the M ( F ) -conjugacy orbit of T ( F ) C -reg . (In particular f M is a C -regular function on M ( F ) .)(2) LS e ( f ) ∈ S ( G e ) vanishes unless G e is split over F . If G e is split over F then LS e ( f ) isrepresented by a C -regular function on G e ( F ) .Proof. (1) We keep writing T ( F ) C -reg for the set of C -regular elements relative to G , which contain C -regular elements relative to M . Since T ( F ) C -reg is invariant under the Weyl group of G , anelement γ ∈ M ( F ) ss is conjugate to an element of T ( F ) C -reg in G ( F ) if and only if it is so in M ( F ).In light of Lemma 3.4.5, it suffices to show the following: if O γ ( C GM ( f )) = 0 for regular semisimple γ ∈ M ( F ) then γ is M ( F )-conjugate to an element of T ( F ) C -reg .If γ is G -regular then we have from § O γ ( f ) = D G/M ( γ ) O γ ( C GM ( f )), which is nonzero onlyif γ is conjugate to an element of T ( F ) C -reg . If γ is regular but outside the M ( F )-orbit of T ( F ) C -reg ,then a sufficiently small neighborhood V of γ does not intersect the M ( F )-orbit of T ( F ) C -reg . Onthe other hand, G -regular elements are dense in V . Since an orbital integral is locally constant onthe regular semisimple set, it follows that O γ ( C GM ( f )) = 0.(2) If T transfers to a maximal torus in G e then G e is split over F since T is a split torus.Thus LS e ( f ) = 0 unless G e is split over F . Now we assume that T transfers to a maximal torus T e ⊂ G e , equipped with an F -isomorphism T ≃ T e (canonical up to the Weyl group action). Viathe isomorphism we transport λ to λ e : G m → T e and identify Φ( T e , G e ) as a subset of Φ( T, G ).By abuse of notation, keep writing T ( F ) C -reg for its image in T e ( F ). Then C -regular elements of T e ( F ) are contained in T ( F ) C -reg .Now the rest of the proof of (2) similar to that of (1), based on Lemma 3.4.5. It suffices to checkthat if SO γ e (LS e ( f )) = 0 for G -regular semisimple γ e ∈ G e ( F ) then γ e is stably conjugate to anelement of T ( F ) C -reg . This is evident from the transfer of orbital integral identity. (cid:3) Corollary 3.4.8.
For f ∈ H ( G ) C -reg and M ∈ L < ( G ) , we have supp O a M ( C GM ( f )) ⊂ a M,C -reg .Proof.
Let f M be as in the preceding lemma. Thensupp O a M ( C GM ( f )) = supp O a M ( f M ) ⊂ supp a M ( f M ) ⊂ H M ( T ( F ) C -reg ) ⊂ a M,C -reg , where the last inclusion comes from Lemma 3.4.3. (cid:3) The following lemma, to be invoked in § C -regular functions detect. Lemma 3.4.9.
Let I be a finite set and let C > . Let π i ∈ Irr( G ( F )) and c i ∈ C for i ∈ I . If X i ∈ I c i Tr π i ( f ) = 0 ∀ f ∈ H ( G ) C -reg then P i ∈ I c i · J P ( π i ) = 0 in Groth( G ( F )) ⊗ Z C .Proof. Fix a regular cocharacter µ : G m → T over F such that P = P op µ . For each φ ∈ H ( T ),we have some integer k such that φ ( k ) are µ -acceptable for all k ≥ k and their µ -ascent f ( k ) arerepresented by C -regular functions by Corollary 3.4.6. Thanks to Lemma 3.1.2,0 = X i ∈ I c i Tr ( f ( k ) | π i ) = X i ∈ I c i Tr ( φ ( k ) | J P ( π i )) , ∀ k ≥ k . We conclude by Lemma 3.1.8. (cid:3) OF IGUSA VARIETIES 33
Fixed central character.
We explain that the facts thus far in § ν : G m → G be a cocharacter over F and ( G e , G e , s e , η e ) an endoscopic datumfor G with G e = L G e . We can view X as a closed subgroup of M ν ( F ), G e ( F ), and G e ω ( F ) of thepreceding sections via the canonical embeddings of Z ( F ) into their centers.As before, H acc ( M ν , χ − ) ⊂ H ( M ν , χ − ) is the subspace of functions which are supported onacceptable elements. Taking the image, we also have I acc ( M ν , χ − ) and S acc ( M ν , χ − ). Since theset of acceptable elements on M ν ( F ) is invariant under translation by Z ( F ), the χ -averaging mapinduces a surjection H acc ( M ν ) → H acc ( M ν , χ − ). The analogous surjectivity holds for I acc and S acc .The earlier results continue to hold with fixed central character, with the following minor modi-fications. We omit the proofs as no new ideas are required; the basic idea is to apply χ -averagingconsistently. To adapt § . Averaging the ν -ascent map, we obtain J ν : I ( M ν , χ − ) → I ( G, χ − ) , J ν : S ( M ν , χ − ) → S ( G, χ − )satisfying the orbital integral and trace identities in Lemma 3.1.2 (with central character of π equal to χ ) and Corollary 3.1.3. The evident analogues of Lemmas 3.1.5 and 3.1.6 hold true (withno changes to the bottom rows in the latter lemma). The map ( · ) ( k ) in (3.1.6) induces linearautomorphisms on I ( M ν , χ − ) and I ( G, χ − ). With this, Lemmas 3.1.7 and 3.1.8 imply theirnatural analogues, restricting π , π in the latter lemma to those with central character χ . To adapt § . Averaging the map H ( G ) → H ( M ) given by f f M , we obtain a map H ( G, χ − ) →H ( M, χ − ), which induces C GM : I ( G, χ − ) → I ( M, χ − )satisfying the same orbital integral identity as in § C GM by the sameformula (3.2.1) from the space of linear functionals on R ( G, χ ) to that on R ( M, χ ). Lemmas 3.2.2and 3.2.1 carry over as written, with χ − -equivariance imposed everywhere. To adapt § . The Langlands–Shelstad transfer with fixed central character was already consideredin § I ( M ν , χ − ) J ν / / ⊕ LS e ,ω ( ( PPPPPPPPPPPPPP I ( G, χ − ) LS e / / S ( G e , χ e , − ) L ω ∈ Ω e ,ν S ( G e ω , χ e , − ) P ω J νω ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ To adapt § . Definition 3.4.4 extends obviously to H ( G, χ − ) by the same support condition. Akey observation is that the notion of C -regularity is invariant under Z ( F )-translation, so that thelatter definition behaves well. More precisely, the χ -averaging map from H ( G ) ։ H ( G, χ − ) isstill surjective when restricted to the respective subspaces of C -regular functions. Using this, wecarry over all results in § χ − -equivariantfunctions and representations with central character χ .3.6. z -extensions. Throughout this section up to now, we assumed G e = L G e on the endoscopicdatum e . When the assumption is not guaranteed, we pass from e and G to e and G via z -extensions and pull back the central character datum from ( X , χ ) to ( X , χ ) as explained in § Lemma 3.6.1.
There exists a cocharacter ν : G m → G over F lifting ν : G m → G . Proof.
It suffices to show the claim that if 1 → T → T → T → F and if T is a finite product of induced tori of the form Res F ′ /F G m , thenHom F ( G m , T ) → Hom F ( G m , T ) is onto. To see this, we apply X ∗ to obtain a short exact se-quence of Gal( F /F )-modules 0 → X ∗ ( T ) → X ∗ ( T ) → X ∗ ( T ) →
0. Since H (Gal( F /F ) , X ∗ ( T ))vanishes by hypothesis via Shapiro’s lemma, we deduce that X ∗ ( T ) Gal(
F /F ) → X ∗ ( T ) Gal(
F /F ) isonto, which is what we wanted to show. (cid:3) Let us fix ν as above. By Definition 3.1.1, γ ∈ M ν ( F ) is ν -acceptable if and only if its imagein M ν ( F ) is ν -acceptable. Everything in this section goes through with e , G , ( X , χ ) , ν playingthe roles of e , G, ( X , χ ) , ν . We will write λ e , λ e ,ω for the characters λ e , λ e ω of § e and G . 4. Asymptotic analysis of the trace formula
We prove key trace formula estimates in this section, to be applied to identify leading terms inthe trace formula for Igusa varieties in §
7. The main estimate is Theorem 4.2.2, whose lengthyproof is presented in § Q -semisimple rank. The point isthat the trace formula appearing in the intermediate steps need not arise from geometry.4.1. Setup and some basic lemmas.
Throughout Section 4, G is a connected quasi-split reductivegroup over Q which is cuspidal, i.e., Z G has the same Q -split rank and R -split rank. Let ( X , χ )be a central character datum as in § ξ be an irreducible algebraic representation of G C and ζ : G ( R ) → C × be a continuous character such that ξ ⊗ ζ has central character χ − ∞ on X ∞ .The restriction χ ∞ | A G ( R ) via A G ( R ) ⊂ X ∞ can be viewed as an element of X ∗ ( A G ) C , whichis again denoted χ ∞ by abuse of notation. Let λ χ ∞ denote the unique character making thefollowing diagram commute. (The existence is obvious since the composition A G ( R ) → a G is anisomorphism.) A G ( R ) (cid:31) (cid:127) / / χ ∞ G ( R ) H G ∞ / / / / a G = X ∗ ( A G ) R λ χ ∞ / / C × We have a canonical identification a G = X ∗ ( A G ) R = Hom( X ∗ ( G ) Q , R ) , a ( χ
7→ h χ | A G , a i ) . (4.1.1)Fix distinct primes p, q . Let ν : G m → G Q p be a cocharacter over Q p . Let ν ∈ Hom( X ∗ Q ( G ) , Q )denote the image of ν ∈ X ∗ ( A M ν ) Q = Hom( X ∗ Q ( M ν ) , Q ) induced by M ν ֒ → G . By definition, ν ( χ ) = ν ( χ ) for χ ∈ X ∗ Q ( G ). Viewing ν as a member of a G , we can compute h χ ∞ , ν i ∈ C via thecanonical pairing X ∗ ( A G ) C × X ∗ ( A G ) C → C . Lemma 4.1.1. λ χ ∞ ( H Gp ( ν ( p ))) = p −h χ ∞ ,ν i .Proof. By definition, H Gp ( ν ( p )) sends χ ∈ X ∗ Q ( G ) to log | χ ( ν ( p )) | p . Similarly for a ∈ A G ( R ) , wehave H G ∞ ( a ) = ( χ log | χ ( a ) | ∞ ). We claim that H Gp ( ν ( p )) = H G ∞ ( ν ( p ) − ). To show this, choose r ∈ Z ≥ such that rν ∈ X ∗ ( A G ). Since a G is torsion-free, it suffices to check that H Gp (( rν )( p )) = H G ∞ (( rν )( p ) − ), or equivalently that | χ (( rν )( p )) | p = | χ (( rν )( p )) | − ∞ , χ ∈ X ∗ Q ( G ) . In the notation of the preceding section, ν = pr G ( ν ). OF IGUSA VARIETIES 35
Since χ (( rν )( p )) ∈ Q is an integral power of p (as both χ and rν are algebraic), we have | χ (( rν )( p )) | ∞ = | χ (( rν )( p )) | − p = | χ (( rν )( p )) | − p . This proves the claim. Now the claim implies that λ χ ∞ ( H Gp ( ν ( p ))) = λ χ ∞ ( H G ∞ ( ν ( p ) − )) = χ ∞ ( ν ( p ) − ) = p −h χ ∞ ,ν i . (cid:3) If G is a connected reductive group over Q and S is a set of Q -places, we write H GS ( γ ) := X v ∈ S H Gv ( γ ) ∈ a G . If S c is the complement of S , we write H G,S c := H GS . Lemma 4.1.2.
Let G be a connected reductive group over Q .(i) Let S be a set of Q -places. Let γ, γ ′ ∈ G ( Q ) . If γ and γ ′ are conjugate in G ( Q ) then wehave H GS ( γ ) = H GS ( γ ′ ) ∈ a G .(ii) Let S be the set of all Q -places. Let γ ∈ G ( Q ) , then H GS ( γ ) = 0 ∈ a G .Proof. ( i ) Let F/ Q be a finite extension such that γ and γ ′ are conjugate in G ( F ). Set G ′ :=Res F/ Q M . The natural embedding i : G → G ′ allows to view γ, γ ′ as elements of G ′ ( Q ), andinduces an injection a G ֒ → a G ′ . Thus it suffices to prove that H G ′ S ( γ ) = H G ′ S ( γ ′ ), since the map H GS : G ( A S ) → a G is functorial with respect to i . By the reduction in the preceding paragraph,we may assume that γ and γ ′ are conjugate in G ( Q ). Then the proof is trivial since H GS is ahomomorphism into an abelian group.( ii ). Using the functoriality for G → Z ′ G from step 1, we may replace G by its cocenter Z ′ G , then Z ′ G by the maximally split torus A inside Z ′ G , and finally we may replace A by G m , in which casethe statement boils down to the usual product formula. (cid:3) The following will be useful when studying Levi terms in the geometric side of the trace formula.
Lemma 4.1.3.
Let M ∈ L cusp ( G ) and let γ ∈ Γ ell , X ( M ) be a regular element. Let P ⊂ G be aparabolic subgroup with Levi component M . Let f ∞ ∈ H ( M ( A ∞ ) , χ ∞ , − ) and let ξ be an irreduciblerepresentation of M C . Then we have vol( X Q \ X /A G, ∞ ) − χ ( I Mγ ) ζ ( γ )Tr ( γ ; ξδ / P ) O Mγ ( f ∞ )= vol( I Mγ ( Q ) A I Mγ , ∞ \ I Mγ ( A ) / X ) O Mγ ( f ξ,ζδ / P f ∞ ) . Proof.
By Equation (2.8.3) we have χ ( I Mγ ) = ( − q ( I Mγ ) τ ( I Mγ )vol( A I Mγ, ∞ \ I M, cmpt γ ( R )) − d ( I Mγ ) = 1 . As γ is regular, I Mγ is a torus and so d ( I Mγ ) = 1. Additionally we have q ( I Mγ ) = 0 (as γ is elliptic)and τ ( I Mγ ) = vol( I Mγ ( Q ) A I Mγ , ∞ \ I Mγ ( A )) . Thus we obtain χ ( I Mγ ) = vol( I Mγ ( Q ) A I Mγ , ∞ \ I Mγ ( A ))vol( A I Mγ, ∞ \ I M, cmpt γ ( R )) − . By (2.4.1) and e ( I Mγ ) = 1, we have ζ ( γ )Tr ( γ ; ξδ / P ) = vol( A M, ∞ \ I Mγ ( R )) O Mγ ( f ξ,ζδ / P ) . We obtain vol( X Q \ X /A G, ∞ ) − χ ( I Mγ ) ζ ( γ )Tr ( γ ; ξδ / P ) O Mγ ( f ∞ ) == vol( I Mγ ( Q ) A I Mγ , ∞ \ I Mγ ( A ))vol( A M, ∞ \ I Mγ ( R ))vol( X Q \ X /A G, ∞ )vol( A I Mγ , ∞ \ I M, cmpt γ ( R )) O Mγ ( f ξ,ζδ / P f ∞ )= vol( I Mγ ( Q ) A I Mγ , ∞ \ I Mγ ( A ))vol( X Q \ X /A G, ∞ ) O Mγ ( f ξ,ζδ / P f ∞ )= vol( I Mγ ( Q ) A I Mγ , ∞ \ I Mγ ( A ) / X ) .O Mγ ( f ξ,ζδ / P f ∞ ) (cid:3) The main estimate and its consequences.
We prove the following bounds for elliptic endoscopicgroups and Levi subgroups of G , to be applied in § O ( f ( k )) (resp. o ( k )) for a nonzero C -valued function f ( k ) on k ∈ Z means thatthe quantity divided by | f ( k ) | has bounded absolute value (resp. tends to 0) as k → + ∞ . Infact we only take f ( k ) to be complex powers of p (but not necessarily real powers; that is why wetake absolute values). As the reader will see, every instance of o ( f ( k )) represents a power-saving,namely it is bounded by a power of p with (the real part of) exponent strictly smaller than theexponent for f ( k ).Let us fix a Q -rational Borel subgroup B with Levi component T ⊂ B (which is a maximal torusin G ). We fix a Levi decomposition B = T N . As before we write A T ⊂ T for the maximal Q -splitsubtorus. Additionally we write S p ⊂ T Q p for the maximal Q p -split subtorus.Part of our setup is a cocharacter ν : G m → G over Q p . By conjugating ν if necessary, we mayand will assume that ν has image in T and that ν is B -dominant. Write ρ ∈ X ∗ ( T ) Q for the halfsum of all B -positive roots of T in G over Q p . Thus we have h ρ, ν i ∈ Z > . We transport variousdata over Q p or Q p to ones over C via a fixed isomorphism ι p : Q p ≃ C . (In the context of geometry, ι p is fixed in § q such that G Q q is a split group. (For the existence, choose a number field F over which G splits. Then any prime q that splits completely in F will do.) If needed for endoscopy,an auxiliary z -extension G of G over Q is always chosen to be split over Q q ; this is possible because G Q q is split. Thus the contents of § z -extensions apply to G and G over Q q . Since all endoscopic groups appearing in the argument will be split over Q q (to be ensured byLemma 3.4.7 in the proof of Corollary 4.2.3), whenever choosing their z -extensions, we take themto be also split over Q q without further comments. Proposition 4.2.1.
Let f ∞ ,p = Q v = ∞ ,p f v ∈ H ( G ( A ∞ ,p ) , ( χ ∞ ,p ) − ) and φ p ∈ H acc ( M ν ( Q p ) , χ − p ) .For k ∈ Z , write f ( k ) p ∈ H ( G ( Q p ) , χ − p ) for a ν -ascent of φ ( k ) p as in § T G disc ,χ ( f ( k ) p f ∞ ,p f ξ,ζ ) = O (cid:16) p k ( h ρ,ν i + h χ ∞ ,ν i ) (cid:17) . Proof.
The left hand side equals X π ∈A disc ,χ ( G ) m ( π )Tr ( f ( k ) p | π p )Tr ( f ∞ ,p | π p )Tr ( f ξ,ζ | π ∞ ) . Write J P op ν ( π p ) = P i c i τ i in Groth( M ν ( Q p )) with τ i ∈ Irr( M ν ( Q p )). Let ω τ i denote the centralcharacter of τ i . ThenTr ( f ( k ) p | π p ) = Tr (cid:16) φ ( k ) p | J P op ν ( π p ) (cid:17) = X i c i Tr ( φ ( k ) p | τ i ) = X i c i ω τ i ( ν ( p )) k Tr ( φ p | τ i ) . OF IGUSA VARIETIES 37
We define a character λ A : G ( Q ) \ G ( A ) → R × > as the composite λ A : G ( Q ) \ G ( A ) H G → a G λ χ ∞ → R × > . Write λ v for the restriction of λ A to G ( Q v ) for a place v of Q . For each π ∈ A disc ,χ ( G ) contributingto the sum, we see that π ⊗ λ − A is a unitary automorphic representation of G ( A ) since π ∞ ⊗ λ − ∞ is unitary (by construction, π ∞ ⊗ λ − ∞ has trivial central character on A G ( R ) ). Thus π p ⊗ λ − p isunitary. Applying Corollary 2.5.2 to π ⊗ λ − at p , we have (cid:12)(cid:12) ω τ i ( ν ( p )) λ − p ( ν ( p )) (cid:12)(cid:12) ≤ δ − / P op ν ( ν ( p )) = p h ρ,ν i , noting that ν ( p ) ∈ A −− P ν . We deduce via Lemma 4.1.1 that | ω τ i ( ν ( p )) | ≤ p h ρ,ν i | λ p ( ν ( p )) | p = p h ρ,ν i | p −h χ ∞ ,ν i | p = p h ρ,ν i + h χ ∞ ,ν i . By [BZ77, Cor. 2.13] the length of J P op ν ( π p ), namely P i c i , can be bounded only in terms of G .This completes the proof. (cid:3) We state the main trace formula estimate of this paper. The proof will be given in § Theorem 4.2.2 (main estimate) . Let G, ( X , χ ) , p, q, ξ, ζ, ν be as defined in the beginning of Section4, and additionally assume that X = Y ( A ) A G, ∞ for a central torus Y ⊂ Z G over Q . Let • f ∞ ,p,q = Q v = ∞ ,p,q f v ∈ H ( G ( A ∞ ,p,q ) , ( χ ∞ ,p,q ) − ) , • φ p ∈ H acc ( M ν ( Q p ) , χ − p ) , and • f ( k ) p ∈ H ( G ( Q p ) , χ − p ) be a ν -ascent of φ ( k ) p , for k ∈ Z ≥ .Then there exists a constant C = C ( f ∞ ,q , φ p ) ∈ R > such that for each f q ∈ H ( G ( Q q ) , χ − q ) C -reg , T G ell ,χ ( f ( k ) p f q f ∞ ,p,q f ξ,ζ ) = T G disc ,χ ( f ( k ) p f q f ∞ ,p,q f ξ,ζ ) + o (cid:16) p k ( h ρ,ν i + h χ ∞ ,ν i ) (cid:17) . As a corollary, we derive the stable analogue of Theorem 4.2.2. We keep the setup of Theorem4.2.2 and let f ∞ ,p,q , φ p , f ( k ) p be as in there. For each e ∈ E < ell ( G ), we choose z -extensions to introduce e = ( G e , L G e , s e , η e ) and a central character datum ( X e , χ e ) as in § e , e such that η e ( W F ) and η e ( W F ) have bounded images, as explained in Lemma2.6.2 and § f ( k ) , e = Y v f ( k ) , e ,v ∈ H ( G e ( A ) , ( χ e ) − )be a transfer of f ( k ) p f q f ∞ ,p,q f ξ,ζ . Then we have the following bound. Corollary 4.2.3.
In the setup of Theorem 4.2.2, there exists a constant C = C ( f ∞ ,q , φ p , ξ, ζ ) ∈ R > such that for every f q ∈ H ( G ( Q q )) C -reg , firstly ST G ell ,χ ( f ( k ) p f q f ∞ ,p,q f ξ,ζ ) = ( T G disc ,χ ( f ( k ) p f ∞ ,p,q f q f ξ,ζ ) + o (cid:0) p k ( h ρ,ν i + h χ ∞ ,ν i ) (cid:1) ,O (cid:0) p k ( h ρ,ν i + h χ ∞ ,ν i ) (cid:1) , and secondly for each e ∈ E < ell ( G ) (note that f ( k ) , e ,q inherits C -regularity from f q ), ST G e ell ,χ e (cid:16) f ( k ) , e (cid:17) = o (cid:16) p k ( h ρ,ν i + h χ ∞ ,ν i ) (cid:17) . Remark . In the inductive proof of the last bound, we only use the fact that its q -componentis C -regular, ∞ -component is a Lefschetz function, and most importantly the p -component is anascent for a suitable cocharacter. We do not rely on the fact that f ( k ) , e is a transfer of a functionon G ( A ). Proof.
The second estimate is immediate from the first via Proposition 4.2.1. Let us prove the firstand third asymptotic formulas, by reducing the former to the latter.We induct on the semisimple rank of G . (For each G , we prove the corollary for all centralcharacter data and all ν .) The estimate is trivial when G is a torus, in which case ST G ell ,χ = T G ell ,χ = T G disc ,χ . We assume that G is not a torus and that Corollary 4.2.3 is true for all groups which havelower semisimple rank than G . Write f ( k ) := f ( k ) p f q f ∞ ,p,q f ξ,ζ . The stabilization (Proposition 2.9.1)tells us that ST G ell ,χ ( f ( k ) ) = T G ell ,χ ( f ) − X e ∈E < ell ( G ) ι ( G, G e )ST G e ell ,χ e (cid:16) f ( k ) , e (cid:17) . In light of Theorem 4.2.2, since the summand is nonzero only for a finite set of e by Lemma 2.9.2(depending only on the finite set of primes v where either G Q v or f v is ramified), it suffices toestablish the last bound of the corollary. This task takes up the rest of the proof.If G e R contains no elliptic maximal torus or if A G e = A G (equivalently if A G e = A G ), then f e , ∞ is trivial as observed in [Kot90, p.182, p.189] so the desired estimate is trivially true. Henceforth,suppose that G e R contains an elliptic maximal torus. Then f ( k ) , e , ∞ is a finite linear combination of f η e ,ζ e over the set of ( η e , ζ e ) such that η e ◦ ̟ η e ,ζ e ≃ ̟ ξ,ζ . Proposition 3.3.2 and its adaptation to z -extensions according to § § f ( k ) , e ,p = X ω λ e ,ω ( ν ( p )) k J ν ,ω (cid:16) φ ( k ) , e ,p,ω (cid:17) = f ( k ) , e ,p = X ω λ e ,ω ( ν ( p )) k f ( k ) , e ,p,ω , where we have put f ( k ) , e ,p,ω := J ν ,ω (cid:16) φ ( k ) , e ,p,ω (cid:17) for a ν ,ω -ascent of φ ( k ) , e ,p,ω ∈ H acc ( G e ,ν ( Q p ) , ( χ e ,p ) − ).Here we applied Lemma 3.3.5 (keeping § § φ ( k ) , e ,p,ω of φ ( k )1 ,p supported on ν ,ω -acceptable elements.Recalling that η e ( W F ) ⊂ L G is a bounded subgroup, we see from Lemma 3.3.1 that λ e ,ω is aunitary character. Thus we are reduced to showing the existence of some C e > ω and ( η e , ζ e ) as above whenever f e ,q is C e -regular:ST G e ell ,χ e (cid:16) f e , ∞ ,q,p f e ,q f ( k ) , e ,p,ω f η e ,ζ e (cid:17) ? = o (cid:16) p k ( h ρ,ν i + h χ ∞ ,ν i ) (cid:17) , k ∈ Z ≥ . (4.2.1)Indeed, take C to be the maximum of all C e over the finite set of e contributing to the sum. Thenfor each C -regular f q , Lemma 3.4.7 tells us either that G e is split over Q q and f e ,q is C -regular(thus also C e -regular), or that G e is non-split over Q q and f e ,q vanishes. Thus the bound (4.2.1)applies, and we will be done.By the induction hypothesis, there exists C e > f e ,q is C e -regular, the lefthand side of (4.2.1) is O (cid:16) p k ( h ρ e ,ν ,ω i + h χ e , ∞ ,ν ,ω i ) (cid:17) , with ν ,ω ∈ X ∗ ( A G e ) defined from ν ,ω in the sameway ν from ν , and where ρ e is the half sum of positive roots of G e for which ν ,ω is a dominantcocharacter. (In other words, ρ e is to ν ,ω as ρ is to ν .) Therefore it is enough to check that(a) h ρ e , ν ,ω i < h ρ, ν i (in Q ).(b) Re h χ e ∞ , ν i = Re h χ , ∞ , ν ,ω i ,Let us begin with (a). Since h ρ, ν i = h ρ , ν i , with ρ defined for G as ρ is for G (recall that ν : G m → G is a lift of ν ), the proof of (a) is reduced to the case when G = G and ν = ν . Wehave an embedding b G e ֒ → b G coming from η e , which restricts to b G e ω ֒ → c M ν . Here we have chosenΓ F -invariant pinnings for the dual groups such that the restriction works as stated. We may andwill arrange that the Borel subgroup of b G restricts to that of b G e . Fix a maximal torus b T ⊂ b G e ω thatis part of the pinning for b G e ω . Viewing b T also as a maximal torus in each of b G e and b G , we writeΦ ∨ ( b T , b G ) and Φ ∨ ( b T , b G e ) for the corresponding sets of coroots. Write b ν ∈ X ∗ ( b T ) for the dominant OF IGUSA VARIETIES 39 member in the Weyl orbit of characters determined by ν . Then h ρ e , ν ω i = X α ∨∈ Φ ∨ ( b T , b G e ω ) h α ∨ ,ν i > h α ∨ , ν i , h ρ, ν i = X α ∨∈ Φ ∨ ( b T, b G ) h α ∨ ,ν i > h α ∨ , ν i . (4.2.2)Thus it suffices to verify that there exists a coroot α ∨ ∈ Φ ∨ ( b T , b G ) outside b G e such that h α ∨ , ν i > b ν in b G is identified with the dual group d M ν (namely h α ∨ , ν i = 0 if and only if α ∨ is a coroot of d M ν ), so we will be done if Lie d M ν + Lie b G e is a proper subspace of Lie b G . Thisis exactly proved in [KST, Lem. 4.5 (ii)] applied to G = b G , M = c M ν , and δ = s e . (The proof of loc. cit. greatly simplifies. One reduces to the case when the Dynkin diagram of G is connected asin the first paragraph in the proof of that lemma. Then argue as in the fourth paragraph of thatlemma, with X n = 0 and with the role of X ss played by the semisimple element s e .)Now we prove (b). Since h χ ∞ , ν i = h χ , ∞ , ν i , we reduce to showing (b) when G = G and e = e (with possibly nontrivial central character data). Thus we drop the 1’s from the subscriptsand check that Re h χ e ∞ , ν i = Re h χ ∞ , ν ω i . We claim that ν = ν ω in X ∗ ( A G ) R = X ∗ ( A G e ) R . In the diagram below, the triangle on the rightcommutes, and we want the triangle on the left commutes as well. G mν (cid:10) (cid:10) ν ω (cid:15) (cid:15) ν " " ❊❊❊❊❊❊❊❊❊ ν ω ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘❘ A G A G e A M ν (cid:31) (cid:127) / / A G e ω We choose maximal tori T ⊂ M ν ⊂ G and T e ⊂ G e ω ⊂ G e with an isomorphism T F ≃ T e F toidentify the absolute Weyl group Ω G e as a subgroup of Ω G . (This is done as in [Kot86, § ν = ν ω . By (3.1.5), we have the equalities ν = | Ω G | − X ω ∈ Ω G ω ( ν ) , ν ω = | Ω G e | − X ω ∈ Ω G e ω ( ν ) . Hence ν = | Ω G / Ω G e | − P ω ∈ Ω G / Ω G e ω ( ν ω ) = ν ω . Indeed, the last equality follows since ν ω ∈ X ∗ ( A G e ) R = X ∗ ( A G ) R , which tells us that ω ( ν ω ) = ν ω for ω ∈ Ω G .Applying (2.6.3) at the archimedean place, we have χ ∞ = λ e ∞ χ e ∞ as characters of A G ( R ).Since λ e ∞ is unitary, | χ ∞ | = | χ e ∞ | . Since ν ∈ X ∗ ( A G ) R (not just in X ∗ ( A G ) C ), we conclude thatRe h χ e ∞ , ν i = Re h χ ∞ , ν i as desired. This verifies (b). (cid:3) Some facts and notation on Weyl groups and Weyl chambers.
In this subsection we fix someadditional notation on Weyl groups, Weyl chambers, which will be needed in the proof of the mainestimate in the next subsection.Recall that we have fixed a maximal torus T ⊂ G and that we wrote A T ⊂ T for the maximal Q -split subtorus in T . We also write S p ⊂ T Q p for the maximal Q p -split subtorus in T Q p . We thenhave A T, Q p ⊂ S p ⊂ T Q p . We will write Ω G ⊂ Ω Gp ⊂ Ω G for the Weyl group of A T , S p , and T in G . Similar notation will be used for other objects relatedto Weyl groups, for instance if M ⊂ G Q p is a Levi subgroup defined over Q p , we write Ω GM,p ⊂ Ω Gp for the set of Kostant representatives for Ω Gp / Ω Mp .Let M be the unique Levi of a standard parabolic P ⊃ B such that M ⊃ T . Write Φ GM =Φ GM ( A M ; G ) for the set of A M roots that appear in Lie( G ). We recall, an element x ∈ a M is called regular if h α, x i 6 = 0 for all α ∈ Φ GM . We write a reg M for the subspace of all regular x ∈ a M . We callthe connected components of a reg M the (open) Weyl chambers of a M . The subset C + M := { x ∈ a reg M | ∀ α ∈ Φ GM : h α, x i > } ⊂ a reg M , is the dominant Weyl chamber . Let Ω GM ⊂ Ω G be the set of Kostant representatives for the quotientΩ G / Ω M . The Weyl chambers C ∈ π ( a reg M ) are parametrized via C = C ω := ω − ( C + M ) ∈ π ( a reg M ),where ω ∈ Ω GM . If C ⊂ a reg M is a Weyl chamber, we write C ∨ ⊂ a ∗ M for the dual chamber , i.e. the setof t ∈ a ∗ M such that t ( x ) > x ∈ C . Remark . In Section 3 we defined similar notions, but using absolute roots in the Q p -relativespace X ∗ ( S p ) R . We explain here how the notions can be compared. Write Φ( A T , B ), Φ( S p , B Q p ),Φ( T, B ) for the various sets of positive roots attached to A T , S p and T . The inclusions A T, Q p ⊂ S p, Q p ⊂ T Q p induce surjections, X ∗ ( T Q p ) → X ∗ ( S p, Q p ) → X ∗ ( A T, Q p ), and the natural mapsΦ( T Q p , B Q p ) → Φ( S p, Q p , B Q p ) → Φ( A T, Q p , B Q p )are surjective as well. Therefore in X ∗ ( A T ) R the following sets are all the same:(1) The set of x ∈ X ∗ ( A T ) R such that for all α ∈ Φ( T, B ) we have h α, x i 6 = 0;(2) The set of x ∈ X ∗ ( A T ) R such that for all α ∈ Φ( S p , B Q p ) we have h α, x i 6 = 0;(3) The set of x ∈ X ∗ ( A T ) R such that for all α ∈ Φ( A T , B Q p ) we have h α, x i 6 = 0.Similarly, the various versions of C -regularity (defined by using Q -relative, Q p -relative, or absoluteroots) all coincide. In place of T the above discussion is also true for any Levi subgroup M and thespace a M . Moreover, the natural maps π ( a reg M ) → π ( X ∗ ( S p ) reg R ) → π ( X ∗ ( T ) reg R ) are injections.Finally, under the inclusion a T = X ∗ ( A T ) R → X ∗ ( S p ) R , the subset ( X ∗ ( S p ) R ) C -reg ⊂ X ∗ ( S p ) R pullsback to the set of x ∈ X ∗ ( A T ) such that conditions (1) and (2) of Definition 3.4.1 hold for relativeroots α ∈ Φ( A T , B ).4.4. Proof of Theorem 4.2.2.
The rest of this section is devoted to establishing the main estimatein Theorem 4.2.2. We note that the argument spans until page 46 and will be interrupted by alemma which establishes small facts that are needed along the way. Before diving into the technicaldetails, we recommend the reader to first have a look at the outline that we sketched below (1.4.2)in the introduction.
Proof of 4.2.2.
We argue by induction on the Q -semisimple rank r G of G . If r G = 0, then we have T G ell ,χ = T G disc ,χ , and the statement follows. Assume now that the theorem is established for allgroups of lower Q -semisimple rank and the accompanying data.We take the constant C = C ( f ∞ ,q , φ p ) := 1log q · max M,x p,q, ∞ ,ε p ,α |h α, x p,q, ∞ + ε p i| , (4.4.1)where M ∈ L cusp ( G ), x p,q, ∞ ∈ supp O a M ( f ∞ ,p,qM ), ε p ∈ pr M supp a ωp ( M ) ∩ Mν ( φ p,ω p ( M ) ∩ M ν ), and α ∈ Φ GM .Define the constants v X := vol( X Q \ X /A G, ∞ ) and c M := ( − dim( A M /A G ) | Ω M || Ω G | , M ∈ L cusp ( G ) . Write f ∞ , ( k ) := f ∞ ,p,q f ( k ) p f q , to indicate the dependence on k at p . The running hypothesis on f q is that it is C -regular for (4.4.1). By Proposition 2.8.1 we have T G disc ,χ ( f ξ,ζ f ∞ , ( k ) ) = X M ∈L cusp c M v − X X γ ∈ Γ ell , X ( M ) χ ( I Mγ ) ζ ( γ )Φ M ( γ, ξ ) O Mγ ( f ∞ , ( k ) M ) | ι M ( γ ) || Stab M X ( γ ) | , (4.4.2)(here we used that X is of the form Y ( A ) A G, ∞ for a central torus Y ⊂ Z G over Q ). OF IGUSA VARIETIES 41
We first compare the term corresponding to M = G ∈ L cusp on the right hand side of Equa-tion (4.4.2) with T G ell ,χ . In Lemma 4.1.3 we checked that in this case we have c G v − X χ ( I Gγ ) ζ ( γ )Φ GG ( γ, ξ ) O Gγ ( f ξ,ζ f ∞ , ( k ) ) | ι G ( γ ) || Stab G X ( γ ) | = vol( I γ ( Q ) \ I γ ( A ) / X ) O γ ( f ξ,ζ f ∞ , ( k ) ) ι ( γ ) − | Stab G X ( γ ) | − (4.4.3)for γ regular elliptic. For γ ∈ Γ ell , X ( G ) non-regular, we have O γ ( f q ) = 0 since f q is C -regular. Thus(4.4.2) (see also (2.8.2)) can be rearranged as T G ell ,χ ( f ξ,ζ f ∞ , ( k ) ) = T G disc ,χ ( f ξ,ζ f ∞ , ( k ) ) − X M ∈L < cusp c M v − X X γ ∈ Γ ell , X ( M ) χ ( I Mγ ) ζ ( γ )Φ M ( γ, ξ ) O Mγ ( f ∞ , ( k ) M ) | ι M ( γ ) || Stab M X ( γ ) | . (4.4.4)By Lemma 3.2.2 we have f ( k ) p,M = X ω p ∈ Ω G Q pM,Mν f ( k ) p,M,ω p ∈ H ( M ( Q p ) , χ − p ) , (4.4.5)where f ( k ) p,M,ω p := J ν ωp ( ω − p φ ( k ) p,M ωp ) , M ω p = ω p ( M ) ∩ M ν . (4.4.6)By Lemma 3.4.7 (1) and Remark 4.3.1 we may arrange that the constant term f q,M is supportedon C -regular elements. Thus f q,M is decomposed according to the chambers of a reg M : f q,M = X ω q ∈ Ω GM f q,M,ω q ∈ H ( M ( Q q ) , χ − q ) C -reg , where f q,M,ω q satisfies supp O a M ( f q,M,ω q ) ⊂ C ω q . We define f ∞ , ( k ) M,ω p ,ω q := f ∞ ,p,qM f ( k ) p,M,ω p f q,M,ω q ∈ H ( M ( A ) , χ − ) . Changing the order of summation (each sum is finite), Equation (4.4.4) becomes T G ell ,χ ( f ξ,ζ f ∞ , ( k ) ) = T G disc ,χ ( f ξ,ζ f ∞ , ( k ) ) − X M,ω p ,ω q c M v − X X γ ∈ Γ ell , X ( M ) χ ( I Mγ ) ζ ( γ )Φ M ( γ, ξ ) O Mγ ( f ∞ , ( k ) M,ω p ,ω q ) | ι M ( γ ) || Stab M X ( γ ) | . (4.4.7)To state the next lemma, we define a constant k = k ( f ∞ ,q , φ p ) := max M,ω p ,ω q ,α,ε p ,x p, ∞ (cid:12)(cid:12)(cid:12)(cid:12) h α, ε p + x p i log( p ) h α, pr M ( ω − p ν ) i (cid:12)(cid:12)(cid:12)(cid:12) ∈ R > , where the maximum is taken over M ∈ L < cusp ( G ), ω p ∈ Ω G Q p M,M ν , ω q ∈ Ω GM , ε p ∈ pr M supp O a Mωp ( ω − p φ p,M ωp ), x p, ∞ ∈ supp O a M ( f ∞ ,pM ), and α ranges over those α ∈ Φ GM such that h α, pr M ( ω − p ν )) i 6 = 0. For each M ∈ L cusp ( G ), denote by P ( M ) the set of parabolic subgroups P of G of which M is a Levicomponent.We have fixed a maximal torus T in G C (we have G C ≃ G Q p via ι p ), along with a Borel subgroup B . We write ρ = ρ G for the half sum of the B -positive roots of T in Lie( B ). Note that we have ρ | A T = ρ . We use similar definitions for ρ M and ρ M if M ⊂ G is a Levi subgroup.For each λ ∈ X ∗ ( T ) + , write ξ Mλ for the irreducible M C -representation with highest weight λ ,and define ω ∞ ⋆ λ := ω ∞ ( λ + ρ ) − ρ for each ω ∞ ∈ Ω GM . Let λ = λ B , λ ∗ B ∈ X ∗ ( T ) denote thehighest weight of ξ and its dual representation ξ ∗ , respectively, relative to B . Write ω ∈ Ω M forthe longest Weyl group element, and λ ω ∞ := − ω ( ω ∞ ⋆ λ ∗ B ) = ω ω ∞ ω λ B − ω ω ∞ ρ − w ρ, so that we have ξ Mλ ω ∞ = ( ξ Mω ∞ ⋆λ ∗ B ) ∗ . Lemma 4.4.1.
Assume that k > k . Consider M, ω p , ω q , γ as in (4.4.7) , such that O Mγ ( f ∞ , ( k ) M,ω p ,ω q ) = 0 . Let x ∞ := H M ∞ ( γ ) ∈ a M . Then the following are true.(i) The element x ∞ ∈ a M is regular and lies in the chamber C = C ( M, ω p , ω q ) ⊂ a reg M whichhas the following set of positive roots { α ∈ Φ GM | h α, pr M ( ω − p ν ) i < } ∪ { α ∈ Φ GM | h α, pr M ( ω − p ν ) i = 0 and α ∈ −C ∨ ω q } . (4.4.8) (ii) There exists an explicit subset Ω G ⋄ M = Ω G ⋄ M ( M, ω p , ω q ) ⊂ Ω GM (see (4.4.18) ) and an explicitsign ε ⋄ = ε ⋄ ( M, ω p , ω q ) (see below (4.4.19) ) such that we have Φ M ( γ, ξ ) = ε ⋄ X P ∈P ( M ) δ − / P ( γ ) X ω ∞ ∈ Ω G ⋄ M ε ( ω ∞ )Tr ( γ ; ξ Mλ ω ∞ ) , where ε ( w ∞ ) ∈ {± } denotes the sign as an element of the Weyl group Ω G .Proof. ( i ) In this proof, if S is a set of places of Q , we write x S := H MS ( γ ) ∈ a M , x S := H M,S ( γ ) ∈ a M . We check that h α, x ∞ i 6 = 0 for all α ∈ Φ GM ( i.e., x is regular). By the product formula in Lemma 4.1.2we have − h α, x ∞ i = h α, x p, ∞ i + h α, x p i . (4.4.9)By assumption O γ ( f ∞ , ( k ) M,ω p ) = 0, so x p, ∞ ∈ supp O ( f ∞ ,pM ). Using Lemma 3.1.7 (and (4.4.6)) we findsupp O a M ( f ( k ) p,M,ω p ) = k · H Mp ( ω − p ν ( p )) + pr M (supp O a Mωp ( ω − p φ p,M ωp )) . Therefore x p = k · H Mp ( ω − p ν ( p )) + ε p (4.4.10)for some ε p ∈ pr M supp O a Mωp ( ω − p φ p,M ωp ). Thus h α, x p i = k · h α, H Mp ( ω − p ν ( p )) i + h α, ε p i = − k (log p ) · h α, pr M ( ω − p ν )) i + h α, ε p i . (4.4.11)We now distinguish cases. First consider α ∈ Φ GM such that h α, pr M ( ω − p ν ) i 6 = 0. By (4.4.9), −h α, x ∞ i = h α, x p, ∞ i + h α, ε p i − k (log p ) · h α, pr M ( ω − p ν ) i . As k > k we have k (log p ) · |h α, pr M ( ω − p ν ) i| > |h α, ε p + x p, ∞ i| . In particular h α, x ∞ i 6 = 0.The second case is when h α, pr M ( ω − p ν ) i = 0. Then − h α, x ∞ i = h α, x p, ∞ i + h α, ε p i . (4.4.12)As f q is C -regular, we have (see (3.4.2) and Remark 4.3.1 |h α, x q i| > C log q ≥ |h α, x p,q, ∞ + ε p i| . for all α ∈ Φ GM . In particular h α, x q i + h α, x p,q, ∞ + ε p i 6 = 0 . Therefore each side of (4.4.12) does not vanish. Hence h α, x ∞ i 6 = 0 for all α ∈ Φ GM . OF IGUSA VARIETIES 43
We now determine for which α ∈ Φ GM we have h α, x ∞ i >
0. If h α, pr M ( ω − p ν ) i 6 = 0, thensign( h α, x ∞ i ) = − sign( h α, pr M ( ω − p ν ) i )by the arguments following (4.4.11). If h α, pr M ( ω − p ν ) i = 0, thensign( h α, x ∞ i ) = − sign( h α, x q i )by C -regularity (see (4.4.12)). We have x q ∈ supp a M ( f q,M,ω q ). Statement ( i ) follows.( ii ) Let us start by recalling a result of Goresky–Kottwitz–MacPherson. We write pr ∗ M : X ∗ ( T ) R → X ∗ ( A M ) R for the restriction mapping. Let P = M N ∈ P ( M ). Write ρ N (resp. ρ N ) for the halfsum of the positive roots of A M (resp. T ) that occur in the Lie algebra of the unipotent radical N of P .Write ω ξ for the central character of ξ . Write α , . . . , α n ∈ a ∗ M for the simple roots of A M inLie ( N ), which form a basis of ( a M / a G ) ∗ . This determines the dual basis consisting of t , . . . , t n ∈ a M / a G . Put I := { , , . . . , n } . Define the subsets (cf. [GKM97, p.534]) I ( γ ) := { i ∈ I | h α i , x i < } ,I ( ω ∞ ) := { i ∈ I | h pr ∗ M ( − ω ∞ ⋆ λ ∗ B ) − ρ N − ω ξ , t i i > } . (4.4.13)By the discussion above Thm. 7.14.B in [GKM97] we have ϕ P ( − x ∞ , pr ∗ M ( ω ∞ ⋆ λ ∗ B ) + ρ N + ω ξ ) = ( ( − dim( A G ) ( − dim( A M /A G ) −| I ( γ ) | , if I ( ω ∞ ) = I ( γ ) , , otherwise . (4.4.14)We define L M ( γ ) ∈ C following [GKM97, p. 511], when x ∞ is regular: L M ( γ ) := ( − dim( A G ) X P ∈P ( M ) δ − / P ( γ ) X ω ∞ ∈ Ω GM ε ( ω ∞ )Tr ( γ − ; ξ Mω ∞ ⋆λ ∗ B ) ·· ϕ P ( − x ∞ , pr ∗ M ( ω ∞ ⋆ λ ∗ B ) + ρ N + ω ξ ) . (4.4.15)As γ is regular, Theorems 5.1 and 5.2 of [GKM97] imply the following identity Φ M ( γ, ξ ) = L M ( γ ) . (4.4.16)By ( i ), x ∞ is regular, so we may use Formula (4.4.15):Φ M ( γ, ξ ) = L M ( γ ) = ( − dim( A G ) X P ∈P ( M ) δ − / P ( γ ) X ω ∞ ∈ Ω GM ε ( ω ∞ )Tr ( γ − ; ξ Mω ∞ ⋆λ ∗ B ) ·· ϕ P ( − x ∞ , pr ∗ M ( ω ∞ ⋆ λ ∗ B ) + ρ N + ω ξ ) . (4.4.17)Since all contributing γ have x ∞ = H M ∞ ( γ ) lying inside the chamber C again by (i), the set I ( γ )does not depend on x ∞ (but does depend on ( M, ω p , ω q )). Write I = I ( M, ω p , ω q ) for I ( γ ), andΩ G ⋄ M = Ω G ⋄ M ( M, ω p , ω q ) := { ω ∞ ∈ Ω GM | I ( ω ∞ ) = I } , (4.4.18)in terms of (4.4.13). Then (4.4.17) simplifies thanks to (4.4.14): L M ( γ ) = ( − dim( A M /A G ) −|I | X P ∈P ( M ) δ − / P ( γ ) X ω ∞ ∈ Ω G ⋄ M ε ( ω ∞ )Tr ( γ − ; ξ Mω ∞ ⋆λ ∗ B ) . (4.4.19)Statement ( ii ) follows by taking ε ⋄ := ( − dim( A M /A G ) −|I | and using Tr ( γ − ; ξ Mω ∞ ⋆λ ∗ B ) = Tr ( γ, ξ Mλ ω ∞ ). (cid:3) We write L M ( γ ) where the authors of [GKM97] write L νM ( γ ). This is because we only need to use the “middleweight profile”, so there is no need to distinguish in our notation. In their formula the symbol ν P appears, as theyallow more general weight profiles. Since we use the middle weight profile, we have ν P = − ρ N − ω ξ . In [GKM97], they write Φ M ( γ, Θ ξ ∗ ) for Φ M ( γ, ξ ∗ ). Their E corresponds to our ξ ∗ . We continue assuming k > k . We write henceforward c ′ M := ε ⋄ ε ( ω ∞ ) c M . We apply Lemma 4.4.1( ii )to Equation (4.4.7) and change the order of summation to obtain T G ell ,χ ( f ξ,ζ f ∞ , ( k ) ) = T G disc ,χ ( f ξ,ζ f ∞ , ( k ) ) + X M,P,ω p ,ω q ,ω ∞ c ′ M v − X X γ ∈ Γ ell , X ( M ) χ ( I Mγ )Tr ( γ ; ξ Mλ ω ∞ ⊗ ζδ − / P ) O Mγ ( f ∞ , ( k ) M,ω p ,ω q ) | ι M ( γ ) || Stab M X ( γ ) | , (4.4.20)where M, ω p , ω q run over the same sets as before, and P, ω ∞ range over P ( M ) , Ω G ⋄ M , respectively.We apply Lemma 4.1.3 to equalize v − X X γ ∈ Γ ell , X ( M ) χ ( I Mγ )Tr ( γ ; ξ Mλ ω ∞ ⊗ ζδ − / P ) O Mγ ( f ∞ , ( k ) M,ω p ,ω q ) | ι M ( γ ) || Stab M X ( γ ) | = X γ ∈ Γ ell , X ( M ) vol( I Mγ ( Q ) A I Mγ , ∞ \ I Mγ ( A ) / X ) O Mγ ( f λ ω ∞ ,ζδ − / P f ∞ , ( k ) M,ω p ,ω q ) | ι M ( γ ) || Stab M X ( γ ) | , (4.4.21)using that every γ with O Mγ ( f ∞ , ( k ) M,ω p ,ω q ) = 0 in (4.4.21) is regular since f q,M,ω q is supported on regularelements. Define X M := X · A M, ∞ , v M := vol( X Q \ X /A G, ∞ ) − vol( X M, Q \ X M /A M, ∞ ) . Write z Mω ∞ : A M, ∞ → C × for the restriction of the central character of ξ Mλ ω ∞ ⊗ ζδ − / P to A M, ∞ . We observe that z Mω ∞ | A G, ∞ = χ − as the restriction of the central character of ξ Mλ ω ∞ to Z G coincides with the central character of theoriginal representation ξ . Note that we have Z G ( R ) ∩ A M, ∞ = A G, ∞ ⊂ Z M ( R ) and therefore X ∩ A M, ∞ = A G, ∞ . Consequently, there exists a unique character χ Mω ∞ : X M → C × such that χ Mω ∞ | A M, ∞ = ( z Mω ∞ ) − and χ Mω ∞ | X = χ. The pair ( X M , χ Mω ∞ ) is a central character datum for M as in § X M is moreover of theform X M = Y ( A ) A M, ∞ as required by the statement of Theorem 4.2.2. Additionally, observe that f λ ω ∞ ,ζδ − / P f ∞ , ( k ) M,ω p ,ω q ∈ H ( M ( A ) , χ M, − ω ∞ ) . The expression in (4.4.21) can be rewritten as X γ ∈ Γ ell , X M ( M ) v M · vol( I Mγ ( Q ) A I Mγ , ∞ \ I Mγ ( A ) / X M ) O Mγ ( f λ ω ∞ ,ζδ − / P f ∞ , ( k ) M,ω p ,ω q ) | ι M ( γ ) || Stab M X M ( γ ) | = v M · T M ell ,χ Mω ∞ ( f λ ω ∞ ,ζδ − / P f ∞ , ( k ) M,ω p ,ω q ) . (4.4.22)Put c ′′ M := c ′ M v M . Combining (4.4.20) and (4.4.22), we obtain T G ell ,χ ( f ξ,ζ f ∞ , ( k ) ) = T G disc ,χ ( f ξ,ζ f ∞ , ( k ) ) + X M,P,ω p ,ω q ,ω ∞ c ′′ M · T M ell ,χ Mω ∞ ( f λ ω ∞ ,ζδ − / P f ∞ , ( k ) M,ω p ,ω q ) . (4.4.23) OF IGUSA VARIETIES 45
We have ω p ( M ∩ B ) ⊂ B and ω − p ( M ν ∩ B ) ⊂ B since ω p ∈ Ω G Q p M,M ν . In particular, for any root α appearing in Lie( M ∩ N ), ω p α also appears in Lie( M ∩ N ). Consequently, h α, w − p ν i = h w p α, ν i ≥ , and hence w − p ν is dominant for M ∩ B . (See the paragraph above Proposition 4.2.1 for dominanceof ν relative to B .) By Proposition 4.2.1 and the induction hypothesis for M ∈ L < cusp , we have T M ell ,χ Mω ∞ ( f λ ω ∞ ,ζδ / P f ∞ , ( k ) M,ω p ,ω q ) = O (cid:16) p k ( h ρ M ,ω − p ν i + h ( χ Mω ∞ ) ∞ , pr M ( ω − p ν ) i ) (cid:17) . (4.4.24)(To apply the induction hypothesis, we need to ensure that the setup of Theorem 4.2.2 applies to theleft hand side. The point is that the conditions at p and q are satisfied. At p , this is a consequenceof (4.4.6) and Lemma 3.2.1; thus each f ( k ) p,M,ω p is an ascent from an acceptable function. At q , thisfollows from Lemma 3.4.7 (1).) Again from Proposition 4.2.1 for G we have T G disc ,χ ( f ( k ) p f ∞ ,p f ξ,ζ ) = O (cid:16) p k ( h ρ,ν i + h χ ∞ , pr G ν i ) (cid:17) . (4.4.25)Now assume that ( M, P, ω p , ω q , ω ∞ ) contribute to (4.4.23), in particular M ∈ L < cusp , and alsoassume that O Mγ ( f λ ω ∞ ,ζδ / P f ∞ , ( k ) M,ω p ,ω q ) = 0 (4.4.26)for some γ ∈ Γ ell , X M ( M ). Then we claim thatRe( h ρ, ν i + h χ ∞ , pr G ν i ) > Re( h ρ M , ω − p ν i + h ( χ Mω ∞ ) ∞ , pr M ( ω − p ν ) i ) . (4.4.27)This claim, together with (4.4.24) and (4.4.25), tells us that the main term for G dominates theproper Levi terms in (4.4.23), thereby implies the theorem. Thus it is enough to prove the claim.To this end, it is sufficient to show that( a ) h ρ, ν i > h ρ M , ω − p ν i ,( b ) Re h χ ∞ , pr G ν i ≥ Re h ( χ Mω ∞ ) ∞ , pr M ( ω − p ν ) i .Moreover, it is enough to prove that (a) and (b) are true for sufficiently large k (note that the setΩ G ⋄ M and thus ω ∞ depends on k ). Item ( a ) follows from h ρ M , ω − p ν i = h ρ ω p M , ν i < h ρ, ν i . To see “ < ”, first observe that the non-strict inequality “ ≤ ” follows from the fact that ω p is a Kostantrepresentative (cf. (4.4.5)). It remains to check that h ρ ω p M , ν i 6 = h ρ, ν i . This argument is similar to the one in the paragraph below Equation (4.2.2): It suffices to find aroot α in Lie( G ) that is not in Lie( ω p M ) such that h α, ν i 6 = 0. As ν is not central, the argumentfor Lemma 4.5(ii) of [KST] shows that Lie( M ν ) + Lie( M ) = Lie( G ). Hence we can find a root α inLie( G ) which does not occur in either Lie( M ) or Lie( M ν ), i.e., h α, ν i 6 = 0.We now focus on ( b ). We first make the following temporary assumption h pr ∗ M ( − ω ∞ ⋆ λ ∗ B ) − ρ N − ω ξ , pr M ( w − p ν ) i 6 = 0 . (4.4.28)From the equality I ( γ ) = I ( ω ∞ ) we obtain (cf. (4.4.13)) h pr ∗ M ( − ω ∞ ⋆ λ ∗ B ) − ρ N − ω ξ , x i ≤ . As O γ ( f ( k ) p,M,ω p ) = 0, we have x = − k (log p )pr M ( w − p ν ) + ε p , where ε p ∈ pr M supp O a Mωp ( ω − p φ p,M ωp )(cf. (4.4.10)). We obtain h pr ∗ M ( − ω ∞ ⋆ λ ∗ B ) − ρ N − ω ξ , − k (log p )pr M ( w − p ν ) + ε p i ≤ . We may (and do) assume that k is sufficiently large so that k |h pr ∗ M ( − ω ∞ ⋆ λ ∗ B ) − ρ N − ω ξ , − (log p )pr M ( w − p ν ) i| > |h pr ∗ M ( − ω ∞ ⋆ λ ∗ B ) − ρ N − ω ξ , ε ′ p i| for all ε ′ p ∈ pr M supp O a Mωp ( ω − , ′ p φ p,M ωp ) and all ω ′ p ∈ Ω G Q p M,M ν (this amounts to a lower bound on k that depends only on the initial data). We deduce using the above temporary assumption and k ≫ h pr ∗ M ( − ω ∞ ⋆ λ ∗ B ) − ρ N − ω ξ , pr M ( w − p ν ) i ≥ . (4.4.29)Observe that if (4.4.28) is false, then (4.4.29) is still true. Hence (4.4.29) is true unconditionally.We now conclude: h ( χ Mω ∞ ) ∞ , pr M ( w − p ν ) i = −h z Mω ∞ , pr M ( w − p ν ) i = −h pr ∗ M ( λ ω ∞ ) , pr M ( w − p ν ) i − h ζδ − / P , pr M ( w − p ν ) i = −h pr ∗ M ( − w ( ω ∞ ⋆ λ ∗ B )) , pr M ( w − p ν ) i − h ζ − ρ N , pr M ( w − p ν ) i = h pr ∗ M ( ω ∞ ⋆ λ ∗ B ) + ρ N + ω ξ , pr M ( w − p ν ) i | {z } ≤ + h− ω ξ − ζ, pr M ( w − p ν ) i | {z } = h χ ∞ , pr M ( w − p ν ) i (cid:3) Shimura varieties of Hodge type
The goal of this section is to set up the scene for the mod p geometry of Shimura varieties andcentral leaves, paving the way for introducing Igusa varieties in the next section. We pay specialattention to the connected components (synonymous to top-dimensional irreducible components inour setting) and phrase their description in the representation-theoretic language via H .5.1. Connected components in characteristic zero.
From here on, let (
G, X ) be a Shimura datumas in [Del79] satisfying axioms (2.1.1.1), (2.1.1.2), and (2.1.1.3) therein. (We assume that (
G, X ) isof Hodge type starting in the next subsection.) Write E = E ( G, X ) for the reflex field [Del79, 2.2.1],which is a finite extension of Q in C . We have the algebraic closure E ⊂ C . Let K be a neat opencompact subgroup of G ( A ∞ ). We write Sh K = Sh K ( G, X ) for the canonical model over E , whichforms a projective system of quasi-projective varieties with finite ´etale transition maps as K varies.We have the E -scheme Sh := lim ←− K Sh K . Put d := dim Sh K (which does not depend on K ). Write G ( Q ) + for the preimage of G ( R ) + (defined in § G ( Q ). The closure of G ( Q ) + in G ( A ∞ ) isdenoted by G ( Q ) − + .Recall some facts about connected components from [Del79, 2.1]. We have a bijection π (Sh K,E ) ∼ → G ( Q ) \ G ( A ) /G ( R ) + K. (5.1.1)This yields a G ( A ∞ )-equivariant bijection π (Sh E ) ∼ → G ( A ) /G ( Q ) ̺ ( G sc ( A )) G ( R ) + upon takinglimit over all K . Taking quotient by a particular K recovers the above bijection (5.1.1). Note that G ( A ) /G ( Q ) ̺ ( G sc ( A )) G ( R ) + is an abelian group quotient of G ( A ), and G ( Q ) \ G ( A ) /G ( R ) + K is afinite abelian group quotient.Fix a prime ℓ and a field isomorphism ι : Q ℓ ≃ C . When V is a Q ℓ -vector space (possibly witha group or algebra action), write ιV := V ⊗ Q ℓ ,ι C . By convention, all instances of cohomologyin this paper are ´etale cohomology. The description of π (Sh E ) translates into a G ( A ∞ )-moduleisomorphism ιH (Sh E , Q ℓ ) ≃ M π π ∞ , (5.1.2)where the sum runs over one-dimensional automorphic representations π such that π ∞ is trivialwhen restricted to G ( R ) + . Indeed, at each prime p , we have dim π p = 1 since π p factors through OF IGUSA VARIETIES 47 G ( Q p ) → G ( Q p ) ab = G ( Q p ) /̺ ( G sc ( Q p )), cf. Corollary 2.3.3. It is clear that each one-dimensionalautomorphic representation has multiplicity one in the space of automorphic forms.Now fix a prime p = ℓ and an open compact subgroup K p ⊂ G ( Q p ). By taking limit of (5.1.1)over open compact subgroups K p ⊂ G ( A ∞ ,p ), writing Sh K p := lim ←− K p Sh K p K p , π (Sh K p ,E ) ∼ → G ( Q ) − + \ G ( A ∞ ) /K p . (5.1.3)We have a G ( A ∞ ,p )-module H i (Sh K p ,E , Q ℓ ) = lim −→ K p H i (Sh K p K p ,E , Q ℓ ) , i ≥ , where K p runs over sufficiently small open compact subgroups of G ( A ∞ ,p ). The degree zero partadmits an automorphic description similar to (5.1.2). Lemma 5.1.1.
There is a G ( A ∞ ,p ) -module isomorphism ιH (Sh K p ,E , Q ℓ ) ≃ M π π ∞ ,p , where the sum runs over discrete automorphic representations π of G ( A ) such that (i) dim π = 1 ,(ii) π p is trivial on K p , and (iii) π ∞ is trivial on G ( R ) + .Proof. This is clear from (5.1.2) by taking K p -invariants. (cid:3) Integral canonical models.
From now on, assume that (
G, X ) is a Shimura datum of
Hodgetype . This means that there is an embedding into the Siegel Shimura datum i V,ψ : (
G, X ) ֒ → (GSp( V, ψ ) , S ± V,ψ ) , where ( V, ψ ) is a symplectic space over Q , and S ± V,ψ denotes the associated Siegel half spaces. Forsimplicity we write GSp = GSp(
V, ψ ) and S ± = S ± V,ψ ).We fix a prime p as earlier. To explain integral canonical models for Sh = Sh( G, X ) at p , we setthings up following [Kis17, (1.3.3)], leaving the details to loc. cit . For the rest of this paper, weassume that G is unramified over Q p and fix a reductive integral model G Z ( p ) over Z ( p ) . We stillwrite G for this model if there is no danger of confusion. Thereby we have a fixed hyperspecialsubgroup K p := G ( Z p ). We refer to this setup by (Unr( G, p, K p )). • (Unr ( G, p, K p ) ) : G is unramified over Q p with a fixed reductive integral model over Z ( p ) ,and K p = G ( Z p ).We may assume that i V,ψ is induced by an embedding G Z ( p ) ֒ → GL( V Z ( p ) ) for a Z ( p ) -lattice V Z ( p ) ⊂ V and that ψ induces a perfect pairing on V Z ( p ) . There exists a finite set of tensors ( s α ) ⊂ V ⊗ Z ( p ) such that G Z ( p ) is the scheme-theoretic stabilizer of ( s α ) in GL( V Z ( p ) ). We may assume that one ofthe tensors is given by ψ ⊗ ψ ∨ ∈ ( V ∨ Z ( p ) ) ⊗ ⊗ V ⊗ Z ( p ) , whose stabilizer is GSp( V Z ( p ) , ψ ). We fix the set( s α ). There is a hyperspecial subgroup K ′ p ⊂ GSp(
V, ψ )( Q p ) extending K p (i.e., K ′ p ∩ G ( Q p ) = K p )such that i V,ψ induces an embedding of Shimura varieties over the reflex field E (so that the mapis induced by i V,ψ : X → S ± V,ψ on C -points)Sh K p ( G, X ) ֒ → Sh(GSp , S ± ) ⊗ Q E. (5.2.1)It is implied by (Unr( G, p, K p )) that p is unramified in E . We fix an isomorphism ι p : C ≃ Q p ,determining an embedding E ֒ → Q p as well as a p -adic place p of E . Thus we may identify E p ≃ Q p . See [Sta16, Tag 03Q4] for the canonical isomorphism, which is G ( A ∞ ,p )-equivariant by a routine check. Alterna-tively, it is harmless to think of the identity as a definition for the left hand side. This way the weak polarization in the sense of [Kis17] is remembered by a geometric incarnation of ( s α ). So weneed not keep track of polarizations on abelian varieties separately. The integer ring O E localized at p is denoted by O E, ( p ) , and its residue field by k ( p ). Identify theresidue field of Q p with F p , thus fixing an embedding k ( p ) ֒ → F p .Kisin [Kis10, Thm. 2.3.8] (for p >
2) and Kim–Madapusi Pera [KMP16, Thm. 4.11] (for p =2) constructed integral canonical models, namely a projective system of smooth quasi-projectiveschemes S K p K p over O E, ( p ) for all sufficiently small open compact subgroups K p ⊂ G ( A ∞ ,p ) withfinite ´etale transition maps S K p K p, ′ → S K p K p for K p, ′ ⊂ K p . The projective system is equippedwith an action of G ( A ∞ ,p ), given by the isomorphism S K p K p ∼ → S K p g − K p g , g ∈ G ( A ∞ ,p ) , K p ⊂ G ( A ∞ ,p ) , extending the isomorphism Sh K p K p ∼ → Sh K p g − K p g giving the action of g on the generic fiber. Theinverse limit S K p := lim ←− K p S K p K p is a scheme over O E, ( p ) with a G ( A ∞ ,p )-action, characterizeduniquely by an extension property [Kis10, Thm. (2.3.8), (2)]. The construction yields a map of O E, ( p ) -schemes S K p → S K ′ p (GSp , S ± ) ⊗ Z ( p ) O E, ( p ) , (5.2.2)whose fiber over E is identified with (5.2.1), where S K ′ p (GSp , S ± ) is the integral model over Z ( p ) for Sh(GSp( V, ψ ) , S ± V,ψ ) parametrizing polarized abelian schemes up to prime-to- p isogenies withprime-to- p level structure, as in [Kis10, (2.3.3)]. Moreover we have universal polarized abelianschemes h : A K p K p → S K p K p compatible with the transition maps in the projective system.Let S K p K p ,k ( p ) := S K p K p ⊗ O E, ( p ) k ( p ) denote the special fiber. Write Sh K p (resp. S K p ,k ( p ) ) forthe inverse limit of Sh K p K p (resp. S K p K p ,k ( p ) ) over K p . By base change to E p , O E p , and k ( p ),respectively, we obtain Sh K p ,E p , S K p , O E p , and S K p ,k ( p ) from Sh K p , S K p , and S K p ,k ( p ) . There arecanonical G ( A ∞ ,p )-equivariant embeddings of generic and special fibersSh K p ,E p ֒ → S K p , O E p ← ֓ S K p ,k ( p ) . These embeddings induce G ( A ∞ ,p )-equivariant bijections by means of arithmetic compactificationas shown in [MP19, Cor. 4.1.11]: π (Sh K p ,E p ) ∼ → π ( S K p , O E p ) ∼ ← π ( S K p ,k ( p ) ) . Proposition 5.2.1.
The G ( A ∞ ,p ) -action is transitive on π (Sh K p ,E p ) and π ( S K p ,k ( p ) ) .Proof. By the preceding proposition, it is enough to check the transitivity on π (Sh K p ,E p ), whichis [Kis10, Lem. 2.2.5] (using the fact that K p is hyperspecial). (cid:3) Let T be a k ( p )-scheme. At each point x ∈ S K p K p ( T ) we have an abelian variety A x over T (up toa prime-to- p isogeny) pulled back from A K p K p . As in [Kis17, (1.3.6)], we have ( s α,ℓ ) ⊂ ( R h ´et ∗ Q ℓ ) ⊗ for each prime ℓ = p . By pullback, we equip the prime-to- p rational Tate module V p ( A x ) of A x with ( s α,ℓ,x ) ℓ = p .When T = Spec k with k/k ( p ) an extension in k ( p ), write D ( A x [ p ∞ ]) for the (integral) Dieudonn´emodule of A x [ p ∞ ], and Φ x for the Frobenius operator acting on it. Following [Kis17, (1.3.10)] wehave crystalline Tate tensors ( s α, ,x ) ⊂ D ( A x [ p ∞ ]) ⊗ coming from ( s α ). Lovering [Lov17], and alsoHamacher [Ham19, § s α, ,x ). Namely there exist crystalline Tate tensors ( s α, )on the Dieudonn´e crystal D ( A K p K p [ p ∞ ]) associated with A K p K p [ p ∞ ] over S K p K p ,k ( p ) such that( s α, ) specializes to ( s α, ,x ) at every x ∈ S K p K p ( k ( p )). Alternatively, this also follows from the weak approximation theorem, which tells us that G ( Q ) ֒ → G ( Q p ) × G ( R )has dense image. For this, apply [PR94, Thm. 7.7] and notice that the set S of the theorem can be taken away from p and ∞ from the discussion in § loc. cit. since G is unramified at p . OF IGUSA VARIETIES 49
Central leaves.
A central leaf in the special fiber of a Shimura variety of Hodge type is the locusof points where the corresponding abelian varieties have p -divisible groups (with extra structure)in a fixed isomorphism class. This notion has been introduced by [Oor04, Man05, Ham19, Zha]. Seealso [HK19].Let B ( G Q p ) denote the set of ( G ( ˘ Q p ) , σ )-conjugacy classes in G ( ˘ Q p ). Fix a Borel subgroup B ⊂ G Z p (here G Z p comes from the fixed model over Z ( p ) ) and a maximal torus T ⊂ B . We havethe set of dominant coweights X ∗ ( T Q p ) + and X ∗ ( T Q p ) + Q . Via the fixed isomorphism ι p : Q p ≃ C ,we obtain T C ⊂ B C ⊂ G C as well as X ∗ ( T C ) + and X ∗ ( T C ) + Q . Since the conjugacy class { µ X } isdefined over E and since G Q p is quasi-split, we have a cocharacter µ p ∈ X ∗ ( T Q p ) + defined over E p in the conjugacy class { ι p µ X } . When there is no danger of confusion, we omit the subscripts Q p and C . Write ρ ∈ X ∗ ( T ) Q for the half sum of all positive roots, and h· , ·i for the canonical pairing X ∗ ( T ) Q × X ∗ ( T ) Q → Q or its extension to C -coefficients.Each b ∈ G ( ˘ Q p ) gives rise to a Newton cocharacter ν b : D → G ˘ Q p (so it is a “fractional” cocharacterof G ˘ Q p ) and a connected reductive group J b over Q p given by J b ( R ) := { g ∈ G ( R ⊗ Q p ˘ Q p ) : g − bσ ( g ) = b } , R : Q p -algebra . (5.3.1) Lemma 5.3.1.
The Newton cocharacter ν b factors through the center of J b . The induced cocharacter D → A J b is Q p -rational.Proof. The centrality follows from [Kot85, (4.4.2)]. The cocharacter D → A J b is σ -invariant by thedefinition of J b , thus Q p -rational. (cid:3) We define an open compact subgroup of J b ( Q p ) (where “int” stands for integral): J int b := J b ( Q p ) ∩ G (˘ Z p ) = { g ∈ G (˘ Z p ) : g − bσ ( g ) = b } . Given b ∈ G ( ˘ Q p ), we denote its ( G ( ˘ Q p ) , σ )-conjugacy class by [ b ] and ( G (˘ Z p ) , σ )-conjugacy classby [[ b ]]. Recall that b ∈ G ( ˘ Q p ), or [ b ] ∈ B ( G Q p ), is basic if ν b : D → G ˘ Q p has image in Z ( G ˘ Q p ), orequivalently if J b is an inner form of G [RR96, Prop. 1.12]. The following condition will appear inour irreducibility results later. The definition makes a difference only when G ad is not Q -simple.See Lemma 5.3.7 below for a relation to § Definition 5.3.2.
Let G ad = Q i ∈ I G ad i be a decomposition into Q -simple factors. An element b ∈ G ( ˘ Q p ), or [ b ] ∈ B ( G Q p ), is said to be Q -non-basic if its image in B ( G i, Q p ) via the natural compositemap G → G ad → G i is non-basic for every i ∈ I . Remark . The definition is not purely local in that it depends on not only G Q p but also G .Compare G = GL × GL with G = Res F/ Q GL , where F is a real quadratic field in which p splits.Let x : Spec k ֒ → S K p K p ,k ( p ) be a geometric point supported at x ∈ S K p K p ,k ( p ) , with k/k ( p )an algebraically closed field extension. Write W := W ( k ) and L := Frac W (so that W = ˘ Z p and L = ˘ Q p if k = F p ). We will still write σ for the canonical Frobenius on W . There exists a W -linearisomorphism V ∗ Z ( p ) ⊗ Z ( p ) W ≃ D ( A x [ p ∞ ]) (5.3.2)carrying ( s α ) to ( s α, ,x ). We transport the Frobenius operator Φ x on the right hand side to b x (1 ⊗ σ )on the left hand side to define b x ∈ G ( L ). Then [[ b x ]] (thus also [ b x ]) is independent of the choiceof isomorphism.To define the central leaf associated with b , consider the set C b,K p := { x ∈ S K p K p ,k ( p ) : ∃ isom. of crystals V ∗ Z ( p ) ⊗ Z ( p ) W ≃ D ( A x [ p ∞ ]) s.t. ( s α ) ( s α, ,x ) } , where the isomorphism of crystals means that the σ -linear map b (1 ⊗ σ ) is carried to Φ x via the W -linear isomorphism. The definition of C b,K p depends only on [[ b ]] since [[ b ]] determines the crystal V ∗ Z ( p ) ⊗ Z ( p ) ˘ Z p with Frobenius operator b (1 ⊗ σ ) and crystalline Tate tensors ( s α ).By [Ham17, Prop. 2, p.1262] and since Newton strata are locally closed ([RR96, Thm. 3.6]), C b,K p is a locally closed subset of S K p K p ,k ( p ) . (The analogous assertion for Kisin–Pappas models is shownin [HK19, Cor. 4.12].) We promote C b,K p to a locally closed k ( p )-subscheme of S K p K p ,k ( p ) equippedwith reduced subscheme structure. We still write C b,K p for the scheme and call it the central leaf associated with b . The projection maps between S K p K p ,k ( p ) as K p varies, which are finite ´etale([Kis10, Thm. 2.3.8]), induce finite ´etale projection maps between C b,K p . Put C b := lim ←− K p C b,K p .The following proposition is due to C. Zhang and Hamacher [Zha, Ham19] independently. Proposition 5.3.4.
The k ( p ) -scheme C b,K p is smooth. If nonempty, its dimension is h ρ, ν b i .Proof. These properties can be checked after extending base to k ( p ). Since C b,K p is reduced, it isstill reduced over k ( p ). Thus the proposition follows from [Ham19, Prop. 2.6]. (cid:3) A finite subset B ( G Q p , µ − p ) ⊂ B ( G Q p ) has been defined in [Kot97, §
6] by a group-theoreticgeneralization of Mazur’s inequality. The set B ( G Q p , µ − p ) contains exactly one basic element, butmay contain several elements that are not Q -non-basic. The significance of B ( G Q p , µ − p ) is clearfrom the following. Proposition 5.3.5.
The central leaf C b,K p is nonempty if and only if [ b ] ∈ B ( G Q p , µ − p ) .Proof. It follows from [KMPS, Prop. 1.3.9] that the Newton stratum for b is nonempty if and onlyif [ b ] ∈ B ( G Q p , µ − p ). Thus the “only if” part of the proposition holds. The “if” part follows fromthis via [HK19, Rem. 5.6], noting that Axiom A there is proven by Kisin; see the paragraph aboveRemark 5.6 therein. (cid:3) Henceforth we always assume that [ b ] ∈ B ( G Q p , µ − p ) in light of the proposition. Other than this,we have not imposed conditions on b ∈ G ( ˘ Q p ) but now we replace b with a better representativein its ( G (˘ Z p ) , σ )-conjugacy class (so that C b,K p does not change). Since C b,K p is a variety of finitetype over k ( p ), there exists x ∈ C b,K p ( F p r ) (with F p r ⊃ k ( p )); in particular [[ b x ]] = [[ b ]]. We mayincrease r to be divisible by [ E p : Q p ] such that σ r ( µ p ) = µ p and that ( s α, ,x ) are all defined over Q p r . As in [Kis17, (1.4.1)], we have a Z p r -linear isomorphism (5.3.2) V ∗ Z ( p ) ≃ D ( A x [ p ∞ ]) (5.3.3)carrying ( s α ) to ( s α, ,x ). We fix such an isomorphism. Let Φ x denote the (absolute) Frobeniusoperator on the right hand side. As before, Φ x is transported to b x (1 ⊗ σ ) so that b x ∈ G ( Z p r ) σµ p ( p ) − G ( Z p r )by [Kis17, (1.4.1)]. We see from [Kis17, (1.4.4)] that b x σ ( b x ) · · · σ r − ( b x ) σ r acting on the left handside of (5.3.3) is transported to the geometric p r -Frobenius on the right hand side. The latter isa semisimple Q p r -linear map by Tate’s theorem. After making r further divisible, we may assumethat the geometric p r -Frobenius is diagonalizable and that rν b x : G m → G ˘ Q p is a cocharacter (asopposed to a fractional cocharacter). Since b x ∈ G ( Q p r ), [Kot85, (4.4.1)] tells us that ν b x is definedover Q p r . In particular the slope decomposition of D ( A x [ p ∞ ]) ⊗ Z pr Q p r is defined over Q p r . Themap ιrν b x : G m → GSp ˘ Q p maps p to the scalar p rλ on each slope λ -component (with rλ ∈ Z ); so infact rν b x is defined over Q p r . The matching of the p r -Frobenius now gives b x σ ( b x ) · · · σ r − ( b x ) = rν b x ( p )(This is called the decency equation, cf. [RZ96, Def. 1.8].) OF IGUSA VARIETIES 51
In summary, replacing b with b x and making r more divisible as above, we may and will assumefrom now on that(br1) F p r ⊃ k ( p ) and C b,K p ( F p r ) = ∅ ,(br2) b ∈ G ( Z p r ) σµ p ( p ) − G ( Z p r ),(br3) bσ ( b ) · · · σ r − ( b ) = rν b ( p ).These conditions have the following implications. (So the running assumption that [ b ] ∈ B ( G Q p , µ − p )is subsumed by (br2).)(br1)’ µ p is defined over Q p r , by (br1).(br2)’ [ b ] ∈ B ( G Q p , µ − p ) and ν b is defined over Q p r , by (br2).Since µ p is defined over E p , which is unramified over Q p , (br1)’ is easy to see. In (br2)’, [ b ] ∈ B ( G Q p , µ − p ) comes from [RR96, Thm. 4.2]; we already explained above that ν b is defined over Q p r if b ∈ G ( Q p r ).Since the G ( Q p r )-conjugacy class of rν b is defined over Q p [Kot85, (4.4.3)], and since G Q p isquasi-split, there exists h ∈ G ( Q p r ) such that h − ( rν b ) h is defined over Q p . Multiplying h on theright by an element of G ( Q p ), we can ensure that h − ( rν b ) h factors through G m → T and is B -dominant, namely h − ( rν b ) h ∈ X ∗ ( T ) + . Fix such a h and put b ◦ := h − bσ ( h ) so that ν b ◦ = h − ( ν b ) h from [Kot85, (4.4.2)]. We also have a Q p -isomorphism J b ∼ → J b ◦ , g h − gh determined by h , which carries rν b to rν b ◦ .Starting from ν b ◦ ∈ X ∗ ( T ) + Q defined over Q p as above, we put P b ◦ := P ν b ◦ in the notation of § P op b ◦ , N b ◦ , N op b ◦ , and M b ◦ . In particular P op b ◦ (resp. M b ◦ ) is a standard Q p -rational parabolic (resp. Levi) subgroup of G Q p , and M b ◦ is the centralizer of ν b ◦ in G Q p . Thereis an inner twist [RZ96, Cor. 1.14] J b ◦ ⊗ Q p Q p r ≃ M b ◦ ⊗ Q p Q p r given by the cocycle Gal( Q p n / Q p ) → M b ◦ ( Q p r ), σ b ◦ . Thus M b ◦ is also an inner twist of J b over Q p (which is independent of the choice of b ◦ up to isomorphism of inner twists by routine check).Under the canonical Q p -isomorphisms Z ( M b ◦ ) ≃ Z ( J b ) and A M b ◦ = A J b , it is readily checked that ν b ◦ is carried to ν b . Example . We have the following for the ordinary strata of modular curves, when G Q p = GL .Take B and T to the subgroup of upper triangular (resp. diagonal) matrices. Then µ is thecocharacter z diag( z,
1) up to conjugacy. We can take b = b ◦ such that ν b ( z ) = diag(1 , z − ),which is visibly B -dominant. Then P op b = B = P − ν b , M b = T , and δ P b ( ν b ( p )) = | p − | = p . Lemma 5.3.7.
The element b ∈ G ( ˘ Q p ) as above is Q -non-basic if and only if ( Q -nb( P b )) of § Write G ad = Q i ∈ I G ad i as in Definition 5.3.2 and b i ∈ G ad i ( ˘ Q p ) for the image of b . Byfunctoriality of Newton cocharacters, the composition of ν b with the natural map G → G ad i is ν b i ,which is Q p -rational since ν b is. This implies that the image of P b in G ad i is P b i , where P b i ⊂ G ad i isdefined analogously as P b in G over Q p . Each b i ∈ G ad i ( ˘ Q p ) is basic if and only if ν b i is central in G ad i (i.e. trivial) if and only if P b i = G ad i . Therefore ( Q -nb( P b )) holds if and only if b i is non-basicfor every i ∈ I , and the latter is the definition for b to be Q -non-basic. (cid:3) Lemma 5.3.8.
Let → Z → G → G → be a z -extension over Q p . Let µ p : G m → G be acocharacter over Q p r as above, and suppose that b and r satisfy (br1)–(br3). Then there exist thefollowing data: • a cocharacter µ p, : G m → G over Q p r lifting µ together with b ∈ G ( Q p r ) lifting b suchthat b ∈ G ( Z p r ) σµ p, ( p ) − G ( Z p r ) (thus the analogues of (br1)’, (br2), and (br2)’ for G hold true), • b ◦ ∈ G ( Q p r ) in the σ -conjugacy class of b such that ν b ◦ is defined over Q p and lifts ν ◦ b .Moreover, we can make r more divisible, while keeping the same µ p , b , µ p, , and b ◦ , such that rν b is a cocharacter of G . (Thus the analogues of (br1)’, (br2), and (br2)’ for G continue to holdfor the new r .)Proof. By the proof of Lemma 3.6.1, there exists µ p, over Q p r lifting µ ; we fix any µ p, . Since G ( Z p r ) → G ( Z p r ) is onto (by the surjectivity on F p r -points and the smoothness of G → G ), themap G ( Q p r ) → G ( Q p r ) induces a surjection G ( Z p r ) σµ p, ( p ) − G ( Z p r ) ։ G ( Z p r ) σµ ( p ) − G ( Z p r ) . Take b ∈ G ( Q p r ) to be any preimage of b under this map. This takes care of the first point inthe lemma. As for the second point, since the G is quasi-split over Q p , there exists b ◦ ∈ G ( Q p r ) σ -conjugate to b such that ν b ◦ is defined over Q p , and also such that ν b ◦ factors through T ⊂ G ,as explained for ν b ◦ above. Then the composite of ν b ◦ with G ։ G is conjugate to ν b ◦ in G , so oneis carried to the other by an element of the Q p -rational Weyl group of G [Kot84a, Lem. 1.1.3 (a)].Identifying the latter group with the Q p -rational Weyl group of G , we can use the same elementto modify ν b ◦ so that ν b ◦ maps to ν b ◦ under G ։ G . Finally, the last point on r in the lemma istrivial. (cid:3) In the setup of the lemma, we introduce Q p -algebraic groups J b , J b ◦ , P b ◦ , M b ◦ , etc. for G bymimicking the definition for G . Let T , B denote the preimages of T, B in G . Since ν b ◦ mapsto ν b ◦ , it is clear that ν ◦ b ∈ X ∗ ( T ) + , where + means B -dominance, and that P b ◦ , M b ◦ mapto P b ◦ , M b ◦ . As before, we can identify Z ( M b ◦ ) = Z ( J b ◦ ), which carries ν b ◦ to ν b . With thisunderstanding, we will abuse notation to write M b , P b , M b , P b etc. for M b ◦ , P b ◦ , M b ◦ , P b ◦ etc. tosimplify notation, and write ν b , ν b for ν b ◦ , ν b ◦ if there is little danger of confusion.5.4. The Hecke orbit conjecture.
We state Oort’s Hecke orbit conjecture for Shimura varietiesof Hodge type with hyperspecial level at p , breaking it down into discrete and continuous partsfollowing [Cha06]. The reader is reminded of the setup (Unr( G, p, K p )) that G Q p is unramified andthat K p ⊂ G ( Q p ) is hyperspecial.Let x ∈ S K p K p ,k ( p ) ( F p ). Denote by ˜ x ⊂ | S K p ,k ( p ) | the preimage of x in the topological space | S K p ,k ( p ) | via the projection map S K p ,k ( p ) → S K p K p ,k ( p ) . Define the prime-to- p Hecke orbit H ( x ) := ˜ x · G ( A ∞ ,p ) ⊂ | S K p ,k ( p ) | . Write H K p ( x ) for the image of H ( x ) in | S K p K p ,k ( p ) | . By C K p ( x ) we mean the central leaf through x , namely C A x [ p ∞ ] ,K p . Since the action of G ( A ∞ ,p ) does not interfere with the isomorphism classof A x [ p ∞ ] with tensors ( s α, ,x ), we see that H K p ( x ) ⊂ | C K p ( x ) | . One of the main concerns of this paper is the following conjecture.
Conjecture 5.4.1 (Hecke Orbit Conjecture) . Let x ∈ S K p K p ,k ( p ) ( F p ) such that [ b x ] is Q -non-basic.Then the subset H K p ( x ) enjoys the following properties. (HO) H K p ( x ) is Zariski dense in the central leaf C K p ( x ) . (HO disc ) H K p ( x ) meets every irreducible component of C K p ( x ) . (HO cont ) The Zariski closure of H K p ( x ) in C K p ( x ) is a union of irreducible components of C K p ( x ) . A priori we only know that rν b is a fractional cocharacter, even though rν b is an (integral) cocharacter of G . OF IGUSA VARIETIES 53
Remark . The hypothesis on [ b x ] cannot be weaken to only requiring that [ b x ] be basic. Forexample, for Shimura varieties arising from ( G × · · · × G, X × · · · × X ), with ( G, X ) a Shimuradatum, we see the necessity to assume [ b x ] to be basic in every copy of G (which is a Q -factor).We introduce a conjecture closely related to (HO disc ). If true, the conjecture would yield acanonical G ( A ∞ ,p )-equivariant bijection π ( C ( x )) ∼ → π (cid:16) S K p ,k ( p ) (cid:17) via limit over K p . Conjecture 5.4.3 (HO ′ disc ) . For every x ∈ S K p K p ( k ( p )) such that [ b x ] is Q -non-basic, the immersion C K p ( x ) ֒ → S K p K p ,k ( p ) induces a bijection π ( C K p ( x )) ∼ → π ( S K p K p ,k ( p ) ) . The logical relationship between the conjectures is as follows. Obviously (HO disc ) and (HO cont )together yield (HO). We will see shortly in the proof of Corollary 5.4.5 that (HO disc ) follows from(HO ′ disc ). (HO) + (HO disc ) + (HO cont ) k k (HO ′ disc ) k s A representation-theoretic reformulation of (HO ′ disc ) is the following theorem, to be implied bythe main theorem of this paper on Igusa varieties. Recall µ p from § Theorem 5.4.4.
Let b be Q -non-basic with [ b ] ∈ B ( G Q p , µ − p ) . As G ( A ∞ ,p ) -modules, H ( C b , Q ℓ ) ≃ H (Sh K p , Q ℓ ) . Proof.
See § (cid:3) Theorem 5.4.4 ensures the existence of an isomorphism between the two G ( A ∞ ,p )-modules. Thisis easily promoted to the assertion that the natural map between them induced by the immersions C b,K p → Sh K p K p (as K p varies) is an isomorphism, as the next corollary shows. Corollary 5.4.5.
Conjectures (HO ′ disc ) and (HO disc ) are true for Q -non-basic b with [ b ] ∈ B ( G Q p , µ − p ) .Proof. Let us prove (HO ′ disc ). The immersions C K p ( x ) ֒ → S K p K p ,k ( p ) as K p varies induce a G ( A ∞ ,p )-equivariant map π ( C ( x )) → π ( S K p ,k ( p ) ) , which is a surjection since G ( A ∞ ,p ) acts transitively on the target (Proposition 5.2.1). Given this,the injectivity follows from the isomorphism of Theorem 5.4.4.Next we verify (HO disc ). If [ b x ] is Q -non-basic, then the G ( A ∞ ,p )-action is transitive on π ( C ( x ))by Proposition 5.2.1 and (HO ′ disc ). Since H ( x ) is invariant under the G ( A ∞ ,p )-action, we deduce(HO disc ). (cid:3) Thanks to Xiao’s work on (HO cont ), we verify some special cases of Conjecture (HO). Just for thenext corollary, suppose that (
G, X ) comes from a PEL datum ( B, O B , ∗ , V, h , i , h ) of type A or C asin loc. cit. or [Kot92b]. Write F for the center of B (which is a field), and F for the fixed field of ∗ in F . We say that x ∈ S K p K p ( k ( p )) is B - hypersymmetric if End B ( A x ) ⊗ Q Q p ≃ End B ⊗ Q p ( A x [ p ∞ ])via the natural map, where End means the space of morphisms in the corresponding isogenycategories. Corollary 5.4.6.
In the setup (Unr(
G, p, K p )) , assume that ( G, X ) comes from a PEL datum of typeA or C as above, and that every prime of F above p is inert in F . (The last condition is vacuousfor type C.) If x ∈ S K p K p ( k ( p )) lies in the connected component of a non-basic Newton stratumsuch that the component contains a B -hypersymmetric point, then Conjecture (HO) is true for x ,i.e., H K p ( x ) is Zariski dense in C K p ( x ) . Proof.
This corollary follows immediately from the preceding one and [Xia, Thm. 7.1], as observedin Theorem 1.8 of loc. cit. (cid:3)
Remark . For simplicity, we used only case (ii) in part (2) of [Xia, Thm. 7.1]. Case (i) thereinextends the above corollary to another case for PEL type A when p splits completely between F and F . 6. Igusa varieties
In this section the proof of the main theorem is reduced to a statement about the cohomologyof Igusa varieties. We will also see that it suffices to consider the completely slope divisible case,when the tower of Igusa varieties admits models over a fixed finite field. This allows us to applyFujiwara’s formula to compute the cohomology in the next section.6.1.
Infinite-level Igusa varieties.
We continue in the setup of § b ∈ G ( ˘ Q p ) satisfying(br1)–(br3). Let b ′ ∈ GSp( ˘ Q p ) denote the image of b . By Dieudonn´e theory, we have a polarized p -divisible group Σ b ′ over F p r such that D (Σ b ′ ) = V ∗ Z ( p ) ⊗ Z ( p ) Z p r with Frobenius operator b ′ (1 ⊗ σ ).By Σ b we mean the p -divisible group Σ b ′ equipped with crystalline Tate tensors ( t α ) on D (Σ b ′ )corresponding to ( s α ) on V Z ( p ) . When there is no danger of confusion, we still write Σ b and Σ b ′ fortheir base changes to F p .Applying the construction of § S K ′ p K ′ ,p (GSp , S ± ) and b ′ , we obtain a central leaf C b ′ ,K ′ ,p ⊂ S K ′ p K ′ ,p (GSp , S ± ). Let R be an F p -algebra. Following [CS17, Sect. 4.3] we have the Igusa variety Ig b ′ ,K ′ ,p → C b ′ ,K ′ ,p , F p whose R -points parametrize isomorphismsΣ b ′ × F p R ≃ A R [ p ∞ ] (6.1.1)compatible with polarizations up to Z × ( p ) -multiples, where A R denotes the pullback of the uni-versal abelian scheme via Spec R → C b ′ ,K ′ ,p F p . Then Ig b ′ ,K ′ ,p is a perfect scheme, which is anAut(Σ b ′ )-torsor over C b ′ ,K ′ ,p F p by [CS, Cor. 2.3.2], where Aut(Σ b ′ ) denotes the group scheme ofautomorphisms of Σ b ′ (preserving the polarization up to Z × p -multiples).The map S K p K p , F p → S K ′ p K ′ ,p , F p clearly induces a map C b,K p , F p → C b ′ ,K ′ ,p , F p . We define thesubscheme Ig b,K p ⊂ ( Ig b ′ ,K ′ ,p × C b ′ ,K ′ ,p, F p C b,K p , F p ) perf = Ig b ′ ,K ′ ,p × C perf b ′ ,K ′ ,p , F p C perf b,K p , F p (6.1.2)to be the locus where (6.1.1) carries ( s α ) to ( s α, ) on the Dieudonn´e modules. Composing withprojection maps, we have F p -morphisms Ig b,K p → Ig b ′ ,K ′ ,p and Ig b,K p → C perf b,K p , F p . The latter givesrise to the composite map Ig b,K p → C perf b,K p , F p → C b,K p , F p → S K p K p , F p . The Hecke action of G ( A ∞ ,p ) on S K p K p , F p as K p varies is similarly defined on the tower of C b,K p , F p (resp. Ig b,K p ). Lemma 6.1.1.
The following are true.(1) The F p -scheme Ig b,K p is perfect and a pro-´etale J int b -torsor over C perf b,K p , F p . (2) The map Ig b,K p → Ig b ′ ,K ′ ,p is a closed embedding, under which the J b ′ ( Q p ) -action on Ig b ′ ,K ′ ,p restricts to an action of J b ( Q p ) on Ig b,K p (via the embedding J b ( Q p ) ֒ → J b ′ ( Q p ) ). It can be shown that Ig b,K p → C b,K p is an Aut(Σ b )-torsor by [CS, Cor. 2.3.2] and adapting the argument there,but we do not need it. OF IGUSA VARIETIES 55
Proof.
This follows from [Ham19, Prop. 4.1, 4.10], noting that our Ig b,K p is his J ( p −∞ ) ∞ (theperfection of his J ∞ ) and that our J int b is his Γ b . Two points require some further explanation.Firstly, we see that J int b = Γ b as follows. Observe that J int b ⊂ J b ( Q p ) ⊂ J b ′ ( Q p ) and J int b ⊂ G (˘ Z p ) ⊂ GSp(˘ Z p ). Thus J int b consists of automorphisms of Σ b ′ which are exactly the stabilizersof ( t α ) via Dieudonn´e theory. Secondly, [Ham19, Prop. 4.1] tells us that J ∞ → C b,K p , F p is apro-´etale J int b -torsor. Since every perfection map (as a limit of absolute Frobenius) is a universalhomeomorphism, which preserves the pro-´etale topology [BS15, Lem. 5.4.2], it follows that theperfection J ( p −∞ ) ∞ → C perf b,K p , F p is also a pro-´etale J int b -torsor. (cid:3) Lemma 6.1.2.
Let R be a perfect F p -algebra. Then Ig b,K p ( R ) is identified with the set of equivalenceclasses of ( x, j ) , where • x ∈ S K p K p ( R ) is an abelian scheme over Spec R and • j : Σ b × F p R → A x [ p ∞ ] is a quasi-isogeny carrying ( s α ) to ( s α, ,x ) ,and A x denotes the pullback of the universal abelian scheme along x . Here ( x, j ) and ( x ′ , j ′ ) areconsidered equivalent if, in the notation of § p -power isogeny i : A x → A x ′ carrying ( s α,ℓ,x ) ℓ = p to ( s α,ℓ,x ′ ) ℓ = p and ( s α, ,x ) to ( s α, ,x ′ ) such that i ◦ j = j ′ . Each ρ ∈ J b ( Q p ) acts on the R -points of Ig b,K p by sending j to j ◦ ρ . Proof.
This is the Hodge-type analogue of [CS17, Lem. 4.3.4] proven in the PEL case. By loc. cit. , Ig b ′ ,K ′ ,p ( R ) is the set of p -power isogeny classes of ( A, j ) with A ∈ S K ′ p K ′ ,p ( R ) and j : Σ b × F p R → A [ p ∞ ] a quasi-isogeny compatible with polarizations up to Z × p . Now we have a commutative diagramfrom the construction of central leaves and Igusa varieties: Ig b,K p / / (cid:127) _ closed (cid:15) (cid:15) C b,K p , F p (cid:31) (cid:127) loc . closed / / (cid:15) (cid:15) S K p K p , F p (cid:15) (cid:15) Ig b ′ ,K ′ ,p / / C b ′ ,K ′ ,p , F p (cid:31) (cid:127) loc . closed / / S K ′ p K ′ ,p , F p Now we prove the first assertion by constructing the maps in both directions, which are easily seento be inverses of each other. Given y ∈ Ig b,K p ( R ), its image gives x ∈ S K p K p ( R ). The j comesfrom the image of y in Ig b ′ ,K ′ ,p ( R ). The compatibility of j with crystalline Tate tensors followsfrom the very definition of Ig b,K p . Conversely, let ( x, j ) be as in the lemma. Modifying by a quasi-isogeny, we may assume that j is an isomorphism. Then ( A, j ) comes from a point y ′ ∈ Ig b ′ ,K ′ ,p ( R )as observed above. Since Spec R and C b,K p are reduced, x ∈ S K p K p ( R ) comes from a point in x ∈ C b,K p ( R ). Then y ′ and x have the same image in C b ′ ,K ′ ,p , F p ( R ), so determine a point y ∈ (cid:16) Ig b ′ ,K ′ ,p × C b ′ ,K ′ ,p, F p C b,K p , F p (cid:17) perf ( R ) = (cid:16) Ig b ′ ,K ′ ,p × C b ′ ,K ′ ,p, F p C b,K p , F p (cid:17) ( R ) . The compatibility of j with crystalline Tate tensors exactly tells us that y ∈ Ig b,K p ( R ).It remains to show the last assertion. In light of Lemma 6.1.1 (2), the assertion on the J b ( Q p )-action follows from the analogue description for J b ′ ( Q p )-action on Ig b ′ ,K ′ ,p as in [CS17, Lem. 4.3.4,Cor. 4.3.5]. (cid:3) We make a right action of J b ( Q p ) on Ig b,K p so that it becomes a left action on the cohomology. In [CS17, § j is reverse to ours, from A [ p ∞ ] to Σ b × F p R . The two conventions are identified via taking the inverse of j (with the understanding that the authors of loc. cit. are also using the right action of J b ( Q p ), though this does notappear there explicitly). The J b ( Q p )-action on Ig b,K p commutes with the Hecke action of G ( A ∞ ,p ) (as K p varies) asit is clear on the moduli description. Now we would like to understand the G ( A ∞ ,p ) × J b ( Q p )-representation H i ( Ig b , Q ℓ ) := lim −→ K p H i ( Ig b,K p , Q ℓ ) , i ≥ , where the limit is over sufficiently small open compact subgroups of G ( A ∞ ,p ).From § J b ( Q p ) ab ։ M b ( Q p ) ab , which comes from the proof of Corollary2.3.3. (The latter also tells us that M b ( Q p ) ab = M b ( Q p ) ♭ and G ( Q p ) ab = G ( Q p ) ♭ .) J b ( Q p ) (cid:15) (cid:15) (cid:15) (cid:15) M b ( Q p ) (cid:15) (cid:15) (cid:15) (cid:15) / / G ( Q p ) (cid:15) (cid:15) (cid:15) (cid:15) J b ( Q p ) ab / / / / M b ( Q p ) ab / / G ( Q p ) ab (6.1.3)Thus every one-dimensional smooth representation of G ( Q p ) (necessarily factoring through G ( Q p ) ab )can be viewed as a one-dimensional representation of M b ( Q p ) or J b ( Q p ) via the above diagram. Weare ready to state the main theorem on the cohomology of Igusa varieties in this paper. Theorem 6.1.3 (Main Theorem) . In the setup of (Unr(
G, p, K p )) , let b ∈ G ( ˘ Q p ) be a Q -non-basicelement such that [ b ] ∈ B ( G Q p , µ − p ) . Then there is a G ( A ∞ ,p ) × J b ( Q p ) -module isomorphism ιH ( Ig b , Q ℓ ) ≃ M π π ∞ ,p ⊗ π p , where the sum runs over discrete automorphic representations π of G ( A ) such that (i) dim π = 1 ,(ii) π ∞ is trivial on G ( R ) + .Proof. This theorem will be reduced to the completely slope divisible case by Corollary 6.2.3 andestablished in § (cid:3) Remark . Since dim π p = 1, we have π p ⊗ δ P b = J P op b ( π p ) ⊗ δ / P b . (The point is that the unipotentradical N op b acts trivially on π p .) The latter appears in Lemma 3.1.2, and naturally occurs whendescribing the cohomology of Igusa varieties, as in [HT01, Thm. V.5.4] and [Shi12, Thm. 6.7].6.2. Finite-level Igusa varieties in the completely slope divisible case.
Let us assume that the p -divisible group Σ b is completely slope divisible. (This condition has nothing to do with G -structures.) We will recall the definition of finite-level Igusa varieties following [Man05, CS17,Ham19]. We continue to fix a sufficiently divisible r ∈ Z ≥ as in § b with its slope decomposition is definedover F p r .We start from the Siegel case. Write Ig b ′ ,m,K ′ ,p → C b ′ ,K ′ ,p for Igusa varieties of level m ∈ Z ≥ as in [Man05, §
4] or [Ham19, § F p r rather than over F p ), defined to parametrizeliftable isomorphisms on the p m -torsion subgroup of each slope component. As shown in loc. cit. ,Ig b ′ ,m,K ′ ,p → C b ′ ,K ′ ,p is a finite ´etale morphism, forming a projective system for varying m via theobvious projection maps. Write Ig b ′ ,K ′ ,p for the projective limit of Ig b ′ ,m,K ′ ,p over m . There aremaps Ig b ′ ,K ′ ,p → Ig b ′ ,m,K ′ ,p , F p for m ≥ Ig b ′ ,K ′ ,p → Ig perf b ′ ,K ′ ,p , F p .See [CS17, Prop. 4.3.8] and the preceding paragraph for details.Following [Ham19, § F p r rather than over F p ), define the F p r -subscheme˜Ig b,K p ⊂ (cid:16) Ig b ′ ,K ′ ,p × C b ′ ,K ′ ,p C b,K p (cid:17) perf OF IGUSA VARIETIES 57 to be the locus given by the same condition as in (6.1.2). Define Ig b,m,K p as the image of thecomposite map ˜Ig b,K p → Ig b ′ ,K ′ ,p × C b ′ ,K ′ ,p C b,K p → Ig b ′ ,m,K ′ ,p × C b ′ ,K ′ ,p C b,K p . The projection onto the second component gives an F p r -morphism Ig b,m,K p → C b,K p , which is finite´etale by [Ham19, Prop. 4.1]. Via the canonical projection Ig b,m +1 ,K p → Ig b,m,K p commuting withthe maps to C b,K p , we take the projective limit and denote it by Ig b,K p .Besides the Hecke action of G ( A ∞ ,p ) on the tower of Ig b,K p as K p varies, we also have Ig b,K p , F p acted on by a submonoid S b ⊂ J b ( Q p ) defined in [Man05, p.586]. (The latter action is defined onlyover F p in general since self quasi-isogenies of Σ b are not always defined over finite fields.) Theprecise definition is unimportant, but it suffices to know two facts. Firstly, S b generates J b ( Q p ) asa group. Secondly, S b contains p − (the inverse of the multiplication by p map on Σ b ) and fr − r := rν b ( p ) ∈ J b ( Q p ) . By Lemma 5.3.1, fr − r ∈ A J b ( Q p ). Let Fr denote the absolute Frobenius morphism on an F p -scheme. Lemma 6.2.1.
The following hold true.(1) fr − r ∈ A −− P op b ⊂ A M b ( Q p ) = A J b ( Q p ) . As an element of M b ( Q p ) , we have fr − r ∈ A −− P op b .(Recall that A −− P op b was defined in § Fr r × on Ig b,K p × F pr F p induces the same action on Ig b,K p as the action of fr − r ∈ J b ( Q p ) .(3) There is a canonical isomorphism Ig b,K p ≃ Ig perf b,K p , F p over C b,K p , F p , compatible with the G ( A ∞ ,p ) × S b -actions as K p varies.Proof. (1) We already know fr − r ∈ A J b ( Q p ) = A M b ( Q p ). (See § rν b is B -dominant ( § rν b ( p ) ∈ A − P op b . Moreover rν b ( p ) ∈ A −− P op b as the centralizer of rν b ( p ) in G is exactly M b .(2) Write Fr Σ for the absolute Frobenius action on Σ b / F p r . Recall that bσ ( b ) · · · σ r − ( b ) = rν b ( p )from § − r = rν b ( p ) acts on Σ b / F p r as (Fr Σ ) r . Thus fr − r sends ( x, j ) to ( x, j ◦ Fr r Σ ) inthe description of R -points in Lemma 6.1.2. On the other hand, Fr r × Ig b,K p sends ( x, j ) to( x ( r ) , j ( r ) ), where x ( r ) corresponds to the p r -th power Frobenius twist of x (so that A x ( r ) = ( A x ) ( r ) ),and j ( r ) is the p r -th power twist of j . Finally we observe that ( x ( r ) , j ( r ) ) is equivalent to ( x, j ◦ Fr r Σ )via the p r -power relative Frobenius A x → A x ( r ) .(3) We have the map Ig b,K p → Ig b,K p , F p over C b,K p , F p from the definition, which factors through Ig b,K p → Ig perf b,K p , F p since Ig b,K p is perfect. This is shown to be an isomorphism exactly as in theproof of [CS17, Prop. 4.3.8], the point being a canonical splitting of the slope decomposition overthe perfect scheme Ig perf b,K p , F p . (cid:3) Now we compare Igusa varieties arising from two central leaves in the same Newton stratum.Let b, b ∈ G ( ˘ Q p ). Assume that b and b satisfy the conditions at the end of § J b ( Q p ) ≃ J b ( Q p ) (induced by a conjugation in the ambient group G ( ˘ Q p )), canonicalup to J b ( Q p )-conjugacy. Proposition 6.2.2.
There exists a G ( A ∞ ,p ) -equivariant isomorphism Ig b ∼ → Ig b , Here is a note on the sign. On slope 0 ≤ λ ≤ r is p λ , but ν b records slope − λ sincewe use the covariant Dieudonn´e theory. which is also equivariant for the actions of J b ( Q p ) and J b ( Q p ) through a suitable isomorphism J b ( Q p ) ≃ J b ( Q p ) in its canonical J b ( Q p ) -conjugacy class.Proof. Since [ b x ] = [ b x ], there exists a quasi-isogeny f : Σ b → Σ b compatible with G -structures.Using the description of Lemma 6.1.2, we can give an isomorphism Ig b ∼ → Ig b on R -points by( A, j ) ( A, j ◦ f ). The equivariance property is evident. (cid:3) Corollary 6.2.3.
In the setting of Theorem 6.1.3, if the theorem is true for all b such that Σ b iscompletely slope divisible, then the theorem is true without the restriction on b .Proof. Let b be arbitrary. The argument of [Zha, Lem. 4.2.8] shows the existence of g ∈ G ( ˘ Q p )such that, setting b := g − b σ ( g ), • b ∈ G (˘ Z p ) σµ ( p ) − G (˘ Z p ), • Σ b is completely slope divisible.(We can further multiply g on the right by an element of G (˘ Z p ), thus not affecting complete slopedivisibility, while ensuring that (br1)–(br3) are satisfied as in § J b ( Q p ) ≃ J b ( Q p ) as in there, we have H ( Ig b , Q ℓ ) ≃ H ( Ig b , Q ℓ ) as G ( A ∞ ,p ) × J b ( Q p )-modules . We are assuming Theorem 6.1.3 for b , so it follows that the same theorem holds for b . (Observethat the transfer of one-dimensional representations via J b ( Q p ) ≃ J b ( Q p ) do not depend on thechoice of isomorphism.) (cid:3) Cohomology of Igusa varieties
The main purpose of this section is to prove Theorem 6.1.3. Throughout we are in the completelyslope divisible case of § § Compactly supported cohomology in top degree.
In 6.2 we have constructed Ig b,K p such that Ig b,K p is the perfection of Ig b,K p , F p (with compatible transition maps as K p varies). Recall thatdim Ig b = h ρ, ν b i . Define for i ∈ Z ≥ , H ic (Ig b,m, F p , Q ℓ ) := lim −→ K p H ic (Ig b,m,K p , F p , Q ℓ ) , H ic (Ig b, F p , Q ℓ ) := lim −→ m ≥ H ic (Ig b,m, F p , Q ℓ ) . As for H ic (Ig b, F p , Q ℓ ), we have a G ( A ∞ ,p ) × J b ( Q p )-module structure on H ic (Ig b, F p , Q ℓ ). This is anadmissible G ( A ∞ ,p ) × J b ( Q p )-module as the cohomology is finite-dimensional at each finite level. Itis convenient to prove the following dual version of Theorem 6.1.3. Theorem 7.1.1.
In the setup of (Unr(
G, p, K p )) , assume that b is Q -non-basic, and that Σ b iscompletely slope divisible. Then there is a G ( A ∞ ,p ) × J b ( Q p ) -module isomorphism ιH h ρ,ν b i c (Ig b, F p , Q ℓ ) ≃ M π π ∞ ,p ⊗ ( π p ⊗ δ P b ) , where the sum runs over discrete automorphic representations π of G ( A ) such that (i) dim π = 1 ,(ii) π ∞ is trivial on G ( R ) + .Proof. The proof will be carried out in § § (cid:3) OF IGUSA VARIETIES 59
Theorem 7.1.1 implies Theorem 6.1.3.
We may put ourselves in the completely slope divisible caseby Corollary 6.2.3. Write d := h ρ, ν b i . Applying Poincar´e duality to finite-level Igusa varietiesIg b,m,K p and taking direct limit over m and K p , we obtain a pairing H (Ig b, F p , Q ℓ ) × H dc (Ig b, F p , Q ℓ ( d )) → Q ℓ , where Q ℓ ( d ) denotes the d -th power Tate twist. The construction of duality (Exp.XVIII, § Rf ! m,K p Q ℓ ≃ Q ℓ ( d )[ − d ] (concentratedin degree 2 d ) over m and K p , where f m,K p : Ig b,m,K p → Spec F p r denotes the structure map. Thusthe action of G ( A ∞ ,p ) × J b ( Q p ) on Ig b = { Ig b,m,K p } induces an action on Q ℓ ( d )[ − d ], through acharacter ς : G ( A ∞ ,p ) × J b ( Q p ) → Q × ℓ . (As in § J b ( Q p ) is defined a priori ona submonoid S b and then extended to J b ( Q p ). Alternatively, this action can be defined directlyafter perfectifying Ig b .) Together with the G ( A ∞ ,p ) × J b ( Q p )-action on Ig b , this yields an actionof G ( A ∞ ,p ) × J b ( Q p ) on H dc (Ig b, F p , Q ℓ ( d )) and H (Ig b, F p , Q ℓ ), respectively. It follows from thefunctoriality of Poincar´e duality that the above pairing is G ( A ∞ ,p ) × J b ( Q p )-equivariant. Thus H (Ig b, F p , Q ℓ ) is isomorphic to the (smooth) contragredient of H dc (Ig b, F p , Q ℓ ( d )), which is isomor-phic to H dc (Ig b, F p , Q ℓ ) ⊗ ς . Therefore Theorem 7.1.1 implies that ιH (Ig b, F p , Q ℓ ) ≃ M π ∈ Π ( G ) (( π ∞ ,p ) ⊗ ( π p ⊗ δ P b )) ∨ ⊗ ς − , (7.1.1)where Π ( G ) denotes the range of π in the summation of that theorem. (Since dim π = 1, the dualsign in (7.1.1) simply means inverse.)On the other hand, H (Ig b, F p , Q ℓ ) is the space of smooth Q ℓ -valued functions on π (Ig b, F p ), onwhich G ( A ∞ ,p ) × J b ( Q p ) acts through right translation. (Here smoothness means invariance underan open compact subgroup of G ( A ∞ ,p ) × J b ( Q p ).) In particular H (Ig b, F p , Q ℓ ) contains the trivialrepresentation, which consists of constant functions on π (Ig b, F p ). Hence ς − = ( π ∞ ,p ) ⊗ ( π ,p ⊗ δ P b )for some π ∈ Π ( G ). Since Π ( G ) is invariant under twist by π , we see that (7.1.1) is still truewithout ς . Moreover Π ( G ) is invariant under taking dual, so we can rewrite (7.1.1) as ιH (Ig b, F p , Q ℓ ) ≃ M π π ∞ ,p ⊗ ( π p ⊗ δ P b ) . Finally, the same holds with Ig b in place of Ig b, F p thanks to Lemma 6.2.1 (3). (cid:3) Remark . It may be possible to compute the character ς in the proof, but we have got aroundit. As we know the Frobenius action on Q ℓ ( d )[ − d ], Lemma 6.2.1 (2) tells us that fr − r ∈ J b ( Q p )acts by p rd . We guess that ς is trivial on G ( A ∞ ,p ) and equal to δ − P b on J b ( Q p ).7.2. The basic setup for harmonic analysis.
Let φ ∞ ,p = ⊗ v = ∞ ,p φ v ∈ H ( G ( A ∞ ,p )) and φ p ∈H ( J b ( Q p )). With a view towards Theorem 7.1.1, we want to computeTr (cid:16) φ ∞ ,p φ ( j ) p (cid:12)(cid:12)(cid:12) ιH c (Ig b, F p , Q ℓ ) (cid:17) := X i ≥ ( − i Tr (cid:16) φ ∞ ,p φ ( j ) p (cid:12)(cid:12)(cid:12) ιH ic (Ig b, F p , Q ℓ ) (cid:17) . We keep T , B , b ∈ G ( ˘ Q p ), and r ∈ Z ≥ as before, so that rν b ∈ X ∗ ( T ) + . Recall that rν b ( p ) ∈ A J b .Given φ p ∈ H ( J b ( Q p )), define φ ( j ) p ∈ H ( J b ( Q p )) by φ ( j ) p ( δ ) := φ p ( jν b ( p ) − δ ) , j ∈ r Z ≥ . This coincides with the analogous definition of φ ( k ) p in § φ ( j ) p = φ ( k ) p via k = j/r and ν = rν b . (The difference is that ν is a cocharacter but ν b is only a fractional cocharacter.)An element δ ∈ J b ( Q p ) is acceptable if its image in M b ( Q p ) is acceptable (Definition 3.1.1) underthe isomorphism J b ( Q p ) ≃ M b ( Q p ) induced by some (thus any) inner twist at the end of § in § H acc ( J b ( Q p )) ⊂ H ( J b ( Q p )) denote the subspace of functions supported on acceptableelements. Choose j ∈ Z ≥ such that φ ( j ) p ∈ H acc ( J b ( Q p )) , j ∈ r Z , j ≥ j . Such a j exists by the argument of Lemma 3.1.7. By Lemma 6.2.1 and the definition of φ ( j ) p , wehave Tr (cid:16) φ ∞ ,p φ ( j ) p (cid:12)(cid:12)(cid:12) ιH c (Ig b, F p , Q ℓ ) (cid:17) = Tr (cid:16) φ ∞ ,p φ p × (Fr j × (cid:12)(cid:12)(cid:12) ιH c (Ig b, F p , Q ℓ ) (cid:17) , (7.2.1)where Fr j is the j/r -th power of the relative Frobenius of Ig b over F p r . Since the action of Fr j isthe same as the action of a central element of J b ( Q p ), it commutes with the action of φ ∞ ,p φ p . Thus(7.2.1) and the Lang–Weil bound tell us that the top degree compactly supported cohomology inTheorem 7.1.1 is captured by the leading term as j → ∞ . This will be the basic idea underlyingthe proof of the theorem in § X , χ ) = ( A G, ∞ , ) for G , which can also be viewedas a central character datum for G ∗ via Z ( G ) = Z ( G ∗ ). (Since we compute the cohomology withconstant coefficients, we do not need to consider nontrivial χ .)Also fixed is a z -extension 1 → Z → G → G → Q once and for all as at the start of § µ p, : G m → G over Q p r , b ∈ G ( Q p r ) , and possibly make r more divisible so that rν b is a cocharacter as in Lemma 5.3.8. Via ι p : Q p ≃ C ,we transport µ p, to µ : G m → G over C . The choice of µ p, and b will affect the test functionsat ∞ and p , respectively.For each e ∈ E ell ( G ), fix a Q -rational minimal parabolic subgroup of G e and its Levi componentas at the start of § G e in place of G there). Call them P ∗ and M ∗ in the case e = e ∗ . Onthe other hand, we have G Q p ⊃ B Q p ⊃ T Q p from § G Q p is quasi-split, there is a canonical G ad ( Q p )-conjugacy class of isomorphisms G Q p ≃ G ∗ Q p . We fix one such isomorphism such that B Q p (resp. T Q p ) is carried into P ∗ , Q p (resp. M ∗ , Q p ). The images of B Q p and T Q p in G ∗ Q p will play theroles of B and T in § e ∈ E < ell ( G ), we have a z -extension 1 → Z → G e → G e → Q and an endoscopicdatum e = ( G e , G e , s e , η e ) for G as in § X e , χ e )for G e .7.3. The test functions away from p . For each e ∈ E ell ( G ), Let us introduce the test functions toenter the statement of Theorem 7.5.1 below. Here we consider the places away from p . The place p will be treated in the next subsection.The first case is away from p and ∞ . When e = e ∗ , we have ( f Ig , ∗ ) ∞ ,p = ⊗ ′ v = ∞ ,p f ∗ v , where f ∗ v ∈ H ( G ∗ ( Q v )) is a transfer of φ v as in § e ∈ E < ell ( G ), at each v = ∞ , p , the function φ v admits a transfer f e ,v ∈ H ( G e ( Q v ) , ( χ e ,v ) − ). Then we take( f Ig , e ) ∞ ,p = ⊗ ′ v = ∞ ,p f e ,v ∈ H ( G e ( A ∞ ,p ) , ( χ e , ∞ ,p ) − ) . The next case is the real place. The construction of the test function f Ig , e , ∞ ∈ H ( G e ( R ) , ( χ e , ∞ ) − )follows [Kot90, §
7] based on Shelstad’s real endoscopy and Clozel-Delorme’s pseudocoefficients. Weadapt it to the case with central characters. In the easier case of e = e ∗ = ( G ∗ , L G ∗ , , id), wetake f Ig , ∗∞ := e ( G ∞ ) f in the notation of § e ∈ E < ell ( G ). In the notation of § ξ and ζ are trivial in the current setup (since we are focusing on the constant coefficient case).Write ξ and ζ for the pullbacks of ξ and ζ from G to G ; they are again trivial. We obtain adiscrete L -packet Π( ξ , ζ ) for G ( R ) along with an L -parameter φ ξ ,ζ : W R → L G as in § OF IGUSA VARIETIES 61
Let Φ ( G e , R , φ ξ ,ζ ) denote the set of discrete L -parameters φ ′ ∈ Φ( G e ( R )) such that η e φ ′ ≃ φ ξ ,ζ .Then define (cf. [Kot90, p.186]) f Ig , e , ∞ := ( − q ( G ) h µ , s e i X φ ′ det( ω ∗ ( φ ′ )) f φ ′ , where f φ ′ is the averaged Lefschetz function for the L -packet of φ ′ defined in § φ ′ ∈ Φ( G e , R , φ ξ ,ζ ). As in [KSZ, § we check that f Ig , e , ∞ is ( χ e , ∞ ) − -equivariantand compactly supported modulo X e , ∞ .7.4. The test functions at p . We apply the contents of § ν := rν b over F = Q p with uniformizer ̟ = p . In particular, we have P b := P ν whose Levi factor is M b = M ν .Consider the case e = e ∗ . Each function φ p ∈ H ( J b ( Q p )) admits a transfer φ ∗ p ∈ H ( M b ( Q p )) asexplained in § φ p ∈ H acc ( J b ( Q p )), we can arrange that φ ∗ p ∈ H acc ( M b ( Q p )) by multiplyingthe indicator function on the set of acceptable elements in M b ( Q p ). (This is possible as the subsetof acceptable elements is nonempty, open, and stable under J b ( Q p )-conjugacy.) The image of φ ∗ p in S ( M b ) depends only on φ p (as an element of S ( J b )). In the notation of § f Ig , ∗ , ( j ) p := J ν (cid:16) δ / P ν · φ ∗ , ( j ) p (cid:17) ∈ S ( G ) , j ∈ Z ≥ , As before, we still write f Ig , ∗ , ( j ) p for a representative in H ( G ( Q p )). Lemma 3.1.2 implies thatTr (cid:16) f Ig , ∗ , ( j ) p | π p (cid:17) = Tr (cid:16) φ ∗ , ( j ) p | J P op ν ( π p ) ⊗ δ / P ν (cid:17) , ∀ π p ∈ Irr( G ( Q p )) . Remark . In the definition of f Ig , ∗ , ( j ) p , we have not multiplied the constant c M H appearingin [Shi10, §
6] (with
H, M H there corresponding to G ∗ , M b here). In the sign convention of Remark6.4 therein, the transfer factor between J b and M b equals e ( J b ), resulting in c M H = e ( J b ). Incontrast, we have taken the transfer factor between inner forms to be 1 (cf. Remark 2.3.6), so c M H = 1 in our convention.Now let e ∈ E < ell ( G ). Recall that b ∈ G ( Q p r ) was chosen. Take ν := rν b . The z -extension1 → Z → G → G → z -extensions of M b and J b over Q p . By pulling back via M b ֒ → G , and from the definition of J b and J b , we indeed obtain exact sequences of Q p -groups1 → Z → M b → M b → , → Z → J b → J b → . (For J b , the point is that the σ -stabilizer subgroup of Res ˘ Q p / Q p G m is simply G m .) We pull back φ ( j ) p ∈ H ( J b ( Q p )) to obtain φ ( j )1 ,p ∈ H ( J b ( Q p ) , χ ,p ). (Recall that χ = Q v χ ,v is the trivialcharacter on X = Z ( A ).) Write φ ∗ p ∈ H ( M b ( Q p )) for a transfer of φ p , and φ ∗ ,p ∈ H ( M b ( Q p ) , χ ,p )for the pullback of φ ∗ p . Then φ ∗ , ( j ) p (defined in § φ ( j ) p (namely φ ∗ , ( j ) = ( φ ( j ) p ) ∗ in S ( J b )), and φ ∗ , ( j )1 ,p is a transfer of φ ( j )1 ,p , for all j ∈ Z .The desired test function f Ig , e ,p is described by the process in [Shi10, § J b , G e , G in placeof J b , H, G therein, followed by averaging on X = X e . We point out that [Shi10] is applicable as G has simply connected derived subgroup. For our purpose, we summarize the construction asfollows: f Ig , e , ( j )1 ,p := X ω ∈ Ω e ,ν f Ig , e , ( j )1 ,p,ω , where (7.4.1) f Ig , e , ( j )1 ,p,ω := c ω · J ν ,ω (cid:0) LS e ,ω ( δ / P ν · φ ∗ , ( j )1 ,p ) (cid:1) ∈ H ( G e ( Q p ) , χ e , − ,p ) , The reference numbering is subject to change.
Here c ω ∈ C are constants (possibly zero) independent of φ p . Note that J ν ,ω and LS e ,ω denotethe maps in the setup with fixed central character as in § δ / P ν · φ ∗ , ( j )1 ,p = δ / P ν ( ν ( p )) (cid:0) δ / P ν · φ ∗ ,p (cid:1) ( j ) = p h ρ,ν i (cid:0) δ / P ν · φ ∗ ,p (cid:1) ( j ) . (7.4.2)7.5. The stable trace formula for Igusa varieties.
Keep the preceding setup and notation fromearlier in this section and also from § Theorem 7.5.1.
Given φ ∞ ,p ∈ H ( G ( A ∞ ,p )) and φ p ∈ H ( J b ( Q p )) , there exists j = j ( φ ∞ ,p , φ p ) ∈ Z ≥ such that φ ( j ) p ∈ H acc ( J b ( Q p )) and for every integer j ≥ j divisible by r , the following formulaholds: Tr (cid:16) φ ∞ ,p φ ( j ) p (cid:12)(cid:12) ιH c (Ig b , Q ℓ ) (cid:17) = ST G ∗ ell ,χ ( f Ig , ∗ , ( j ) ) + X e ∈E < ell ( G ) ι ( G, G e )ST G e ell ,χ e (cid:16) f Ig , e , ( j )1 (cid:17) . Proof.
The point is to stabilize the main result of [MC] (generalizing that of [Shi09]), which obtainsthe following expansion for Tr (cid:16) φ ∞ ,p φ ( j ) p (cid:12)(cid:12) ιH c (Ig b , Q ℓ ) (cid:17) : X γ ∈ Σ R -ell ( G ) c ( γ )Tr ξ ( γ )¯ ι G ( γ ) X c =( γ ,a, [ b ]) c ( c ) O G ( A ∞ ,p ) γ ( φ ∞ ,p ) O J b ( Q p ) δ ( φ ( j ) p ) , (7.5.1)where the inner sum is over the set of acceptable b -admissible Kottwitz parameters c , and theconjugacy class ( γ, δ ) in G ( A ∞ ,p ) × J b ( Q p ) is determined by c as in loc. cit. We do not recallthe notation further, as it suffices for our purpose to observe that the right hand side resemblesthe analogous formula for Shimura varieties [KSZ, Thm. 5.4.3] except that different terms appearat p . Indeed, the stabilization of (7.5.1) is very close to the stabilization for Shimura varietiesin [KSZ, § p following [Shi10, § The initial steps in stabilization are identical to [KSZ, § X = A G, ∞ ; in this case τ X ( · ) = τ ( · ), the usual Tamagawa measure). Stabilization away from p and ∞ is based on the Langlands–Shelstad transfer as in [KSZ, § ∞ are stabilizedaccording to Shelstad’s real endoscopy as detailed in [Kot90, § p , we use [Shi10, Lem. 6.5] (adapted to the fixed central charactersetup) instead of [KSZ, § G der is assumed to be simply connected in [Shi10],we can reduce to that case via z -extensions when e = e ∗ , at the expense of introducing a centralcharacter datum; this is done in the same way as explained in § § f Ig , e , ( j )1 ∈ H ( G e ( Q p ) , χ e , − ,p ) is exactly as described in (7.4.1), which is simply an adaptation of theprocess in [Shi10, § § z -extensions when e = e ∗ . This completes the sketchof stabilization of (7.5.1). (cid:3) Remark . In the argument above, we do not need a precise local normalization of transferfactors and Haar measures over all places of Q when e = e ∗ , since it will only affect error terms byconstant factors in the estimate. For instance, we need not know what normalization of transferfactors should be taken at p and ∞ (as well as away from p and ∞ ) to satisfy the product formulafor transfer factors, cf. Remark 2.6.1. We intend to work out precise normalizations in a futurepaper, thus removing ambiguity in the coefficients c ω in (7.4.1). The stabilization in [KSZ] is written for arbitrary Shimura varieties (assuming the Shimura variety analogue of(7.5.1) when (
G, X ) is not of abelian type). In [Shi10], the main assumptions are that (
G, X ) is of certain PEL typeand that G der is simply connected. The former is irrelevant for stabilization, and the latter is removed as we explainin the current proof. OF IGUSA VARIETIES 63
Completion of the proof of Theorem 7.1.1.
The main term in the right hand side of Theorem7.5.1 will turn out to be the following.
Proposition 7.6.1.
Fix φ ∞ ,p φ p ∈ H ( G ( A ∞ ,p ) × J b ( Q p )) , from which f ∗ , ( j ) ∈ H ( G ∗ ( A )) is given asin § j ∈ Z ≥ such that j ≥ j = j ( φ ∞ ,p φ p ) . As j ≥ j varies over positive integersdivisible by r , we have the estimate T G ∗ disc ,χ (cid:0) f Ig , ∗ , ( j ) (cid:1) = X π dim π =1 Tr ( φ ∞ ,p | π ∞ ,p ) · Tr (cid:16) φ ( j ) p | π p ⊗ δ P b (cid:17) + o (cid:16) p j h ρ,ν b i (cid:17) , where the sum runs over one-dimensional automorphic representations π of G ( A ) such that π p isunramified and π ∞ | G ( R ) + = 1 .Proof. We have T G ∗ disc ,χ (cid:16) f Ig , ∗ , ( j ) (cid:17) = X π ∗ m ( π ∗ )Tr (cid:0) f Ig , ∗ ,p | π ∗ ,p (cid:1) Tr (cid:16) f Ig , ∗ , ( j ) p | π ∗ p (cid:17) . (7.6.1)Let J H ( J P op b ( π ∗ p )) denote the multi-set of irreducible subquotients of J P op b ( π ∗ p ) (up to isomorphism).The central character of τ ∈ J H ( J P op b ( π ∗ p )) is denoted ω τ . We see from Lemma 3.1.2 (ii) thatTr (cid:16) f Ig , ∗ , ( j ) p | π ∗ p (cid:17) = Tr (cid:16) δ / P b φ ∗ , ( j ) p | J P op b ( π ∗ p ) (cid:17) = Tr (cid:16) φ ∗ , ( j ) p | J P op b ( π ∗ p ) ⊗ δ / P b (cid:17) = X τ ∈ JH ( J P op b ( π ∗ p )) ω τ ( jν b ( p )) δ / P b ( jν b ( p )) Tr ( φ ∗ p | τ ) . (7.6.2)We have jν b ( p ) ∈ A −− P op b . (See the paragraph above Definition 3.1.1.) By Corollary 2.5.2 (ourrunning assumption that b is Q -non-basic implies ( Q -nb( P op b )) by Lemma 5.3.7), the largest growthof ω τ ( jν b ( p )) as a function in j is achieved exactly when dim π ∗ = 1. In that case, we have m ( π ∗ ) = 1 and π ∗ p is a unitary character. Via Lemma 2.5.3, π ∗ corresponds to a unique one-dimensional automorphic representation π of G ( A ). We have π ∗ p ≃ π p via G ∗ ( Q p ) ≃ G ( Q p ). ThusTr ( φ ∗ , ( j ) p | J P op b ( π ∗ p ) ⊗ δ / P b ) = Tr ( φ ∗ , ( j ) p | π ∗ p ⊗ δ P b ) = Tr ( φ ( j ) p | π p ⊗ δ P b )= δ P b ( jν b ( p ))Tr ( φ p | π p ⊗ δ P b ) = p j h ρ,ν b i Tr ( φ p | π p ⊗ δ P b ) . (7.6.3)See Remark 6.1.4 for the first equality. We used Lemma 2.3.7 for the second equality above. Indeed, π ∗ p ⊗ δ P b as a character of M b ( Q p ) and π p ⊗ δ P b as a character of J b ( Q p ) correspond to each othervia the diagram (6.1.3).Let f denote the averaged Lefschetz function on G ( R ) as in § ξ = and ζ = . Write e ( G ∞ ,p ) := Q v = ∞ ,p e ( G v ) for the product of Kottwitz signs. We can rewrite (7.6.1) as T G ∗ disc ,χ (cid:0) f Ig , ∗ , ( j ) (cid:1) = X π ∗ dim π ∗ =1 Tr (cid:0) f Ig , ∗∞ | π ∗∞ (cid:1) Tr (cid:0) f Ig , ∗ , ∞ ,p | π ∗ , ∞ ,p (cid:1) Tr (cid:0) f Ig , ∗ , ( j ) p | π ∗ p (cid:1) + o (cid:0) p j h ρ,ν b i (cid:1) = X π dim π =1 Tr ( f | π ∞ )Tr ( φ ∞ ,p | π ∞ ,p )Tr (cid:0) φ ( j ) p | π p ⊗ δ P b (cid:1) + o (cid:0) p j h ρ,ν b i (cid:1) , where the last equality was obtained from (7.6.2) at p , Lemma 2.4.2 at ∞ , and Lemma 2.3.7 at theplaces away from p . To conclude, we invoke Lemma 2.4.3 to see that Tr ( f | π ∞ ) = 1 if π ∞ | G ( R ) + = and Tr ( f | π ∞ ) = 0 otherwise. (cid:3) Finally we complete the proof of Theorem 7.1.1 employing the main estimates of § Corollary 7.6.2.
Theorem 7.1.1 is true.
Proof.
Let q = p be an auxiliary prime such that G Q q is split. Fix φ ∞ ,p,q φ p ∈ H ( G ( A ∞ ,p,q ) × J b ( Q p )).Write A ( G ) for the set of one-dimensional automorphic representations π of G ( A ) such that π p is unramified and π ∞ | G ( R ) + = 1. Let e ∈ E < ell ( G ). There exists a constant C e = C e ( φ ∞ ,p,q , φ p ) > φ q ∈ H ( G ( Q q )) C e -reg , we have the following bound on endoscopic terms in thestabilization of Theorem 7.5.1 by applying the last bound in Corollary 4.2.3 to k = j/r , ν = rν ,ω , φ ( k ) p = c ω ( δ / P ν φ ∗ ,p ) ( k ) , and χ = χ for each ω ∈ Ω e ,ν . Notice that f Ig , e , ( j )1 ,p,ω is p h ρ e ,ν ,ω i times f ( k ) p ofthat corollary, in light of (7.4.1) and (7.4.2). We haveST G e ell ,χ e (cid:16) ( f Ig , e ,p ) f Ig , e , ( j )1 ,p,ω (cid:17) = O (cid:16) p k ( h ρ e ,rν ,ω i + h χ e ,rν ,ω i ) (cid:17) = O (cid:16) p j ( h ρ e ,ν ,ω i + h χ e ,ν ,ω i ) (cid:17) . To turn this into a more manageable bound, we use (a) and (b) from the proof of Corollary 4.2.3and the fact that h χ , ∞ , ν i = 0 since χ (which plays the role of χ there) is trivial. Thereby we seethat the right hand side is o (cid:0) p j h ρ,ν b i (cid:1) . Taking the sum over ω ∈ Ω e ,ν , we obtainST G e ell ,χ e (cid:16) f Ig , e , ( j )1 (cid:17) = o (cid:16) p j h ρ,ν b i (cid:17) , e ∈ E < ell ( G ) . (7.6.4)By Lemma 2.9.2, there are only finitely many e contributing to the sum in Theorem 7.5.1 fora fixed choice of φ ∞ ,p,q φ p . Thus the coefficients ι ( G, G e ) are bounded by a uniform constant(depending on φ ∞ ,p,q φ p ). We deduce the following by applying Theorem 7.5.1, (7.6.4), Corollary4.2.3 (the first estimate therein), and Proposition 7.6.1 in the order: there exists a constant C = C ( φ ∞ ,p , φ p ) > C e over the set of finitely many e which contribute) suchthat for every φ q ∈ H ( G ( Q q )) C -reg , we haveTr (cid:16) φ ∞ ,p φ ( j ) p | ιH c (Ig b , Q ℓ ) (cid:17) = ST G ∗ ell ,χ (cid:0) f Ig , ∗ , ( j ) (cid:1) + o (cid:16) p j h ρ,ν b i (cid:17) = T G ∗ disc ,χ (cid:0) f Ig , ∗ , ( j ) (cid:1) + o (cid:16) p j h ρ,ν b i (cid:17) = X π ∈A ( G ) Tr ( φ ∞ ,p | π ∞ ,p ) · Tr (cid:16) φ ∗ , ( j ) p | π p ⊗ δ P b (cid:17) + o (cid:16) p j h ρ,ν b i (cid:17) . We have seen in (7.6.3) that Tr (cid:16) φ ∗ , ( j ) p | π p ⊗ δ P b (cid:17) is either 0 or a nonzero multiple of p j h ρ,ν b i as j varies over multiples of r . Since dim Ig b = h ρ, ν b i , it is implied by (7.2.1) and the Lang–Weilbound that the leading term should be of order p j h ρ,ν b i . ThereforeTr (cid:16) φ ∞ ,p φ ( j ) p | ιH h ρ,ν b i c (Ig b , Q ℓ ) (cid:17) = X π ∈A ( G ) Tr ( φ ∞ ,p | π ∞ ,p ) · Tr (cid:16) φ ∗ , ( j ) p | π p ⊗ δ P b (cid:17) . Let B q be a Borel subgroup of G Q q over Q q with a Levi component T q . By Lemma 3.4.9, we havean isomorphism of G ( A ∞ ,p,q ) × J b ( Q p ) × T q ( Q q )-representations J B q (cid:16) H h ρ,ν b i c (Ig b , Q ℓ ) (cid:17) ≃ X π ∈A ( G ) π ∞ ,p,q ⊗ ( π p ⊗ δ P b ) ⊗ J B q ( π q ) . (A priori the isomorphism exists up to semisimplification, but distinct one-dimensional represen-tations have no extensions with each other.) Repeating the same argument for another prime q ′ / ∈ { p, q } such that G ( Q q ′ ) is split, the above isomorphism exists with q ′ in place of q . Comparingthe two consequences, we deduce Theorem 7.1.1, which asserts that ιH h ρ,ν b i c (Ig b , Q ℓ ) ≃ M π ∈A ( G ) π ∞ ,p ⊗ ( π p ⊗ δ P b ) as G ( A ∞ ,p ) × J b ( Q p )-modules . (cid:3) In fact, the Lang–Weil bound proves that dim Ig b = h ρ, ν b i even if we did not know it a priori. This gives analternative proof of the dimension formula in Proposition 5.3.4. OF IGUSA VARIETIES 65 Applications to geometry
This section is devoted to interesting geometric consequences of Theorem 6.1.3, continuing inthe setup (Unr(
G, p, K p )) for Hodge-type Shimura varieties.8.1. The discrete Hecke orbit conjecture.
To see that Theorem 6.1.3 implies Theorem 5.4.4, wecheck a group-theoretic lemma. Recall that we fixed a maximal torus and a Borel subgroup T ⊂ B ⊂ G over Z p and that M b is a Q p -rational Levi subgroup containing T . The hyperspecial vertexfor G determining K p (contained in the apartment for T ) gives a reductive model for M b over Z p such that M b ( Z p ) = M b ( Q p ) ∩ G ( Z p ). Earlier we defined J int b ⊂ J b ( Q p ). Lemma 8.1.1.
The image of J int b in J b ( Q p ) ab is carried onto the image of M b ( Z p ) in M b ( Q p ) ab under the canonical isomorphism J b ( Q p ) ab ∼ → M b ( Q p ) ab .Proof. The validity of the lemma is invariant when b is changed to a σ -conjugate element b ′ ∈ G ( ˘ Q p ).Indeed, under the canonical identifications J b ′ ( Q p ) ab = J b ( Q p ) ab , the images of J int b ′ and J int b areequal. (In this proof, we need not worry about preserving conditions (br1)–(br3) as they play norole.) To avoid confusion, recall from § M b for M b ◦ with a fixed b ◦ . So M b does not change when σ -conjugating b .By [Kot85, Prop. 6.2] we may replace b with a σ -conjugate to assume that b ∈ M b ( ˘ Q p ) and that b is basic in M b . We further reduce to the case when G der = G sc . Indeed, take an unramified z -extension 1 → Z → G → G → Q p and a lift b ∈ G ( ˘ Q p ) of b as in Lemma 5.3.8. Then J int b → J int b is onto. (Use the fact that Z (˘ Z p ) → Z (˘ Z p ) given by z σ ( z ) − z is onto.) Since M b ( Z p ) → M b ( Z p ) is onto as well (as M b → M b is a smooth morphism over Z p ), we see that theproof for G is reduced to that for G .Thus we assume that b ∈ M b ( ˘ Q p ) is basic and that G der = G sc . So M b, der = M b, sc as well. Thenatural map M b → M ab b is surjective on Z p -points, by smoothness over Z p and the surjectivity on F p -points by Lang’s theorem. It suffices to verify that the image of J int b in J ab b ( Q p ) = M ab b ( Q p ) isequal to M ab b ( Z p ).Since M b is unramified over Q p , it contains an unramified elliptic maximal torus T ′ over Q p . (Seethe last paragraph in [DeB06, § T ′ extends to a torus over Z p , which we still denote by T ′ . By σ -conjugating b , we may assume that b ∈ T ′ ( ˘ Q p ) by [Kot85, Prop. 5.3]. By the definitionof J b and J int b , we have T ′ ( Q p ) ⊂ J b ( Q p ) and T ′ ( Z p ) ⊂ J int b . Moreover we have a commutativediagram T ′ ( Z p ) (cid:31) (cid:127) / / (cid:127) _ (cid:15) (cid:15) J int b (cid:31) (cid:127) / / (cid:15) (cid:15) J b ( Q p ) (cid:15) (cid:15) (cid:15) (cid:15) M b ( Z p ) / / M ab b ( Z p ) (cid:31) (cid:127) / / M ab b ( Q p ) = J ab b ( Q p ) . (The second vertical map is given by the composite J int b ⊂ M b (˘ Z p ) → M ab b (˘ Z p ), whose image isclearly σ -invariant. With this, the commutativity is elementary to check.) The proof will be doneonce we show that the map T ′ ( Z p ) → M ab b ( Z p ) is surjective in the diagram.We have an exact sequence of unramified tori 1 → T ′ ∩ M b, der → T ′ → M ab b →
1, which extendsto an exact sequence of tori over Z p . Then T ′ ( Z p ) → M ab b ( Z p ) is surjective, for the same reasonthat M b ( Z p ) → M ab b ( Z p ) was surjective above. This finishes the proof. (cid:3) Remark . The above proof also works for a Z p -subtorus T ′ of M b that is not unramified andelliptic over Q p . What we need is that b ∈ T ′ ( ˘ Q p ), that T ′ → M ab b is surjective, and that T ′ ∩ M b, der is connected over F p (to apply Lang’s theorem). Under this condition, the same argument showsthat the images of T ′ ( Z p ) and J int b are equal in M ab b ( Q p ) = J ab b ( Q p ). An exemplary situation iswhen T ′ is a (maximally split) maximal torus of M b over Z p with b central in M b . Corollary 8.1.3.
Theorem 5.4.4 is true. The discrete Hecke orbit conjecture holds when b is Q -non-basic.Proof. We start by noting that perfection does not change topological information of a scheme suchas the set of connected components or ´etale cohomology.By Part (1) of Lemma 6.1.1 and Theorem 6.1.3, we have isomorphisms of G ( A ∞ ,p ) × J b ( Q p )-modules ιH ( C b , Q ℓ ) = ιH ( C perf b , Q ℓ ) = ιH ( Ig b , Q ℓ ) J int b ≃ M π π ∞ ,p ⊗ ( π p ⊗ δ P b ) J int b , (8.1.1)where the sum runs over the same set of π as in Theorem 6.1.3.By Lemma 8.1.1, the left hand side is fixed (pointwise) under J int b if and only if the right handside is fixed under M b ( Z p ). The latter condition holds if and only if π p is fixed under G ( Z p ).(The if direction is trivial. For the other implication, note that G ( Z p ) is generated by M b ( Z p )and unipotent subgroups of G ( Z p ), but unipotent subgroups act trivially as they lie in G der ( Z p ).)Thereby we deduce Theorem 5.4.4 from (8.1.1) and Lemma 5.1.1. Finally (HO disc ) and (HO ′ disc )follow from Corollary 5.4.5. (cid:3) Irreducibility of Igusa varieties. In § µ -ordinary Newton strata of certain PEL-type Shimura varieties. Now we explainthat our main theorem implies a generalization thereof to Hodge-type Shimura varieties and tonon- µ -ordinary strata.We continue in the same setup as in the preceding subsections; thus (Unr( G, p, K p )) is assumed,and b is Q -non-basic. Define J ( Q p ) ′ := ker( J b ( Q p ) → G ( Q p ) ab ) using the composite map in thediagram (6.1.3). Recall that pr : Ig b → C perf b, F p is a pro-´etale J int b -torsor. Accepting Theorem 6.1.3 fornow (to be proved in the next section), we show that Igusa varieties are ‘as irreducible as possible’. Theorem 8.2.1 (Irreducibility of Igusa varieties) . The stabilizer of each connected component of Ig b under the J b ( Q p ) -action is equal to J b ( Q p ) ′ .Proof. Fix a component I ⊂ Ig b and write Stab( I ) for the stabilizer of I in J b ( Q p ). Since the J b ( Q p )-action on every π p in Theorem 6.1.3 factors through J b ( Q p ) /J b ( Q p ) ′ , we see that Stab( I ) ⊃ J b ( Q p ) ′ . To prove the reverse inclusion, we show that every δ ∈ J b ( Q p ) \ J b ( Q p ) ′ acts nontriviallyon H ( Ig b , Q ℓ ). Write δ ab ∈ G ( Q p ) ab for the image of δ . Then δ ab = 1 by assumption. It sufficesto show that there exists π as in the summation of Theorem 6.1.3 such that π p is nontrivial on δ ab when π p is viewed as a character of G ( Q p ) ab via Corollary 2.3.3. This follows from Lemma2.5.4. (cid:3) Corollary 8.2.2.
Let S be a connected component of C perf b, F p . Then the set π (pr − ( S )) ⊂ π ( Ig b ) isa torsor under the group J int b / ( J int b ∩ J b ( Q p ) ′ ) . Every component of pr − ( S ) is a pro-´etale torsorunder J int b ∩ J b ( Q p ) ′ , and conversely, if I ⊂ Ig b is an open subscheme such that I → S is a pro-´etale J int b ∩ J b ( Q p ) ′ -torsor via pr , then I is irreducible.Proof. As pr is a J int b -torsor, J int b acts transitively on π (pr − ( S )). Theorem 8.2.1 implies that theaction factors through a simply transitive action of J int b /J int b ∩ J b ( Q p ) ′ , proving the first assertion.The second assertion again follows from the same theorem and the fact that pr is a J int b -torsor. (cid:3) Since similar irreducibility results are stated in the literature over µ -ordinary Newton strata, itis worth verifying that a µ -ordinary Newton stratum is itself a central leaf; thus Ig b is a pro-´etaletorsor over the entire µ -ordinary Newton stratum (after taking perfection). This should be wellknown, but we have not found a convenient reference for the general statement.For the remainder of this subsection, we compare with similar irreducibility results in the µ -ordinary case. Thus we specialize to the case when [ b ] ∈ B ( G, µ − p ) is µ -ordinary , meaning eitherof the following equivalent conditions [Wor, Rem. 5.7 (2)]: OF IGUSA VARIETIES 67 • [ b ] = [ µ − p ( p )] in B ( G ) (which implies [ b ] ∈ B ( G, µ − p )). • [ b ] is the unique minimal element in B ( G, µ − p ) for the partial order (cid:22) therein.In this case, we may and will take b = b ◦ = µ − p ( p ). Indeed, we can change b within its σ -conjugacyclass thanks to Proposition 6.2.2. Put r := [ k ( p ) : F p ]. By the convention of § µ p is definedover Q p r . With this choice of r, µ p , b , we have ν b = r P r − i =0 σ i µ − p (this follows from (4.3.1)–(4.3.3)of [Kot85] with n = r and c = 1), which is defined over Q p , and conditions (br2) and (br3) aresatisfied.We define the µ -ordinary Newton stratum N b,K p as in [Wor], that is, by changing the definition of C b,K p ( § p . Then C b,K p ⊂ N b,K p isclosed by [Ham17, § C b,K p = N b,K p , so that Ig b is a pro-´etaletorsor over N perf b,K p (not just C perf b,K p ); this is the perfection of the usual setup in the literature. Lemma 8.2.3.
In the µ -ordinary setup above, C b,K p = N b,K p .Proof. Proposition 5.3.4 (for the first equality) and the σ -invariance of ρ (since positive roots arecoming from a Borel subgroup over Q p ; this is used for the third equality) imply thatdim C b,K p = h ρ, ν b i = r − r − X i =0 h ρ, σ i µ p i = h ρ, µ p i = dim S K p K p ,k ( p ) . Combined with (HO ′ disc ), this tells us that C b,K p is dense in S K p K p ,k ( p ) . It follows that C b,K p isdense and closed in N b,K p . Since N b,K p is reduced (as it is open in the smooth variety S K p K p ,k ( p ) ),we conclude that C b,K p = N b,K p as schemes. (cid:3) Remark . Van Hoften informed us that the lemma is also a consequence of the following twofacts: that the µ -ordinary Newton stratum is an Ekedahl-Oort stratum [Wor, Thm. 6.10], and thatan Ekedahl-Oort stratum is a central leaf [SZ, Thm. D].We explain Corollary 8.2.2 gives another proof for the irreducibility of Igusa towers in the ( µ -)ordinary case, in the setting of unitary similitude PEL-type Shimura varieties as in [CEF + § § { Ig µ -ord m,K p } mge for the Igusa tower { (Ig µ ) m, } m ≥ over the µ -ordinary stratum N b,K p in [EM, § K p ) with finite ´etale transition maps. The scheme Ig µ -ord K p = lim ←− m Ig µ -ord m,K p is a pro-´etale J int b -torsor over N b,K p . Then Ig b,K p ≃ (Ig µ -ord K p ) perf compatibly with the actionsof G ( A ∞ ,p ) × S b (see Prop. 4.3.8 and the paragraph above Cor. 4.3.9 in [CS17]; see also [CS,Rem. 2.3.7]), and we have a J b ( Q p )-equivariant bijection π ( Ig b,K p ) ≃ π ((Ig µ -ord K p ) perf ) ≃ π (Ig µ -ord K p ) . Therefore each connected component of Ig µ -ord K p has stabilizer J b ( Q p ) ′ in J b ( Q p ).The Z p -group J µ in [EM, Rem. 2.9.3] has the property that J int b = J µ ( Z p ). Let I ⊂ Ig µ -ord K p denote the open subscheme Ig SUµ over a fixed component S of N b,K p as defined in [EM, § SUµ over F p ). Then I is a pro-´etale J int b ∩ J ( Q p ) ′ -torsorover S by construction. (The determinant map of [EM, § J int b -torsor to a torsorunder G ab ( Z p ), so the fiber is a torsor under ker( J int b → G ab ( Z p )).) Hence I is irreducible by thepreceding paragraph, cf. the proof of Corollary 8.2.2.If [ b ] is moreover ordinary , that is if [ b ] is µ -ordinary and ν b is conjugate to µ − p , then µ p is definedover Q p (since the conjugacy class of ν b is always defined over Q p ) and r = [ E p = Q p ] = 1. By our In fact we have not understood the definition of the determinant map in [EM, § B is a field, so weshould restrict our comparison with loc. cit. to this setup. choice b = µ p ( p ) − , we have ν b = µ − p (not just conjugate) in this case. The following lemma ishandy when comparing with results in the ordinary case such as [Hid09, Hid11]. Note that trivially ̺ ( G sc ( Q p )) = G der ( Q p ) when G der = G sc . Lemma 8.2.5. If µ is ordinary, we have J b = M b , J int b = M b ( Z p ) , and J b ( Q p ) ′ = M b ( Q p ) ∩ ̺ ( G sc ( Q p )) .Proof. By definition, M b is the centralizer of ν b = µ − p in G . From the definition (5.3.1) with b = µ − p ( p ), we see that M b is a closed Q p -subgroup of J b . On the other hand, M b is an inner formof J b , so we conclude M b = J b . Then J int b = J b ( Q p ) ∩ G (˘ Z p ) = M b ( Q p ) ∩ G (˘ Z p ) = M b ( Z p ). Thedescription of J b ( Q p ) ′ is obvious from J b = M b . (cid:3) References [ACC + ] Patrick Allen, Frank Calegari, Ana Caraiani, Toby Gee, David Helm, Bao Le Hung, JamesNewton, Peter Scholze, Richard Taylor, and Jack Thorne, Potential automorphy over CM fields , https://arxiv.org/abs/1812.09999 .[AIP16] Fabrizio Andreatta, Adrian Iovita, and Vincent Pilloni, The adic, cuspidal, Hilbert eigenvarieties , Res.Math. Sci. (2016), Paper No. 34, 36. MR 3544971[AIP18] , Le halo spectral , Ann. Sci. ´Ec. Norm. Sup´er. (4) (2018), no. 3, 603–655. MR 3831033[Art89] James Arthur, The L -Lefschetz numbers of Hecke operators , Invent. Math. (1989), no. 2, 257–290.MR 1001841 (91i:22024)[Art96] , On local character relations , Selecta Math. (N.S.) (1996), no. 4, 501–579. MR 1443184(2000a:22017)[BDK86] J. Bernstein, P. Deligne, and D. Kazhdan, Trace Paley-Wiener theorem for reductive p -adic groups , J.Analyse Math. (1986), 180–192. MR 874050[Bor79] A. Borel, Automorphic L -functions , Automorphic forms, representations and L -functions (Proc. Sympos.Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer.Math. Soc., Providence, R.I., 1979, pp. 27–61. MR 546608 (81m:10056)[Boy] Pascal Boyer, On the irreductibility of some Igusa varieties , https://arxiv.org/abs/math/0702329 .[BS15] Bhargav Bhatt and Peter Scholze, The pro-´etale topology for schemes , Ast´erisque (2015), no. 369, 99–201.MR 3379634[BZ77] I. N. Bernstein and A. V. Zelevinsky,
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