Congruences of algebraic automorphic forms and supercuspidal representations
aa r X i v : . [ m a t h . N T ] S e p CONGRUENCES OF ALGEBRAIC AUTOMORPHIC FORMS ANDSUPERCUSPIDAL REPRESENTATIONS
JESSICA FINTZEN AND SUG WOO SHIN, WITH AN APPENDIX BY VYTAUTAS PAˇSK ¯UNAS
Abstract.
Let G be a connected reductive group over a totally real field F which iscompact modulo center at archimedean places. We find congruences modulo an arbitrarypower of p between the space of arbitrary automorphic forms on G ( A F ) and that ofautomorphic forms with supercuspidal components at p , provided that p is larger thanthe Coxeter number of the absolute Weyl group of G . We illustrate how such congruencescan be applied in the construction of Galois representations.Our proof is based on type theory for representations of p -adic groups, generalizingthe prototypical case of GL in [Sch18, §
7] to general reductive groups. We exhibit aplethora of new supercuspidal types consisting of arbitrarily small compact open subgroupsand characters thereof. We expect these results of independent interest to have furtherapplications. For example, we extend the result by Emerton–Paˇsk¯unas on density ofsupercuspidal points from definite unitary groups to general G as above. Contents
Introduction 2Guide for the reader 7Notation and Conventions 71. Construction of supercuspidal types 91.1. Construction of supercuspidal types from 0-toral data 101.2. Construction of supercuspidal types from 1-toral data 112. Construction of a family of omni-supercuspidal types 142.1. Abundance of 0-toral data and 1-toral data 142.2. Construction of omni-supercuspidal types from 0-toral and 1-toral data 203. Congruence to automorphic forms with supercuspidal components 273.1. Constant coefficients 283.2. Non-constant coefficient 313.3. An application to Galois representations 323.4. Density of supercuspidal points in the Hecke algebra 38Appendix A. Calculations for D N +1 E Introduction
Congruences between automorphic forms have been an essential tool in number theory sinceRamanujan’s discovery of congruences for the τ -function, for instance in Iwasawa theoryand the Langlands program. Over time, several approaches to congruences have been de-veloped via Fourier coefficients, geometry of Shimura varieties, Hida theory, eigenvarieties,cohomology theories, trace formula, and automorphy lifting.In this paper we construct novel congruences between automorphic forms in quite a generalsetting using type theory of p -adic groups, generalizing the argument in [Sch18, §
7] forcertain quaternionic automorphic forms. More precisely, we produce congruences mod p m (in the sense of Theorem A below) between arbitrary automorphic forms of generalreductive groups G over totally real number fields that are compact modulo center atinfinity with automorphic forms that are supercuspidal at p under the assumption that p is larger than the Coxeter number of the absolute Weyl group of G . In order to obtainthese congruences, we prove various results about supercuspidal types that we expect tobe helpful for a wide array of applications beyond those explored in this paper. Global results
In order to describe our global results in more details, let G be a connected reductivegroup over a totally real field F whose R -points are compact modulo center under everyreal embedding. Fix an open compact subgroup U p ⊂ G ( A ∞ ,pF ). Let U p ⊂ Q w | p G ( F w ) bean open compact subgroup, let A denote a commutative ring with unity, and ψ p : U p → A × a smooth character that yields an action of U p on A . Write M ( U p U p , A ) for the space of A -valued automorphic forms of level U p U p equivariant for the U p -action on A via ψ p . See § T ( U p U p,m , A ) actingon it. When A = Z p and ψ p is trivial, the corresponding space is denoted by M ( U p U p , Z p ).Now for each m ∈ Z ≥ , put A m := Z p [ T ] / (1 + T + T + · · · + T p m − ). There is an obviousring isomorphism A m / ( T − ≃ Z /p m Z induced by T G ) ∈ Z ≥ denotes the maximum ofthe Coxeter numbers of the irreducible subsystems of the absolute root system for G . (Thetable of Coxeter numbers is given above Proposition 2.1.2. If G is a torus, set Cox( G ) = 1.). Theorem A (Theorem 3.1.1) . Assume p >
Cox( G ) . Then there exist • a basis of compact open neighborhoods { U p,m } m ≥ of ∈ Q w | p G ( F w ) such that U p,m ′ is normal in U p,m whenever m ′ ≥ m , and • a smooth character ψ m : U p,m → A × m for each m ≥ ,such that we have isomorphisms of Z p / ( p m ) -modules (where the U p,m -action in M ( · ) istrivial on the left hand side and through ψ m on the right hand side) M ( U p U p,m , Z p / ( p m )) ≃ ( M ( U p U p,m , A m / ( T − ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 3 that are compatible with the action of T S ( U p U p,m , Z p / ( p m )) on both sides via the Z p -algebraisomorphism T S ( U p U p,m , Z p / ( p m )) ≃ T S ( U p U p,m , A m / ( T − . (A.ii) Moreover, every automorphic representation of G ( A F ) that contributes to ( M ( U p U p,m , A m )) ⊗ Z p Q p is supercuspidal at all places above p . In fact it is technically convenient to allow self-direct sums on both sides of (A.i), see The-orem 3.1.1 below. We also prove the analogue of the theorem for non-constant coefficientsinstead of the constant coefficient Z p . See Theorem 3.2.1 for the precise statement. Thenormal subgroup property of { U p,m } m ≥ in Theorem A is not used in the applications inthis paper, but might be helpful in some settings, e.g., see [EP20, § p between the space M ( U p U p,m , Z p ) that represents automorphic forms of arbitrary level (as one can choosesmaller U p and larger m ) and the space M ( U p U p,m , A m ) representing automorphic formsthat are supercuspidal at p . As such we expect Theorem A to be widely applicable, byreducing a question about automorphic representations to the case when a local componentis supercuspidal, for instance in the construction of automorphic Galois representations asobserved in [Sch18, Rem. 7.4]. Indeed we illustrate such an application in § p -adic Galois representations associated with regular C-algebraic con-jugate self-dual automorphic representations Π of GL N over a CM field, by reducing to theanalogous result of [Clo91, HT01] which assumes that Π has a discrete series representa-tion at a finite prime. (Compare with Theorems 3.3.1 and 3.3.3.) We achieve this via thecongruences of Theorem A, assuming p > N . Although this kind of argument is standard(cf. [Tay91, 1.3]), we supply details as a guide to utilize our theorem in an interestingcontext.We also mention a related result of Emerton–Paˇsk¯unas [EP20, Thm. 5.1] that in the spec-trum of a localized Hecke algebra of a definite unitary group, the points arising fromautomorphic representations with supercuspidal components at p are Zariski dense (when p is a prime such that the unitary group is isomorphic to a general linear group locally at p ). They start from the notion of “capture” [CDP14, § n due to Bushnell–Kutzko. While their theorem and our Theorem A do notimply each other, Paˇsk¯unas suggested to us that our local Theorem C below should providesufficient input for their argument to go through for general G as above. We confirm hissuggestion to extend their density result.To explain the statement, we assume that the center of G has the same Q -rank and R -rankas in [EP20, § H ( U p ) := lim ←− m ≥ lim −→ U p M ( U p U p , Z p / ( p m )) , JESSICA FINTZEN AND SUG WOO SHIN where the second limit is over open compact subgroups of G ( F ⊗ Q Q p ). The space˜ H ( U p ) is acted on by the “big” Hecke algebra T S ( U p ) defined as a projective limit of T S ( U p U p , Z p / ( p m )) over U p and m . Let m be an open maximal ideal of T S ( U p U p , Z p / ( p m ))for the profinite topology. It follows from the definition that classical forms are dense in˜ H ( U p ), which consists of p -adic automorphic forms, but we show that the density state-ment still holds when the component at p is required to be supercuspidal. (See § Theorem B (cf. Corollary 3.4.7 and Theorem 3.4.9) . Assume p >
Cox( G ) . Then classicalautomorphic forms with fixed weight which are supercuspidal at p form a dense subspace in ˜ H ( U p ) . In the spectrum of the m -adic completion of T S ( U p ) , such classical automorphicforms are Zariski dense. This theorem has a potential application to a torsion and p -adic functoriality result follow-ing the outline of [EP20, § G toanother group G is available for automorphic representations on G which are supercusp-idal at p . (We thank Paˇsk¯unas for pointing this out to us.) For a Jacquet–Langlands-typeexample, let G and G be the unit groups of central quaternion algebras over Q with G unramified at p but G ramified at p , and assume that the set of ramified primes awayfrom p for G is contained in that for G . Then loc. cit. constructs a transfer from G to G on the level of completed cohomology, overcoming the local obstruction at p inthe classical Jacquet–Langlands that principal series of G ( Q p ) do not transfer to G ( Q p ).See [Eme14, 3.3.2] for a related discussion. (A similar transfer of torsion classes for Shimuracurves is obtained in [Sch18, Cor. 7.3] by a somewhat different argument based on a versionof Theorem A; this approach should extend to more general groups by using our TheoremA and its variants.)Paˇsk¯unas kindly wrote Appendix C for us in which he shows that the big Hecke algebrasare Noetherian in the setup of definite unitary groups. He also constructs automorphicGalois representations for Hecke eigensystems in the completed cohomology only from theanalogous result by Clozel [Clo91] via the density result of [EP20]. In particular this givesyet another argument to remove the local hypothesis from [Clo91], which has the advantagethat no restriction on p is required, as it is the case for Bushnell–Kutzko’s type theory forGL n . For general reductive groups, we hope that Theorem B will be similarly useful. Local results
The data in Theorem A are constructed via a new variant of types for representations of p -adic groups that we call omni-supercuspidal types . The aim of the theory of types is toclassify complex smooth irreducible representations of p -adic groups up to some naturalequivalence in terms of representations of compact open subgroups. Theorems about theexistence of s -types (types that single out precisely one Bernstein component s ) lie at theheart of many results in the representation theory of p -adic groups and play an importantrole in the construction of an explicit local Langlands correspondence and the study ofits fine structure. The idea of omni-supercuspidal types is that it is harmless for some ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 5 applications if a type cuts out a potentially large family of supercuspidal representations,not just a single supercuspidal Bernstein component, but that it is important to controlthe shape of the types better.Let us elaborate. For our global application, we need to be able to choose the compact opensubgroups arbitrarily small in our types. It is also desirable to require the representationof our compact open subgroup detecting supercuspidality to be one-dimensional. Unfor-tunately the irreducible representations of the compact open subgroups that form s -typeshave neither properties in general. Nevertheless, using the theory of s -types, we show thatthere exists a plethora of omni-supercuspidal types that satisfy the two desiderata andtherefore exhibit a much easier structure than s -types. For readers who are mainly inter-ested in determining if a certain representation is supercuspidal, our omni-supercuspidaltypes (Definition 2.2.1 and Theorem 2.2.15) and the supercuspidal types arising from ourintermediate result, Proposition 2.2.4, allow therefore significantly more flexibility and eas-ier detection.To be more precise, let us introduce some notation. Let F be a finite extension of Q p , and G a connected reductive group over F with dim G ≥
1. A supercuspidal type for G ( F )means a pair ( U, ρ ), where ρ is an irreducible smooth complex representation of an opencompact subgroup U of G ( F ) such that every irreducible smooth complex representation π of G ( F ) for which π | U contains ρ is supercuspidal. Note that a supercuspidal type maypick out several Bernstein components.We define an omni-supercuspidal type of level p m (with m ∈ Z ≥ ) to be a pair ( U, λ ), where λ is a smooth Z /p m Z -valued character on an open compact subgroup U of G ( F ) such that( U, χ ◦ λ ) is a supercuspidal type for every nontrivial character χ : Z /p m Z → C ∗ . The flex-ibility allowed for χ is extremely helpful for producing congruences of automorphic forms.Our main novelty is the following theorem about the existence of such omni-supercuspidaltypes, where Cox( G ) is defined as before. Theorem C (Theorem 2.2.15) . Suppose p >
Cox( G ) . Then there exists an infinite se-quence of omni-supercuspidal types { ( U m , λ m ) } m ∈ Z ≥ such that ( U m , λ m ) has level p m and { U m } m ∈ Z ≥ forms a basis of open neighborhoods of and such that U m ′ is normal in U m whenever m ′ ≥ m . Our proof provides an explicit description of U m and λ m using the Moy–Prasad filtration.To give an outline of our approach, suppose for simplicity of exposition that G is anabsolutely simple group. (The reduction to this case is carried out in the proof of Theorem2.2.15.) Then G is tamely ramified thanks to the assumption that p > Cox( G ). Let T bea tamely ramified elliptic maximal torus of G , and φ : T ( F ) → C ∗ a smooth character ofdepth r ∈ R > that is G -generic of depth r in the sense of [Yu01, § T, r, φ ) a .Then our approach is to show (a) that there is an infinite supply of 0-toral data (with sametorus T ) with r → ∞ and (b) that each 0-toral datum gives rise to an omni-supercuspidal JESSICA FINTZEN AND SUG WOO SHIN type whose level grows along with r . (For general G we need to consider 1-toral data,introduced in Definition 1.2.1, as there may not be enough 0-toral data.)To prove the abundance of 0-toral data (Proposition 2.1.2 below; see Proposition 2.1.6for a result on 1-toral data), we exhibit a G -generic element (of some depth) in the dualof the Lie algebra of T . After reducing to the case that G is quasi-split, we construct T by giving a favorable Galois 1-cocycle to twist a maximally split maximal torus. Toexhibit a G -generic element, we use the Moy–Prasad filtration and eventually demonstratea solution to a certain system of equations; here an additional difficulty comes from aGalois-equivariance condition. While we treat most cases uniformly, we carry out someexplicit computations for types D N +1 and E that can be found in Appendices A and B.The second step is to construct omni-supercuspidal types from 0-toral data. By makingsome additional choices one can enlarge the 0-toral datum to an input for the constructionof Adler ( [Adl98]) and Yu ( [Yu01]). The construction of Adler and Yu yields then asupercuspidal type ( K, ρ ) such that the compact induction c-ind G ( F ) K ρ is irreducible andsupercuspidal. Unfortunately, ρ may not be a character, and the groups K (as r and φ vary) do not form a basis of open neighborhoods because K ⊃ T ( F ). However, ρ restrictedto the Moy–Prasad filtration subgroup G y,r ⊂ K , where y denotes the point of the Bruhat–Tits building of G corresponding to T , is given by a character ˆ φ . We show that ( G y,r , ˆ φ )is a supercuspidal type (which in the 0-toral datum case essentially follows from Adler( [Adl98]), but we also treat the more complicated 1-toral datum case, see Proposition2.2.4). The character ˆ φ can also be defined on the larger group G y, r + , and ˆ φ | G y, r factorsthrough a surjective Z /p m Z -valued character λ , where m is proportional to r . Finally wededuce that ( G ( F ) y, r + , λ ) is an omni-supercuspidal type of level p m .Let us remark on the case when F is a local function field in characteristic p . In fact weprove most intermediate results for arbitrary nonarchimedean local fields. In particularProposition 2.2.4 also holds for local function fields. This proposition of independentinterest provides a rather small compact open subgroup together with a character (thepair ( G y,r , ˆ φ ) in the case of 0-toral data) detecting supercuspidality. It is only in thelast crucial step when moving from these one-dimensional supercuspidal types to omni-supercuspidal types that we require the field F to have characteristic zero. The reason isthat for function fields F every element of G ( F ) y, r + /G ( F ) y,r + has order p (as the quotient isa vector space over a finite field) while we require elements of order p m for our constructionof omni-supercuspidal types in order to achieve that λ surjects onto Z /p m Z . From the local results to the global results
We sketch how to obtain Theorem A from Theorem C by adopting an idea from [Sch18, § We use the notation from Theorem A and assume for simplicity that F = Q . From TheoremC we obtain a sequence of omni-supercuspidal types ( U p,m , λ m ) m ∈ Z ≥ of level p m for the This is a variant of the older idea to produce congruences of group cohomology via two coefficientmodules which contain common factors modulo p m . See [Tay88, p.5] for instance. ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 7 p -adic group G × Q Q p . We define the character ψ m to be the composite map ψ m : U p,m λ m ։ Z /p m Z ֒ → A × m , where the second map sends a mod p m to T a . Then ψ m mod ( T −
1) is the trivial characteron A m / ( T − ≃ Z p / ( p m ), giving the isomorphism (A.i) of Theorem A, which can bechecked to be Hecke equivariant. The omni-supercuspidal property of ( U p,m , ψ m ) impliesthat M ( U p U p,m , A m ) ⊗ Z p Q p is accounted for by automorphic representations with super-cuspidal components at p . In fact this consideration initially motivated our definition ofomni-supercuspidal types. Guide for the reader
The structure of the paper should be clear from the table of contents, but we guide thereader to navigate more easily. At a first reading, the reader might want to concentrate on0-toral data ( § § § §
3, taking Theorem 2.2.15 on faith. Sections 3.1 and3.2 are concerned with the application of Theorem 2.2.15 to build congruences betweenautomorphic forms. While § § § § Notation and Conventions
We assume some familiarity with [Yu01] in that we do not recall every definition or notionused in [Yu01] (e.g. Bruhat–Tits buildings, Moy–Prasad subgroups). However we doprovide precise reference points for various facts we import from the paper.The symbol for the trivial representation (of any group) is . Write Z > (resp. Z ≥ ) forthe set of positive (resp. nonnegative) integers. JESSICA FINTZEN AND SUG WOO SHIN
Every reductive group is assumed to be connected and nontrivial (so that dim ≥
1) in themain text without further comments.Let k be a field. Let k (resp. k sep ) denote an algebraic (resp. separable) closure of k . Whenconsidering algebraic field extensions of k , we consider them inside the algebraic closure k ,which we implicitly fix once and for all. (For instance, this applies when k is the base field F of the main text, which is local in the first two sections and global in the last section.)Let X be an affine k -scheme, and k ′ /k be a field extension and k ′′ a subfield of k . Thenwe write X × k k ′ or X k ′ for the base change X × spec( k ) spec( k ′ ), and Res k/k ′′ X for the Weilrestriction of scalars (which is represented by a k ′′ -scheme).Let F be a nonarchimedean local field. Then we write O F for the ring of integers and k F forthe residue field. We use v : F → Q ∪ {∞} to designate the additive p -adic valuation mapsending uniformizers of F to 1, and we write | · | F : F × → R × > for the modulus charactersending uniformizers to ( k F ) − . We fix a nontrivial additive character Ψ on F which isnontrivial on O F but trivial on elements with positive valuations. If E is a finite extensionof F , then we can extend Ψ to E . We fix such an extension and denote it by Ψ as well.Let G be a reductive group over F (connected by the aforementioned convention). We say G is tamely ramified (over F ) if some maximal torus of G splits over a tamely ramifiedextension of F . We write G ad for the adjoint quotient of G , and g for the Lie algebra of G . By Z ( G ) we mean the center of G . Denote the absolute Weyl group of G by W = W G ,and its Coxeter number by Cox( G ). For a (not necessarily maximal) split torus T ⊂ G ,we denote by Φ( G, T ) the set of ( F -rational) roots of G with respect to T . For each α ∈ Φ( G, T ) we write ˇ α : G m → T for the corresponding coroot, and g α for the subspaceof g on which T acts via α .We write B ( G, F ) for the (enlarged) Bruhat–Tits building of G over F . When F ′ is afinite tamely ramified extension of F , and T a maximally F ′ -split maximal torus of G F ′ (defined over F ′ ), we write A ( T, F ′ ) for the apartment of T over F ′ . Both B ( G, F ) and A ( T, F ′ ) are embedded in B ( G, F ′ ) so their intersection as in D2 of § y ∈ B ( G, F ′ ) and s ∈ R ≥ (resp. s ∈ R ), let G ( F ′ ) y,s (resp. g ( F ′ ) y,s )denote the Moy–Prasad filtration in G ( F ′ ) (resp. g ( F ′ )). All Moy–Prasad filtrations arenormalized with respect to the fixed valuation v . Write G ( F ′ ) y,s + := ∪ r>s G ( F ′ ) y,r and g ( F ′ ) y,s + := ∪ r>s g ( F ′ ) y,r . Moreover, we denote by [ y ] the image of the point y in thereduced Bruhat–Tits building, and we write G ( F ′ ) [ y ] for the stabilizer of [ y ] in G ( F ′ )(under the action of G ( F ′ ) on the reduced Bruhat–Tits building). We often abbreviate G ( F ) y,s , g ( F ) y,s , etc as G y,s , g y,s , etc, when F is the base field. Similarly we may abusenotation and write g for g ( F ). We denote by g ∗ the F -linear dual of g ( F ), and write g ∗ y,s for its Moy–Prasad filtration submodule at y of depth s . More generally, if V is an F ′ -vector space, then V ∗ denotes its F ′ -linear dual. Let K be an open and closed subgroupof G ( F ′ ). For a smooth representation ρ of K , we write c-ind G ( F ′ ) K ρ for the compactlyinduced representation, defined to be the subspace of the usual induction consisting ofsmooth functions on G ( F ′ ) whose supports are compact modulo K . ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 9
Let A denote the ring of ad`eles of Q . Write A k := A ⊗ Q k when k is a finite extension of Q . When S is a set of places of k , we denote by A Sk the subring of elements in A k whosecomponents are zero at the places in S . E.g. A ∞ k denotes the ring of finite ad`eles over k . Acknowledgments
Both authors heartily thank the organizers (Francesco Baldassarri, Stefano Morra, MatteoLongo) of the 2019 Padova School on Serre conjectures and the p -adic Langlands program,where the collaboration got off the ground, and the organizers (Brandon Levin, RebeccaBellovin, Matthew Emerton and David Savitt) of the Modularity and Moduli Spaces work-shop in Oaxaca in 2019 for giving them a valuable opportunity to discuss with MatthewEmerton and Vytautas Paˇsk¯unas. The authors are grateful to Emerton and Paˇsk¯unasfor explaining their paper [EP20]. Special thanks are owed to Paˇsk¯unas for writing anappendix and helping with § Construction of supercuspidal types
Let F be a nonarchimedean local field with residue field k F . The characteristic of k F is aprime p >
0. Let G be a connected reductive group over F . (Most results in the first twosections hold in this generality, sometimes under the condition that p exceeds the Coxeternumber of G , except that some key statements in § F ) = 0.) To fix theidea, every representation is considered on a C -vector space in the first two sections, buteverything goes through without change with an algebraically closed field of characteristiczero (e.g. Q p ) as the coefficient field. In § Definition 1.0.1.
Let ρ be a smooth finite-dimensional representation of an open compactsubgroup U ⊂ G ( F ). We call such a pair ( U, ρ ) a supercuspidal type if every irreduciblesmooth representation π of G ( F ) with Hom U ( ρ, π ) = { } is supercuspidal.If ( U, ρ ) is a supercuspidal type, then it cuts out (possibly several) supercuspidal Bernsteincomponents, thus deserving the name. In type theory, it is typical to construct a supercus-pidal type singling out each individual supercuspidal Bernstein component when possible.We do not insist on this but instead ask for other properties (cf. § G splits over a tamely ramified extensionof F . Construction of supercuspidal types from 0-toral data.Definition 1.1.1. A (for G ) consists of a triple ( T, r, φ ), where • T ⊂ G is a tamely ramified elliptic maximal torus over F , • r ∈ R > , • φ : T ( F ) → C ∗ is a smooth character of depth r . If G = T , then we require φ to be G -generic in the sense of [Yu01, §
9] (relative to any, or, equivalently, every point in B ( G, F ) ∩ A ( T, F ′ ), where F ′ denotes a finite tame extension of F over which T is split).A 0-toral-datum gives rise to the following input for Yu’s construction of supercuspidalrepresentations, where we use the notation of [Yu01, §
3] (see also [HM08, § G = T is an elliptic maximal torus of G = G over F , and T is split over a finitetamely ramified extension F ′ /F .D2: y ∈ B ( G, F ) ∩ A ( T, F ′ ) is a point, which we fix once and for all. (Note that theimage [ y ] of the point y in the reduced Bruhat–Tits building does not depend onthe choice of y .)D3: r = r = r > ρ is the trivial representation of K := T ( F ).D5: φ = φ : T ( F ) → C ∗ is a character that is G -generic (relative to y ) of depth r inthe sense of [Yu01, § φ is trivial. Remark . Note that if G = T , then this datum is strictly speaking not satisfyingCondition D1 of [Yu01, §
3] because Condition D1 requires G ( G . However, as Yu alsopoints out in [Yu01, §
15, p.616], we can equally well work with this “generalized” datum.Later we will vary r and φ for a given T . The point y will remain fixed. Following Yu,we write K := T ( F ) G ( F ) y, r and K := T ( F ) y G ( F ) y, r . In [Yu01], Yu constructs anirreducible smooth representation ρ of K , and letting ρ := ρ | K , shows the following. Theorem 1.1.3 ( § . The com-pactly induced representation π := c-ind G ( F ) K ρ , is irreducible and supercuspidal of depth r . Moreover, every irreducible smooth representa-tion π ′ of G ( F ) with Hom K ( ρ , π ′ ) = { } is in the same Bernstein component as π . Inparticular, ( K , ρ ) is a supercuspidal type.Remark . These representations were already among those constructed by Adler in[Adl98]. However, we have used Yu’s notation here to be consistent with Section 1.2, inwhich we introduce a class of representations that is more general than that arising froma 0-toral-data. We disregard r = 0 as it is enough to consider positive depth for our intended applications. ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 11
Remark . The representations obtained from 0-toral data are the same as the ones con-structed from what DeBacker and Spice call “toral cuspidal G -pairs” in [DS18]. DeBackerand Spice refer to the resulting representations as “toral supercuspidal representations” intheir introduction.During Yu’s construction of ρ , he constructs a smooth character ˆ φ : T ( F ) G ( F ) y, r + → C ∗ (denoted ˆ φ in [Yu01]), which is trivial on ( T, G )( F ) y, ( r + , r +) . Let J := ( T, G )( F ) y, ( r, r +) .Then we show Proposition 1.1.6.
Assume p does not divide the order of the absolute Weyl group of G .Then the pair ( J , ˆ φ | J ) is a supercuspidal type.Proof. This is a special case of Proposition 1.2.6 proved in the next section. (cid:3)
Construction of supercuspidal types from 1-toral data.
Since non-simple re-ductive groups might not admit a 0-toral datum in general, we introduce a slightly moregeneral notion.
Definition 1.2.1. A is a tuple (( G , . . . , G d ) , ( r , . . . , r d − ) , ( φ , . . . , φ d − )),where • G is an elliptic, maximal torus of G and either – G = G = G , or – G ( G ( · · · ( G d − ( G d = G is a sequence of twisted Levi subgroups thatsplit over some tamely ramified extension F ′ of F , • < r < r < . . . < r d − < r + 1 is a sequence of real numbers, • φ i is a smooth character of G i ( F ) of depth r i that is G i +1 -generic relative to y ofdepth r i (if G = G ) for 0 ≤ i ≤ d − y ∈ B ( G, F ) ∩ A ( G , F ′ ).Since G is a torus, we will also write T instead of G .A 1-toral datum gives rise to a tuple(( G , . . . , G d ) , y, ( r , . . . , r d − , r d = r d − ) , ρ, ( φ , . . . , φ d − , φ d = 1)) (1.2.2)as in [Yu01, §
3] (or as in [Yu01, §
15] in the case of G = T ) by setting • y to be a point in B ( G, F ) ∩ A ( G , F ′ ), which we fix once and for all (note that[ y ] does not depend on the choice of y in B ( G, F ) ∩ A ( G , F ′ )), • ρ to be the trivial representation. Remark . A 0-toral datum is a 1-toral datum with the additional condition that d = 1.For the reader interested in our choice of nomenclature: The “0” in “0-toral” refers to thedifference between r d and r being 0, while the “1” in “1-toral” is motivated by the differencebetween r d and r being smaller than 1. Remark . The decision to set φ d = 1 is not a serious restriction (and could be removedif desired). By setting φ d = 1 we only exclude additional twists by characters of G ( F ) if G is not a torus. This convention has the advantage that the translation between the notionin [Yu01] and the notation in [Finb] is easier, i.e. we do not have to distinguish the cases φ d = 1 and φ d = 1.For 0 ≤ i ≤ d −
1, we write H i for the derived subgroup of G i . We set H d := G d (not thederived subgroup of G d unless G d is semisimple). We abbreviate for 1 ≤ i ≤ d , G iy,r i − , ri − + := ( G i − , G i )( F ) y, ( r i − , ri − +) ,H iy,r i − , ri − + := H i ( F ) ∩ G iy,r i − , ri − + . and define h iy,r i − , ri − + analogously, where g i and h i denote the Lie algebras of G i and H i ,respectively, and set J := H y,r , r + H y,r , r + . . . H d − y,r d − , rd − + H dy,r d − , rd − + , = H y,r , r + H y,r , r + . . . H d − y,r d − , rd − + G dy,r d − , rd − + .K d := G y ] H y,r , r H y,r , r . . . H d − y,r d − , rd − H dy,r d − , rd − , K d := G y H y,r , r H y,r , r . . . H d − y,r d − , rd − H dy,r d − , rd − . From the tuple (1.2.2), Yu constructs in [Yu01, §
4] an irreducible representation ρ d of K d such that the following theorem holds, letting ρ d := ρ d | K d . Theorem 1.2.5 (Prop 4.6, Thm 15.1, Cor 15.3 of [Yu01] and Thm 3.1 of [Fina]) . Thecompactly induced representation π := c-ind G ( F ) K d ρ d , is irreducible and supercuspidal of depth r d . Moreover, every irreducible smooth represen-tation π ′ of G ( F ) with Hom K d ( ρ d , π ′ ) = { } is in the same Bernstein component as π .In particular, ( K d , ρ d ) is a supercuspidal type. The representation ρ d restricted to J is given by the character ˆ φ = Q ≤ i ≤ d − ˆ φ i | J (times identity), where ˆ φ i is defined as in [Yu01, § φ i is the unique character of( G ) [ y ] ( G i ) y, G y, ri + that satisfies • ˆ φ i | ( G ) [ y ] ( G i ) y, = φ i | ( G ) [ y ] ( G i ) y, , and • ˆ φ i | G y, ri factors through G y, ri + /G y,r i + ≃ g y, ri + / g y,r i + = ( g i ⊕ r i ) x, ri + / ( g i ⊕ r i ) x,r i + → ( g i ) x, ri + / ( g i ) x,r i + ≃ ( G i ) x, ri + / ( G i ) x,r i + , ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 13 on which it is induced by φ i . Here r i is defined to be g ∩ L α ∈ Φ( G F ′ ,T F ′ ) \ Φ( G iF ′ ,T F ′ ) g ( F ′ ) α for some maximal torus T of G i that splits over a tame extension F ′ of F with y ∈ A ( T, F ′ ), and the surjection g i ⊕ r i ։ g i sends r i to zero. Proposition 1.2.6.
Assume p does not divide the order of the absolute Weyl group of G .Then the pair ( J , ˆ φ | J ) is a supercuspidal type.Remark . The hypothesis on p may not be optimal, but imposed here to importresults from [Finb, Kal19] which assume it. The condition clearly holds if p is larger thanthe Coxeter number of the Weyl group. Proof of Proposition 1.2.6.
The statement is obvious if G is a torus, hence we assume that G is not a torus for the remainder of the proof. Let ( π, V ) be an irreducible smoothrepresentation of G ( F ) such that V contains a one-dimensional subspace V ′ on which thesubgroup J ⊂ G ( F ) acts via ˆ φ . We need to show that π is supercuspidal.Our strategy consists of deriving from the action of J ⊂ G ( F ) via ˆ φ on V ′ a maximaldatum for ( π, V ) (in the sense of [Finb, Def. 4.6]). This can be achieved because thecharacter ˆ φ on J encodes the information of a truncated extended datum in the senseof [Finb, Def. 4.1]. The fact that G is a torus, hence has trivial derived group, allows us tocomplete the truncated extended datum to a maximal datum for ( π, V ) by adding the trivialrepresentation of ( G ) der = { } . Then we can apply [Finb, Cor. 8.3], which is a criterion todeduce that ( π, V ) is supercuspidal from properties of the previously constructed maximaldatum for ( π, V ). Let us provide the details.Since p does not divide the order of the absolute Weyl group of G , the character φ j is trivial on H j ( F ) ∩ G j ( F ) y, ( [Kal19, Lemma 3.5.1]). Hence ˆ φ j | H i +1 y,ri, ri is trivial for0 ≤ i < j < d . Moreover, ˆ φ j | H i +1 y,ri, ri is trivial for d > i > j ≥ φ j . Thus ˆ φ | H i +1 y,ri, ri = ˆ φ i | H i +1 y,ri, ri . We let X i ∈ g ∗ y, − r i such thatthe character ˆ φ | H i +1 y,ri, ri = ˆ φ i | H i +1 y,ri, ri viewed as a character of H i +1 y,r i , ri + /H i +1 y,r i + ≃ h i +1 y,r i , ri + / h i +1 y,r i + is given by Ψ ◦ X i . Since φ i is G i +1 -generic relative to y of depth r i , by Yu’s definitionof genericity [Yu01, § §
9] we can choose X i to have the following extra properties:firstly X i ∈ (Lie ( Z ( G i ))( F )) ∗ ⊂ ( g i ) ∗ (see [Yu01, §
8] for the definition of this inclusion),and secondly v ( X i ( H ˇ α )) = − r i for all α ∈ Φ( G i +1 F sep , T F sep ) \ Φ( G iF sep , T F sep ) , where H ˇ α := d ˇ α (1) ⊂ Lie T ( F sep ) ⊂ g i ( F sep ). (Here d ˇ α : G a → Lie T denotes the Liealgebra morphism arising from ˇ α : G m → T , and recall that we write T = G .) Since X i ∈ (Lie ( Z ( G i ))( F )) ∗ , we have that X i ( H ˇ α ) = 0 for all α ∈ Φ( G iF sep , T F sep ) and that the G i +1 -orbit of X i is closed.Thus X i is almost stable and generic of depth − r i at y as an element of ( g i +1 ) ∗ in thesense of [Finb, Definition 3.1 and Definition 3.5]. By [Finb, Cor. 3.8.] this implies that X i is almost strongly stable and generic of depth − r i at y . Moreover, since φ i is G i +1 -generic,we have G i = Cent G i +1 ( X i ). Hence the tuple ( y, ( r i ) d − ≥ i ≥ , ( X i ) d − ≥ i ≥ , ( G i ) d ≥ i ≥ ) is atruncated extended datum of length d in the sense of [Finb, Def. 4.1]. Since J acts on V ′ via ˆ φ , since ( G ) der = T der = { } , and since B ( G , F ) consists of a single facet, we concludethat ( y, ( X i ) d − ≥ i ≥ , ) is a maximal datum for ( π, V ) in the sense of [Finb, Def. 4.6].Since y is a facet of minimal dimension of B ( G , F ) = B ( T, F ) and Z ( G ) /Z ( G ) isanisotropic, [Finb, Cor. 8.3] implies that ( π, V ) is supercuspidal. (cid:3) Construction of a family of omni-supercuspidal types
We have seen that 0-toral and 1-toral data yield supercuspidal types. Upgrading this,we will see in this section that 0-toral data and 1-toral data give rise to what we callomni-supercuspidal types (see Definition 2.2.1 below), which eventually lead to interestingcongruences of automorphic forms in the global setup. Thus it is crucial to have a familyof omni-supercuspidal types of level tending to infinity, whose existence we prove in thissection.2.1.
Abundance of 0-toral data and 1-toral data.
The current subsection is devotedto show that 1-toral data exist in abundance under a mild assumption on p . We constructan infinite family of 0-toral data of increasing level for absolutely simple groups as anintermediate step.We begin with a preliminary lemma that will be useful for explicit constructions later. Lemma 2.1.1.
Let n ∈ Z ≥ and let E be the degree n unramified extension of F . Let σ be a generator of Gal(
E/F ) . Suppose that p ∤ n . Then there exists an element e in E withthe following properties: • v ( e ) = 0 • P ≤ i ≤ n σ i ( e ) = 0 • The image ¯ e of e in the residue field k E of E is a generator for the field extension k E /k F .Proof. Let e ∈ E \ F be an element of valuation zero whose image e in k E generatesthe extension of finite fields k E /k F . If P ≤ i ≤ n σ i ( e ) = 0, then we are done, so suppose P ≤ i ≤ n σ i ( e ) = 0. Set e := P ≤ i ≤ n σ i ( e ) and e := n · e − e . Then σ ( e ) = e and e ∈ O F . Hence the image ¯ e of e := n · e − e in k E is a generator for the field extension k E /k F and therefore v ( e ) = 0. Moreover P ≤ i ≤ n σ i ( e ) = n P ≤ i ≤ n σ i ( e ) − ne = 0, so theproof is finished. (cid:3) Beware that the sequence of twisted Levi subgroups is increasing in Yu’s data, but decreasing in [Finb].
ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 15
Recall that we write Cox( G ) ∈ Z ≥ for the Coxeter number of the absolute Weyl group W of G . Note that if G is an absolutely simple reductive group and p > Cox( G ), then G splits over a tamely ramified extension of F . For the reader’s convenience, we recall thetable of Coxeter numbers for irreducible root systems of all types:type of W A s B s ( s ≥ C s ( s ≥ D s ( s ≥ E E E F G Cox( G ) s + 1 2 s s s − Proposition 2.1.2.
Let G be an absolutely simple reductive group over F . Assume that p > Cox( G ) . Then there exists a tamely ramified elliptic maximal torus T of G that enjoysthe following property: For every n ∈ Z ≥ , there exist a real number r with n < r ≤ n + 1 and a character φ : T ( F ) → C ∗ of depth r such that ( T, r, φ ) is a 0-toral datum.Proof. We first claim that it suffices to show that G contains an elliptic maximal torus T such that for every n ∈ Z ≥ , there exists a G -generic element X n ∈ t ∗ of depth − r in thesense of [Yu01, §
8] with n < r ≤ n + 1. Suppose that T is such a torus accommodatingsuch an X n for every n . Let y ∈ A ( T, F ′ ) ∩ B ( G, F ) as in § X n with the fixed additive character Ψ to obtain a generic character φ of t r / t r + = t y,r / t y,r + ≃ T y,r /T y,r + , which we can view as a character of T y,r . Since the character takes image in thedivisible group C ∗ , we can extend it to a character of T ( F ), which we also denote by φ .Then φ is a G -generic character of T ( F ) of depth r and ( T, r, φ ) is a 0-toral datum.Let us show the existence of T and X n as above. For simplicity we write X for X n whenthere is no danger of confusion. Since over a non-archimedean local field every anisotropic,maximal torus transfers to all inner forms (see [Kot86, §
10] and [Kal19, Lemma 3.2.1]),we may assume without loss of generality that G is quasi-split . (See [Kal19, 3.2] to reviewwhat transfer of maximal tori means.) Let T sp be a maximal torus of G that is maximallysplit. Denote by X ∗ ( T sp ) the group of cocharacters of T sp × F F sep . Let E ′ be the splittingfield of T sp , which is finite Galois over F . Then the action of Gal( F sep /F ) on X ∗ ( T sp )factors through Gal( E ′ /F ). We denote by W sp = N ( T sp ) /T sp the Weyl group (scheme) of T sp ⊂ G .Before we go further, we outline the strategy for the rest of the proof. Firstly, we will choosea finite Galois extension E/F (in F sep ) as well as a 1-cocycle of Gal( E/F ) with values in W sp . A careful choice will allow us to “twist” T sp to an elliptic maximal torus T over F .Put ˜ E := EE ′ . (In fact we will have ˜ E = E except possibly for type D N .) Secondly,we choose r ∈ ( n, n + 1] in such a way that the remaining steps will work and describean O ˜ E -linear functional X on the finite free O ˜ E -module t ( ˜ E ) r by fixing a convenient basisfor t ( ˜ E ) r . Thirdly, we make explicit the conditions that X ∈ t ∗ (namely X is Gal( ˜ E/F )-equivariant as a linear functional) and that X is G -generic of depth − r . This yields a list ofconstraints for the coordinates of X for the fixed (dual) basis. Finally we exhibit a choiceof coordinates satisfying all constraints.We are about to divide the proof into two cases. The case of type D N is to receive aspecial treatment. Assuming G is not of type D N , we define δ ∈ Aut Z ( X ∗ ( T sp )) encoding the action of Gal( E ′ /F ) on X ∗ ( T sp ) as follows. If G is split, then E ′ = F anddefine δ to be the identity automorphism. If G is not split (still excluding D N ), then[ E ′ : F ] = 2. Write σ for the nontrivial element of Gal( E ′ /F ), and let δ stand for theaction of σ in this case.Viewing the absolute Weyl group W := W sp ( F sep ) as a subgroup of Aut Z ( X ∗ ( T sp )), wedistinguish two cases as follows. Case 1: G is of type D N , or − ∈ W δ
First suppose that G is not of type D N . Let E be a quadratic extension of F thatcontains E ′ . Then there exists w ∈ W sp ( E ) such that wδ = − − X ∗ ( T sp )). Let ¯ f : Gal( F sep /F ) → W be the group homomorphism that factors throughGal( E/F ) and sends the non-trivial element σ of Gal( E/F ) to w . Then ¯ f is a 1-cocyclegiving an element of the Galois cohomology H ( F, W sp ), because σ ∈ Gal(
E/F ) acts on W via conjugation by δ .If G is of type D N , observe that Gal( E ′ /F ) is one of the following groups: { } , Z / Z , Z / Z ,or S . By [Fin19, Lemma 2.2] there exists a quadratic extension E of F such that E ∩ E ′ = F , i.e. Gal( EE ′ /F ) = Gal( E/F ) × Gal( E ′ /F ) canonically. Since the cen-ter of W is {± } in this case ([Hum90, 3.19 Cor., Table 3.1, 6.3 Prop.(d)]), we have − ∈ W sp ( F ). Then the group homomorphism ¯ f : Gal( F sep /F ) → W defined by factoringthrough Gal( E/F ) and sending the non-trivial element σ of Gal( E/F ) to − H ( F, W sp ).By [Rag04, Main Theorem 1.1] every element of H ( F, W sp ) lifts to an element in H ( F, N ( T sp ))that is contained in ker( H ( F, N ( T sp )) → H ( F, G )). Let us denote by f such a lift of ¯ f (inboth cases). Then f gives rise to (the conjugacy class of) a maximal torus T in G over F .The torus T is split over EE ′ , and the nontrivial element σ of Gal( E/F ) ≤ Gal( EE ′ /F )acts on X ∗ ( T ) via f ( σ ) δ = −
1, where we set δ = 1 if G is of type D N .Hence T is an elliptic maximal torus of G .To avoid convoluted notation, we separate cases according as the quadratic extension E/F is unramified or ramified. Write e E = EE ′ .Suppose first that E/F is unramified, and let ̟ F be a uniformizer of F . Since p > Cox( G ),hence p ∤ | W | , the O e E -module t ( e E ) y,n +1 is spanned by { ̟ n +1 F H ˇ α } ˇ α ∈ ˇ∆ as a free module,where ˇ∆ is a choice of simple coroots of T × F e E and H ˇ α = d ˇ α (1), where d ˇ α denotes themap G a → Lie ( T × F e E ) induced by the coroot ˇ α : G m → T . If G is of type D N weassume in addition that ˇ∆ is preserved under the action of Gal( ˜ E/E ), which is possible bythe definition of T and G being quasi-split. (Choose ˇ∆ corresponding to a Borel subgroupof G E over E containing T E , noting that T E ≃ T sp E .) Let a ∈ E with v ( a ) = 0 such that σ ( a ) = − a , which exists by Lemma 2.1.1. Let X be the O e E -linear functional on t ( e E ) y,n +1 defined by X ( ̟ n +1 F H ˇ α ) = a for ˇ α ∈ ˇ∆. Then X ( σ ( ̟ n +1 F H ˇ α )) = X ( ̟ n +1 F H − ˇ α ) = X ( − ̟ n +1 F H ˇ α ) = − a = σ ( a ) = σ ( X ( ̟ n +1 F H ˇ α )) , ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 17 and X is also stable under Gal( E ′ /F ). Hence X defines an O F -linear functional on t x,n +1 =( t ( e E ) x,n +1 ) Gal( e E/F ) . Note that for every coroot ˇ β (of G e E with respect to T e E ), we have X ( H ˇ β ) = m ˇ β a̟ − ( n +1) F for some non-zero integer m ˇ β with − Cox( G ) < m ˇ β < Cox( G ),because the sum of the coefficients of the highest coroot in terms of simple coroots isCox( G ) −
1. Hence, since p >
Cox( G ), we have v ( X ( H ˇ β )) = − ( n + 1) for all coroots ˇ β of G e E with respect to T e E . Thus X is G -generic of depth n + 1 in the sense of [Yu01, §
8] byour assumption on p and [Yu01, Lemma 8.1].It remains to treat the case that E/F is totally ramified. Since p = 2, the quadratic exten-sion E/F is tamely ramified and we may choose a uniformizer ̟ E of E such that σ ( ̟ E ) = − ̟ E . Then, as p > Cox( G ), the O e E -module t ( e E ) y, n +12 is generated by { ̟ n +1 E H ˇ α } ˇ α ∈ ˇ∆ ,where ˇ∆ is chosen as above. We define an O e E -linear functional X on t ( e E ) y, n +12 by setting X ( ̟ n +1 E H ˇ α ) = 1 for ˇ α ∈ ˇ∆. Then X ( σ ( ̟ n +1 E H ˇ α )) = 1, and, if G is of type D N , then X ( τ ( ̟ n +1 E H ˇ α )) = X ( ̟ n +1 E H τ (ˇ α ) ) = 1 for τ ∈ Gal( ˜
E/E ). Thus X descends to an O F -linear functional on t y, n +12 . Moreover, X is G -generic of depth n + , because p > Cox( G ).This concludes Case 1. Case 2: G is not of type D N and − / ∈ W δ
We claim that in this case G is a split group of type A N , D N +1 or E for some integer N ≥
1. (In the D N +1 -case, N ≥ G is split, this follows from [Hum90, 3.19 Cor. andTable 3.1]. If G is non-split and not of type D N , then G is of type A N , D N +1 or E and δ is induced by the non-trivial Dynkin diagram automorphism with respect to someset of simple coroots ˇ∆. Let w be the longest element in the Weyl group W (for simplereflections corresponding to ˇ∆). Then w ( ˇ∆) = − ˇ∆. Since − / ∈ W for type A N , D N +1 and E , we have w = − − w is a nontrivial automorphism of X ∗ ( T sp ) thatpreserves ˇ∆. Hence − w = δ , i.e. − w δ ∈ W δ . Thus none of the non-split groupsappears in Case 2.We treat the three cases separately. When G is a split group of type D N +1 or E , weexhibit the existence of an appropriate G -generic element X in the Lie algebra of a suitabletorus by explicit calculations; the details are deferred to Appendix A and Appendix B.The remaining case is when G is a split group of type A N . We denote by ˇ α , . . . , ˇ α N simple coroots such that ˇ α i and ˇ α j commute if and only if | i − j | 6 = 1. Then the Weylgroup W contains an element w of order N + 1 such that w ( ˇ α i ) = ˇ α i +1 for 1 ≤ i < N and w ( ˇ α N ) = − P ≤ i ≤ N ˇ α i . We denote by E the unramified extension of F of degree N + 1,and let σ be a generator of Gal( E/F ). Then the map f : Gal( F sep /F ) ։ Gal(
E/F ) → W defined by sending σ to w is an element of H ( F, W ) that gives rise to (the conjugacyclass of) an elliptic maximal torus T in G over F . Since p > Cox( G ) = N + 1, the set { ̟ n +1 F H ˇ α i } ≤ i ≤ N forms an O E -basis for t ( E ) y,n +1 . Let a ∈ E be an element of valuationzero such that P ≤ i ≤ N +1 σ i ( a ) = 0 and the image ¯ a of a in the residue field k E is a generatorfor the field extension k E /k F (see Lemma 2.1.1). Then the linear functional X on t ( E ) y,n +1 defined by X ( ̟ n +1 F H ˇ α i ) = σ i − ( a ) descends to an O F -linear functional on t y,n +1 . Moreover we claim that v ( a + σ ( a ) + . . . + σ j ( a )) = 0 , for 1 ≤ j ≤ N − . Suppose this is false. Then ¯ a + σ (¯ a ) + . . . + σ j (¯ a ) = 0, where ¯ a denotes the image of a in the residue field k E . We apply σ to the last equation and subtract the originalequation to obtain that σ j +1 (¯ a ) = ¯ a . This contradicts that ¯ a is a generator of k E /k F because j + 1 < N + 1. Since all coroots of A N are of the form ˇ α i + ˇ α i +1 + . . . + ˇ α j for1 ≤ i ≤ j ≤ N , we deduce that X is G -generic of depth n + 1. (cid:3) Corollary 2.1.3.
Let G be an adjoint simple reductive group over F that is tamely ramified.Assume that p > Cox( G ) . Then there exists a tamely ramified elliptic maximal torus T of G that enjoys the following property: For every n ∈ Z ≥ , there exists a real number r with n < r ≤ n + 1 and a character φ : T ( F ) → C ∗ of depth r such that ( T, r, φ ) is a 0-toraldatum.Proof. As in the proof of Proposition 2.1.2, we may and will show the existence of G -genericelements of depth − r in the Lie algebra instead of G -generic characters of depth r .By assumption, we can take G = Res F ′ /F G ′ for a finite tamely ramified extension F ′ /F and an absolutely simple adjoint group G ′ over F ′ . As usual F ′ is a subfield of F sep ; theinclusion is denoted by ι . By Proposition 2.1.2 there exists a tamely ramified ellipticmaximal torus T ′ of G ′ over F ′ such that for each n ′ ∈ Z ≥ , there exists X ∈ t ′ ( F ′ ) whichis G ′ -generic of depth − r ′ with n ′ < r ′ ≤ n ′ + 1 when the depth is normalized with respectto the valuation v ′ := e v : F ′ ։ Z ∪ {∞} , where e denotes the ramification index of F ′ /F .Writing ˇΦ ′ for the set of coroots of G ′ F sep with respect to T ′ F sep , we have v ′ ( X ( H ˇ β )) = − r ′ , ∀ ˇ β ∈ ˇΦ ′ . (2.1.4)Then T := Res F ′ /F T ′ is an elliptic maximal torus of G over F . Under the canonicalisomorphism t ( F ) ≃ t ′ ( F ′ ), we will show that X viewed as an element of t ( F ) is G -genericof depth − r ′ /e . Clearly this finishes the proof. (For each n , consider n ′ = en so that n < r ′ /e ≤ n + e ≤ n + 1.)We need some preparation. Over F sep we have compatible direct sum decompositions g ( F sep ) = g ′ ( F ′ ) ⊗ F F sep = L ι ∈ Hom F ( F ′ ,F sep ) g ′ ( F ′ ) ⊗ F ′ ,ι F sep ∪ ∪ ∪ t ( F sep ) = t ′ ( F ′ ) ⊗ F F sep = L ι ∈ Hom F ( F ′ ,F sep ) t ′ ( F ′ ) ⊗ F ′ ,ι F sep Correspondingly the set ˇΦ of coroots of G F sep with respect to T F sep decomposes asˇΦ = a ι ∈ Hom F ( F ′ ,F sep ) ˇΦ ′ ι , where ˇΦ ′ ι is the set of coroots of G ′ × F ′ ,ι F sep with respect to T ′ × F ′ ,ι F sep , so that ˇΦ ′ = ˇΦ ′ ι . ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 19
We are ready to verify that X is G -generic of depth − r ′ /e . Namely let us show v ( X ( H ˇ α )) = − r ′ /e, ∀ ˇ α ∈ ˇΦ . (2.1.5)Recall that v ′ = ev . So the condition holds for ˇ β ∈ ˇΦ ′ = ˇΦ ′ ι by (2.1.4). But Gal( F sep /F )acts on ˇΦ in a way compatible with its action on the index set Hom F ( F ′ , F sep ). Thus theGal( F sep /F )-orbit of ˇΦ ′ exhausts ˇΦ. On the other hand, since the valuation is Galois-invariant, − r ′ /e = v ( X ( H ˇ β )) = v ( σ ( X ( H ˇ β ))) = v ( X ( H σ ( ˇ β ) )) , σ ∈ Gal( F sep /F ) , ˇ β ∈ ˇΦ ′ . This completes the proof of (2.1.5). We are done. (cid:3)
Proposition 2.1.6.
Let G be a tamely ramified reductive group over F . Assume that p > Cox( G ) . Then there exists an elliptic maximal torus T of G that splits over a tamelyramified extension with the following property: For every n ∈ Z ≥ , there exists a 1-toraldatum with G = T and n < r ≤ r d − ≤ n + 1 .Proof. If G is a torus, then T = G , and for every n ∈ Z ≥ there exists n < r ≤ n + 1 suchthat T r = T r + . (This follows, for example, from the Moy–Prasad isomorphism from theanalogous assertion for the Lie algebra of T ; the latter is easy since an increase of 1 in theindex corresponds to multiplication by a uniformizer.) Hence there exists a (non-trivial)depth- r character φ of T , and (( T, G ) , ( r ) , ( φ )) is a 1-toral datum.So assume for the remainder of the proof that G is not a torus. Let G , . . . , G N be the simplefactors of the adjoint quotient G ad of G . Then there exists a surjection pr : G → Q ≤ i ≤ N G i whose kernel is the center of G . For 1 ≤ i ≤ N , let T i be an elliptic maximal torus of G i , and φ i : T i ( F ) → C ∗ a character of depth r i as provided by Proposition 2.1.3. Set T := pr − ( T × . . . × T N ). Since p > Cox( G ), the prime p does not divide the orderof π ( G ad ), and hence the depth of φ i ◦ pr i : T ( F ) → C ∗ is r i , where pr i denotes thecomposition of pr with the projection Q ≤ m ≤ N G m ։ G i . We assume without loss ofgenerality that r ≤ r ≤ . . . ≤ r N . Let 1 = i < i < . . . < i j ≤ N be integers such that r i = . . . = r i − < r i = . . . = r i − < r i = . . . < r i j = . . . = r N . For 0 ≤ k ≤ j −
1, set e r k = r i k +1 , and define G k := T pr − ( Q ≤ m ≤ i k +2 − G m ) ⊂ G and φ Tk := Q i k +1 ≤ m ≤ i k +2 − ( φ m ◦ pr m ) | G k . Then(( G , . . . , G j − , G j := G ) , ( e r , . . . , e r j − ) , ( φ T , . . . , φ Tj − ))is a desired 1-toral datum. (cid:3) As a corollary of the above proof when N = 1, we strengthen Corollary 2.1.3 to allownon-adjoint groups. Corollary 2.1.7.
Let G be a tamely ramified reductive group over F whose adjoint quotient G ad is simple. Assume that p > Cox( G ) . Then there exists a tamely ramified ellipticmaximal torus T of G that enjoys the following property: For every n ∈ Z ≥ , there existsa real number r with n < r ≤ n + 1 and a character φ : T ( F ) → C ∗ of depth r such that ( T, r, φ ) is a 0-toral datum. Construction of omni-supercuspidal types from 0-toral and 1-toral data.
Inthis subsection we introduce the notion of omni-supercuspidal types and construct suchtypes from 0-toral and 1-toral data. The results of the last subsection then allow us todeduce the main theorem that there exists a large family of omni-supercuspidal types inan appropriate sense. The following is the key definition of this section.
Definition 2.2.1.
Let m ∈ Z ≥ . An omni-supercuspidal type of level p m is a pair( U, λ ), where U is an open compact subgroup of G ( F ) and λ : U ։ Z /p m Z is a smoothsurjective group morphism such that ( U, ψ ◦ λ ) is a supercuspidal type for every nontrivialcharacter ψ : Z /p m Z → C ∗ . Remark . Let m > m ′ ≥ m,m ′ : Z /p m Z ։ Z /p m ′ Z for the canonicalprojection. If ( U, λ ) is an omni-supercuspidal type of level p m , then ( U, pr m,m ′ ◦ λ ) is anomni-supercuspidal type of level p m ′ .We assume from here until Corollary 2.2.14 that G is tamely ramified over F . (In Theorem2.2.15 the group G need not be tamely ramified.) Fix a 1-toral datum( G , . . . , G d ) , ( r , . . . , r d − ) , ( φ , . . . , φ d − )) (2.2.3)which yields the following input for Yu’s construction(( G , . . . , G d ) , y, ( r , . . . , r d − , r d = r d − ) , ρ = , ( φ , . . . , φ d − , φ d = 1)) . Recall that we introduced an open compact subgroup J and a smooth character ˆ φ in § J , ˆ φ | J ) wehad before. Combined with Proposition 2.1.6, it provides an explicit construction of acharacter on a rather small compact open subgroup that detects supercuspidality. Weexpect this result to be of independent interest. Proposition 2.2.4.
Assume p does not divide the order of the absolute Weyl group of G .Then for every group U such that G ( F ) y,r ⊂ U ⊂ G ( F ) y, rd + , the pair ( U, ˆ φ | U ) is a supercuspidal type.Remark . We have G ( F ) y,r ⊂ G ( F ) y, rd + (so that the proposition is not vacuous) ifand only if r > r d /
2, which is always satisfied if r > Remark . When the 1-toral data come from 0-toral data (so that d = 1, r = r ), theproposition essentially follows from Adler’s work. More precisely, it suffices to handle thecase U = G ( F ) y,r . This case follows immediately from the case G ( F ) y,r ⊂ U = J ⊂ G ( F ) y, r + (Proposition 1.1.6) thanks to [Adl98, 2.3.4] . For the reader who likes to check the proof, “ X ∈ m ⊥ x, − r/ ” in the proof of [Adl98, 2.3.3] (using Adler’snotation) should have been “ X ∈ m ⊥ x, − r + ”, and “ g ∈ G x, r ” should have been “ g ∈ G x, ”. ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 21
Proof.
We may and will assume that r > r d /
2. It suffices to consider the case U = G ( F ) y,r . If G is a torus, there is nothing to prove, so we assume that G is not a torus forthe remainder of the proof. Let ( π, V ) be an irreducible smooth representation of G ( F )and suppose that ( π | G ( F ) y,r , V ) contains the character ˆ φ | G ( F ) y,r . By Proposition 1.2.6 itis enough to show that then ( π | J , V ) contains the character ˆ φ | J . Define J = G y,r , J i = H d − iy,r H d − i +1 y,r d − i , rd − i + · · · H d − y,r d − , rd − + H dy,r d − , rd − + (1 ≤ i ≤ d − , J d = J . We show by induction on j that ( π | J j , V ) contains the character ˆ φ | J j . For j = 0 thestatement is true by assumption. Thus let 1 ≤ j ≤ d and assume the induction hypothesisthat ( π | J j − , V ) contains the character ˆ φ | J j − . We denote by V j − the largest subspace of V on which J j − acts via ˆ φ . Let X i ∈ g ∗ y, − r i be such that ˆ φ i | H i +1 y,ri viewed as a character of H i +1 y,r i /H i +1 y,r i + ≃ h i +1 y,r i / h i +1 y,r i + is given by Ψ ◦ X i . Since φ i is G i +1 -generic of depth r i relative to y , we can choose X i to be almost stable and generic of depth − r i at y (as an element of ( g i +1 ) ∗ ) in the senseof [Finb, Definition 3.1 and Definition 3.5]; compare with the proof of Propositions 1.2.6.By [Finb, Cor. 3.8.] this implies that X i is almost strongly stable and generic of depth − r i at y . Hence the tuple ( y, ( X i ) d − ≥ i ≥ ) is a truncated datum in the sense of [Finb, § φ k is trivial on H ky, and hence alsotrivial on H d − i +1 y, for d − i + 1 ≤ k . Thus ˆ φ k is trivial on H d − i +1 y,r d − i , rd − i + if k = d − i . Thereforethe action of H d − i +1 y,r d − i , rd − i + on V j − is given by ˆ φ d − i for 1 ≤ i ≤ j −
1, which means that theaction is as illustrated below: H d − i +1 y,r d − i , rd − i + ։ H d − i +1 y,r d − i , rd − i + /H d − i +1 y,r d − i + ≃ h d − i +1 y,r d − i , rd − i + / h d − i +1 y,r d − i + Ψ ◦ X d − i y V j − . We can now apply [Finb, Cor. 5.2] repeatedly as in the proof of [Finb, Cor. 5.4]. Moreprecisely, we use the following assignment of notation, where the left hand side denotes theobjects in [Finb, Cor. 5.2] using (only here) the notation of [Finb] and the right hand sidedenotes the objects defined above: n := d, X i := X d − i , r i := r d − i , j := d − j, H i := H d − i +1 ,T j := T ∩ H d − j +1 = G ∩ H d − j +1 , ϕ := Ψ . Choose ǫ ∈ (0 , r ) such that H d − j +1 y,r d − j − ǫ = H d − j +1 y,r d − j . To avoid confusion we write d ′ (insteadof d ) for the positive number d that occurs in [Finb, Cor. 5.2], and we set d ′ := max (cid:16) r d − j , r d − j − n ǫ (cid:17) , where n is the smallest integer in Z ≥ such that r d − j − n ǫ < r . Then either d ′ + ǫ ≥ r or d ′ + ǫ ≥ r d − j − ǫ , which implies together with the assumption H d − j +1 y,r d − j − ǫ = H d − j +1 y,r d − j (needed in the latter case that d ′ + ǫ ≥ r d − j − ǫ ) that H d − j +1 y,r d − j + ,d ′ + ǫ + ⊂ H d − j +1 y,r d − j + ,r . Since H d − j +1 y,r d − j + ,r acts via ˆ φ d − j on V j − , and ˆ φ d − j is trivial when restricted to H d − j +1 y,r d − j + ,r (by construction/definition the group H d − j +1 y,r d − j + ,d ′ + ǫ + acts trivially on V j − as well. Moreover,using that r d − j > r > r d − j , we see that the commutator subgroup (cid:20) J j − , H d − j +1 y,r d − j , rd − j + (cid:21) iscontained in H d − j +1 y,r d − j + ,r + H d − j +2 y,r d − j +1 + , rd − j +12 + · · · H d − y,r d − + , rd − + H dy,r d − + , rd − + ⊂ J j − and hence acts trivially on V j − . Therefore H d − j +1 y,r d − j ,d ′ + preserves V j − and we can find anonzero subspace V ′ on which the action of H d − j +1 y,r d − j ,d ′ + factors through H d − j +1 y,r d − j ,d ′ + /H d − j +1 y,r d − j + and is given by Ψ ◦ ( X d − j + C n ) for some C n ∈ ( h d − j +1 ) ∗ y, − ( d ′ + ǫ ) that is trivial on t j ⊕ h d − j .Moreover, 2 d ′ − r d − j + 2 ǫ ≤ d ′ − min (cid:16) r d − j , ǫ (cid:17) + 2 ǫ < d ′ < r . Hence, applying [Finb, Cor. 5.2] we obtain a nonzero subspace V ′′ of V on which • the action of H d − jy,r is given by ˆ φ ([Finb, Cor. 5.2(iv)]), • for 1 ≤ i ≤ j −
1, the action of H d − i +1 y,r d − i , rd − i + is given by H d − i +1 y,r d − i , rd − i + ։ H d − i +1 y,r d − i , rd − i + /H d − i +1 y,r d − i + ≃ h d − i +1 y,r d − i , rd − i + / h d − i +1 y,r d − i + Ψ ◦ X d − i y V ′′ (this implies that H d − i +1 y,r d − i , rd − i + acts via ˆ φ , because ˆ φ k is trivial on H d − i +1 y,r d − i , rd − i + if k = d − i ) ([Finb, Cor. 5.2(ii)]), • the action of H d − j +1 y,r d − j ,d ′ + is given by H d − j +1 y,r d − j ,d ′ + ։ H d − j +1 y,r d − j ,d ′ + /H d − j +1 y,r d − j + ≃ h d − j +1 y,r d − j ,d ′ + / h d − j +1 y,r d − j + Ψ ◦ X d − j y V ′′ ([Finb, Cor. 5.2(iii)]).If d ′ = r d − j /
2, then J j acts on V ′′ via ˆ φ so we are done. Otherwise, we achieve d ′ = r d − j / V ′′ to be as large as possible with the above properties,then we use the same reasoning as above with V ′′ in place of V j − , and apply [Finb, Cor. 5.2]repeatedly (at each step replacing V j − by the newly obtained subspace, with d ′ playingthe role of d in [Finb, Cor. 5.2]) as the value of d ′ goes through: d ′ = r d − j − ( n + 1) ǫ, r d − j − ( n + 2) ǫ, . . . , r d − j − ( n + N ) ǫ, r d − j , where N is the largest integer for which r d − j − ( n + N ) ǫ > r d − j . (If N ≤
0, we onlyconsider the case d ′ = r d − j .) After the final step of recursion, we obtain a nonzero subspaceof V on which J j acts via ˆ φ | J j . This completes the inductive proof. (cid:3) Write e F for the absolute ramification index of F . Let m be an arbitrary positive integerand set n := 2 e F m − . ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 23
Assume that the 1-toral datum (2.2.3) satisfies n < r ≤ r d − = r d ≤ n + 1 . (2.2.7)Such a 1-toral datum always exists by Proposition 2.1.6 under our current assumption that G is tamely ramified. To produce an omni-supercuspidal type on the group G ( F ) y, rd + , wewant to know: Lemma 2.2.8.
Assume that char( F ) = 0 , that r d is as in (2.2.7) , and that G is tamelyramified. Then the image of ˆ φ : G ( F ) y, rd + /G ( F ) y,r d + → C ∗ is a cyclic group of order p m .Remark . If char( F ) = p , then the last isomorphism in (2.2.10) in the proof belowbreaks down, and the lemma is false for m > Proof.
Recall that we write T = G . Since G is tamely ramified, we have T ( F ) r d T ( F ) r d + ⊂ T ( F ) rd + T ( F ) r d + ⊂ G ( F ) y, rd + G ( F ) y,r d + ≃ g ( F ) y, rd + g ( F ) y,r d + . (2.2.10)Since r d ≤ n + 1 = 2 me F , and hence r d − r d ≤ me F , we see that every element in g ( F ) y, rd + / g ( F ) y,r d + has order dividing p m . Therefore the image of ˆ φ is contained in acyclic subgroup of C ∗ of order p m . (Every finite subgroup of C ∗ is cyclic.)The image of T ( F ) r d /T ( F ) r d + in g ( F ) y, rd + / g ( F ) y,r d + is a p -torsion subgroup containedin g ( F ) y, ( r d − e F )+ / g ( F ) y,r d + . Now suppose that im( ˆ φ ) is contained in a cyclic subgroupof order p m − . Viewing ˆ φ as an additive character on g ( F ) y, rd + / g ( F ) y,r d + , we then haveˆ φ ( p m − X ) = 0 for X ∈ g ( F ) y, rd + / g ( F ) y,r d + . As p m − g ( F ) y, rd + = g ( F ) y, ( rd +( m − e F )+ , itfollows that ˆ φ is trivial on g ( F ) y, ( rd +( m − e F )+ / g ( F ) y,r d + , thus also trivial on T ( F ) r d /T ( F ) r d + (since r d +( m − e F < r d ). This contradicts that ˆ φ | T ( F ) rd = ˆ φ d − | T ( F ) rd − = 1. We concludethat the image of ˆ φ is exactly a cyclic group of order p m . (cid:3) The lemma allows us to factor ˆ φ as G ( F ) y, rd + /G ( F ) y,r d + λ ։ Z /p m Z exp ֒ → C ∗ (2.2.11)for some λ (assuming char( F ) = 0). Here exp stands for n exp(2 πin/p m ). When λ iscomposed with any other nontrivial character of Z /p m Z , the composite map has the formˆ φ i with i p m ). The following lemma shows that ˆ φ i is still a supercuspidal type. Lemma 2.2.12.
Assume that char( F ) = 0 , that p does not divide the order of the ab-solute Weyl group of G , and that G is tamely ramified. Let r and r d be as in (2.2.7) .Then ( G ( F ) y, rd + , λ ) is an omni-supercuspidal type of level p m , i.e. ( G ( F ) y, rd + , ˆ φ i ) is asupercuspidal type for all integers i with i (mod p m ). Proof.
Since the proof is trivial for G = T , we assume that G is not a torus. Recall thatˆ φ = Q ≤ j ≤ d − ˆ φ j , where φ j is a G j +1 -generic character of G j ( F ) of depth r j . We mayassume that 0 < i < p m , and hence0 ≤ v ( i ) ≤ ( m − e F = n + 12 − e F ≤ n − . (2.2.13)We claim that φ ij is a G j +1 -generic character of G j ( F ) of depth r j − v ( i ). To see this, let X j ∈ ( g j ) ∗ y, − r j be a G j +1 -generic element of depth r j such that the character φ j | G jy,ri viewedas a character of G jy,r j /G jy,r j + ≃ g jy,r j / g jy,r j + is given by Ψ ◦ X j . Note that the i -th powermap sends G jy,r j − v ( i ) into G jy,r j and G jy, ( r j − v ( i ))+ into G jy,r j + . The resulting map G jy,r j − v ( i ) /G jy, ( r j − v ( i ))+ → G jy,r j /G jy,r j + corresponds to the map g jy,r j − v ( i ) / g jy, ( r j − v ( i ))+ · i −→ g jy,r j / g jy,r j + induced by multiplication by i . (This can be seen from the binomial expansion andthe fact that r j − v ( i ) >
1. Hence φ ij | G jy,ri − v ( i ) factors through G jy,r j − v ( i ) /G jy,r j − v ( i )+ ≃ g jy,r j − v ( i ) / g jy,r j − v ( i )+ on which it is given by Ψ ◦ ( i · X j ). The claim has been verified.It follows from the claim that(( G , . . . , G d ) , ( r − v ( i ) , . . . , r d − − v ( i )) , ( φ i , . . . , φ id − ))is a 1-toral datum. It is implied by (2.2.13) that r − v ( i ) > n − n −
12 = n + 12 ≥ r d . Thus G ( F ) y,r − v ( i ) ⊂ G ( F ) y, rd + ⊂ G ( F ) rd − v ( i )2 + . Applying Proposition 2.2.4 to the new 1-toral datum above (thus the role of r j in the proposition is played by r j − v ( i )) and observingthat ( ˆ φ ) i | G ( F ) y, rd = b φ i | G ( F ) y, rd , we deduce that ( G ( F ) y, rd + , ˆ φ i ) is a supercuspidal type. (cid:3) The upshot is the following.
Corollary 2.2.14.
Assume char( F ) = 0 , p > Cox( G ) and G is tamely ramified. Thenthere exists a sequence { ( U m , λ m ) } m ≥ such that(1) each ( U m , λ m ) is an omni-supercuspidal type of level p m ,(2) U ⊃ U ⊃ · · · , and { U m } m ≥ forms a basis of open neighborhoods of ,(3) U m ′ is normal in U m whenever m ′ ≥ m .Proof. For m ≥
1, set n := 2 e F m − G , . . . , G d , ( r , . . . , r d − ) , ( φ , . . . , φ d − )) with n < r ≤ r d − ≤ n + 1.We may and will choose the same G and y for every m . Set U m := G ( F ) y, rd + and let λ m be as defined in Equation (2.2.11). Then ( U m , λ m ) is an omni-supercuspidal type of ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 25 level p m by Lemma 2.2.12. Moreover, U ⊃ U ⊃ · · · , and { U m } m ≥ forms a basis of openneighborhoods at 1. Property (3) follows from U m being Moy–Prasad subgroups at thesame point y . (cid:3) We now drop the assumption that G splits over a tamely ramified extension of F , i.e. G isan arbitrary (connected) reductive group defined over F . Theorem 2.2.15.
Assume char( F ) = 0 and p > Cox( G ) . Then there exists a sequence { ( U m , λ m ) } m ≥ such that(1) each ( U m , λ m ) is an omni-supercuspidal type of level p m ,(2) U ⊃ U ⊃ · · · , and { U m } m ≥ forms a basis of open neighborhoods of .(3) U m ′ is normal in U m whenever m ′ ≥ m .Remark . In contrast to the proof of Corollary 2.2.14, the following proof only relieson the omni-supercuspidal types constructed from 0-toral data and not the more generalomni-supercuspidal types constructed from 1-toral data. We have nevertheless includedthe construction of supercuspidal types from 1-toral data in our discussion above as thisconstruction might be useful for other applications.Thus we are presenting the reader with two approaches to prove the theorem. One is toprove the preceding results only in the simpler case of 0-toral data, and then deduce thetheorem via multiple reduction steps. The other is to establish the intermediate results inthe generality of 1-toral data. Then the final theorem (Corollary 2.2.14) is immediate, asfar as G is further assumed to be tamely ramified. Proof of Theorem 2.2.15.
It is enough to find a sequence { ( U m , λ m ) } m ≥ m as in the theoremstatement for some fixed m , since we can decrease level of an omni-supercuspidal typewithout changing the open compact subgroup as explained in Remark 2.2.2.The basic idea of proof is to reduce to the case where G is either a torus or an (absolutely)simple adjoint group, and then handle the two base cases. Below we use ψ m to denote anarbitrary nontrivial character of Z /p m Z . Step 1. Proof when G is a torus .Let G = T be a torus over F (possibly wildly ramified). We consider the filtration subgroup T , which is a pro- p group. Since T ( F ) is dense in T (see e.g. [Bor91, III.8.13]), the group T ( F ) contains infinitely many elements. If T is anisotropic, then the compact group T has finite index in the compact group T ( F ), and hence T has infinitely many elements. If T is not anisotropic, then T contains a copy of the multiplicative group G m and therefore T contains infinitely many elements as well. (An alternative way to see that T ( F ) has apro- p subgroup with infinitely many elements is by considering the exponential map froma small neighborhood of 0 in Lie T ( F ) to T ( F ).)Take U = T and U i +1 = { u p : u ∈ U i } recursively. Then { U i } i ≥ satisfies (2). For1 ≤ i < j , as a finite abelian p -group, U i /U j is a product of cyclic groups of p -power orders. We claim that at least one of the cyclic groups has exact order p j − i . Suppose not,then we would have U j − = U j , which would in turn imply U j = U j +1 = · · · . Since U is pro- p , we deduced that U j = U j +1 = · · · = { } , which contradicts the infinitude of U .Therefore there exists a projection λ m : U m /U m ։ Z /p m Z for m ≥
1. Then condition (1)obviously holds for ( U m , λ m ). Step 2. Proof when G is a simple simply connected group. In this case G = Res F ′ /F G ′ for a finite extension F ′ /F and an absolutely simple simplyconnected group G ′ over F ′ . Since G ( F ) = G ′ ( F ′ ), we reduce to the case when G isabsolutely simple. By Proposition 2.1.2, there is a tamely ramified elliptic maximal torus T ⊂ G along with a sequence of 0-toral data { ( T, r m , φ m ) } m ≥ with 2 e F m − < r m ≤ e F m for all m . Fix y as in the paragraph below Definition 1.1.1. For each m , set U m := G ( F ) y,r m / and let λ m : U m ։ Z /p m Z be defined by Equation (2.2.11). Then Condition(1) holds thanks to Lemma 2.2.12, and Condition (2) is obviously satisfied. Condition (3)follows because the groups are Moy–Prasad filtration subgroups for the same point y . Step 3. Proof when G is a simply connected group. Then G = G × · · · × G N , where G i are simple simply connected groups. By Step 2, we canfind a sequence of omni-supercuspidal types { ( U i,m , λ i,m ) } m ≥ satisfying (1), (2), and (3) foreach G i , for all i . Take U m := U ,m ×· · ·× U N,m for each m ≥ λ m : U m ։ Z /p m Z by λ m ( u , ..., u m ) = P i λ i,m ( u i ). Clearly (2) and (3) hold for { ( U m , λ m ) } . To verify (1),suppose that π is an irreducible smooth representation of G ( F ) containing ψ m ◦ λ m as a U m -subrepresentation for some arbitrary nontrivial character ψ m of Z /p m Z . By [Fla79, Thm. 1], π ≃ ⊗ Ni =1 π i for irreducible, smooth representations π i of G i ( F ). Since λ m pulls back to λ i along the natural inclusion G i ֒ → G , we see that π i | U i,m contains ψ m ◦ λ i,m . By Step 2, π i is supercuspidal for every 1 ≤ i ≤ N . We conclude that π is also supercuspidal. Step 4. Proof of the general case. If G is a torus, we are done by Step 1, so we assume that G is not a torus for the remainderof the proof. Let Z denote the maximal torus of Z , and G sc the simply connected coverof the derived subgroup of G . The multiplication map f : Z × G sc → G has finite kerneland cokernel (either as algebraic groups or as topological groups of F -points). This impliesthat f induces an isomorphism from a small enough open subgroup in Z ( F ) × G sc ( F ) ontoan open subgroup of G ( F ). By Steps 1 and 3, we have omni-supercuspidal types ( U ′ m , λ ′ m )for Z ( F ) and ( U ′′ m , λ ′′ m ) for G sc ( F ) as in the theorem, for all m ≥
1. There exists m suchthat ker( f ) ∩ ( U ′ m ′ × U ′′ m ′′ ) = { } for all m ′ , m ′′ ≥ m . Take U m := f ( U ′ m × U ′′ m ) and define λ m : U m ։ Z /p m Z by λ m ( f ( u ′ , u ′′ )) = λ ′ ( u ′ ) + λ ′′ ( u ′′ ) for u ′ ∈ U ′ m , u ′′ ∈ U ′′ m and m ≥ m .This is well-defined.Condition (2) is satisfied by ( U m , λ m ) as just defined, since { U ′ m × U ′′ m } ≥ forms a basis ofopen neighborhoods of 1 in Z ( F ) × G sc ( F ). Condition (3) is obviously true. ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 27
To check (1), suppose that an irreducible smooth representation π of G ( F ) contains ψ m ◦ λ m as a U m -representation. Let N be the unipotent radical of an F -rational proper parabolicsubgroup of G . We need to show that π N ( F ) = 0.By [Xu16, Lem. 6.2], π | f ( Z ( G ( F )) × G sc ( F )) decomposes as a finite direct sum ⊕ ni =1 π i of irre-ducible f ( Z ( G ( F )) × G sc ( F ))-representations. Without loss of generality, we may assumethat π | U m contains ψ m ◦ λ m . For i >
1, there exists g i ∈ G ( F ) such that π i ≃ g − i π , where g − i π ( γ ) = π ( g i γg − i ) for all γ ∈ f ( Z ( F ) × G sc ( F )) (e.g. see the proof of Lemma 6.2and Corollary 6.3 in [Xu16]). For g ∈ G ( F ), let N ′ g be the unipotent radical of a proper F -rational parabolic subgroup of G sc such that f induces an isomorphism from 1 × N ′ g onto g − N g (and also on the F -points). By Step 3, the irreducible G sc ( F )-representation π | f ( G sc ( F )) ◦ f is supercuspidal and so ( π ◦ f ) N ′ g ( F ) = 0. Thus we obtain as vector spaces π N ( F ) = n M i =1 ( π i ) N ( F ) = n M i =1 ( π ) g i N ( F ) g − i = n M i =1 ( π ◦ f ) N ′ gi ( F ) = 0 . (cid:3) Congruence to automorphic forms with supercuspidal components
Now we switch to a global setup for algebraic automorphic forms as studied in [Gro99].The following notation will be used. • G is a reductive group over a totally real field F such that G ( F ⊗ Q R ) is compactmodulo center. • v is a place of F above p ; p v is the maximal ideal of the ring of integers O F v in thecompletion F v of F with respect to v . • e v := e ( F v / Q p ) is the absolute ramification index of F v (so p e v v = ( p ) as ideals in O F v ). • G v := G × F F v , G p := (Res F/ Q G ) × Q Q p , G ∞ := (Res F/ Q G ) × Q R . • S is a finite set of places of F containing all p -adic and infinite places as well as allramified places for G ; we fix a reductive model for G over spec O F \{ finite places in S } (still denoted by G for convenience), and we use this model to identify G ( A F ) = Q ′ v G ( F v ) as a restricted product. • U p := Q w ∤ p ∞ U w is a (fixed) compact open subgroup of G ( A ∞ ,pF ) such that U w = G ( O F w ) hyperspecial for w / ∈ S , and we write U S := Q w / ∈ S U w .If U p is a compact open subgroup of G p ( Q p ) = Q w | p G ( F w ), and Λ a finitely generated Z p -module with a continuous action of U p , then we write U := U p U p , and M ( U, Λ) := (cid:26) cont. functions f : G ( F ) \ G ( A F ) /U p G ∞ ( R ) ◦ → Λ , such that f ( gu p ) = u − p f ( g ) , for g ∈ G ( A F ) , u p ∈ U p (cid:27) . Here G ∞ ( R ) ◦ denotes the connected component of G ∞ ( R ) that contains the identity. Wewrite Z Np for the free Z p -module of rank N with trivial U p -action, and we drop the ex-ponent if N = 1. Hence M ( U, Z p ) is the space of Z p -valued functions on the finite set G ( F ) \ G ( A F ) /U p U p G ∞ ( R ) ◦ . Note that M ( U, Λ) is a T S := Z [ U S \ G ( A SF ) /U S ]-module un-der the usual double coset action. It is routine to check that the association of T S -modulesΛ M ( U, Λ)is functorial in Λ. In particular, if Λ ′ ≃ Λ ⊕ N as Z p [ U p ]-modules with N ∈ Z ≥ , then M ( U, Λ ′ ) ≃ M ( U, Λ ⊕ N ) = M ( U, Λ) ⊕ N as T S -modules, and T S acts on the last direct sumby the diagonal action. We define T S ( U, Λ) ⊂ End Z p ( M ( U, Λ)) (3.0.1)to be the Z p -subalgebra generated by the image of T S . (Thus T S ( U, Λ) is commutative.) IfΛ ′ ≃ Λ ⊕ N then we have an induced isomorphism T S ( U, Λ ′ ) ≃ T S ( U, Λ), acting equivariantlyon M ( U, Λ ′ ) ≃ M ( U, Λ) ⊕ N . This observation will be used a few times in § § π ( G ∞ ( R )) by right translation on the spaces of automor-phic forms considered here, cf. Proposition 8.6 and the paragraph below (4.1) in [Gro99],commuting with the Hecke algebra actions. Every isomorphism between spaces of auto-morphic forms below is compatible with the π ( G ∞ ( R ))-action. That said, we will notmention π ( G ∞ ( R ))-actions again.3.1. Constant coefficients.
Our goal is to define congruences between arbitrary auto-morphic forms with constant coefficients and automorphic forms that are supercuspidal at p . In order to define the coefficients of the latter space, we denote by A m the Z p -algebra Z p [ T ] / (1 + T + . . . + T p m − ) for m a positive integer. Then we have a canonical Z p -algebraisomorphism A m / ( T − ≃ Z p / ( p m ). Theorem 3.1.1.
Assume p >
Cox( G ) and let N be a positive integer. Then there exist • a basis of compact open neighborhoods { U p,m } m ≥ of ∈ G p ( Q p ) = Q w | p G ( F w ) such that U p,m ′ is normal in U p,m whenever m ′ ≥ m and • a smooth character ψ m : U p,m → A × m for each m ≥ such that we have isomorphisms of Z p / ( p m ) -modules (where the U p,m -action in M ( · ) istrivial on the left hand side and through ψ m on the right hand side) M ( U p U p,m , Z Np / ( p m )) ≃ ( M ( U p U p,m , A m / ( T − ⊕ N (3.1.2) that are compatible with the action of T S ( U p U p,m , Z Np / ( p m )) on the left hand side andthe diagonal action of T S ( U p U p,m , A m / ( T − on the right hand side via the Z p -algebraisomorphism T S ( U p U p,m , Z Np / ( p m )) ≃ T S ( U p U p,m , A m / ( T − . (3.1.3) Moreover, every automorphic representation of G ( A F ) that contributes to ( M ( U p U p,m , A m )) ⊕ N ⊗ Z p Q p is supercuspidal at all places above p . ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 29
Remark . In (3.1.2), we can evidently take the quotients after forming the spaces offunctions. That is, (3.1.2) is equivalent to the isomorphism M ( U p U p,m , Z Np ) / ( p m ) ≃ ( M ( U p U p,m , A m )) ⊕ N / ( T − . However we cannot take the quotients outside the Hecke algebras. The abstract situationis as follows. Let M be a finite free Z p -module, α ∈ End Z p ( M ). Write M := M/ ( p m ) and α ∈ End Z p ( M ) for the image of α . Put T := Z p [ α ] and T := Z / ( p m )[ α ] for the subalgebraof End Z p ( M ) (resp. End Z p ( M )) generated over Z p . Then the obvious map T / ( p m ) → T need not be an isomorphism: consider arbitrary α such that α is the multiplicative unity.Despite the apparent defect, it is readily checked that T S ( U p U p,m , Z Np ) = lim ←− m ≥ m T S ( U p U p,m , Z Np / ( p m ))for each integer m ≥
1, with compatible actions on the spaces of functions with coefficientsin Z Np and Z Np / ( p m ), respectively. The analogous isomorphism holds with non-constantcoefficient V Z p that we will introduce in § Remark . Theorem 3.1.1 also holds if we replace U p by a compact open subgroup U v = Q w ∤ v ∞ U w of G ( A ∞ ,vF ) such that U w is hyperspecial for w / ∈ S , and U p,m by compact opensubgroups U v,m of G ( F v ) for a place v above p . The conclusion in this case includes thatevery automorphic representation G ( A F ) that contributes to ( M ( U v U v,m , A m )) ⊕ N ⊗ Z p Q p is supercuspidal at v . The proof works in a completely analogous way.Before presenting a proof, let us comment on the meaning of the proposition. As m growsto infinity, the space M ( U p U p,m , Q p ), or more precisely its extension of scalars to Q p ,exhausts all automorphic forms on G ( A F ) with constant coefficients, if we also allow U p to shrink arbitrarily. Thus the left hand side of (3.1.2) represents arbitrary automorphicforms on G with Z p / ( p m )-coefficients. Loosely speaking, Theorem 3.1.1 can be thought ofas a congruence modulo a power of p between arbitrary automorphic forms and those withsupercuspidal components at p .A caveat is that the congruence here is between spaces of automorphic forms. It does notfollow from our result that for an individual automorphic representation π , there exists π ( m ) which is supercuspidal at p such that “ π ≡ π ( m ) mod p m ” in terms of Hecke eigenval-ues outside S . To see this, let c π : T S ( U p U p,m , Z p ) → Z p be the Z p -algebra morphism ac-counting for π . Suppose that T S ( U p U p,m , Z p ) / ( p m ) is isomorphic to T S ( U S U v,m , A m ) / ( T − c π mod p m , we obtain a morphism T S ( U S U v,m , A m ) / ( T − → Z p / ( p m ), but now the problem is that it is unclear whetherthe latter lifts to a morphism T S ( U S U v,m , A m ) → Z p . (When m = 1, this is often possibleby the Deligne–Serre lifting lemma [DS74, Lemma 6.11], for instance.) However, we oftendo not need such a lift for applications, see e.g. § § Proof of Theorem 3.1.1 .
We let { ( U p,m , λ m ) } m ≥ be omni-supercuspidal types of level p m for G p as in Theorem 2.2.15, i.e. such that U p, ⊃ U p, ⊃ . . . , the groups { U p,m } m ≥ form abasis of open neighborhoods of 1 and U p,m ′ is normal in U p,m whenever m ′ ≥ m .For each m ≥
1, we have the following commutative diagram of maps, where ζ p m denotesa primitive p m -th root of unity in Q p . A m ⊗ Z p Q p ≃ Q p m − p { ζ aip m } p m − i =1 Z /p m Z (cid:31) (cid:127) / / A × m (cid:31) (cid:127) / / (cid:15) (cid:15) (cid:15) (cid:15) A m mod T − (cid:15) (cid:15) (cid:15) (cid:15) ?(cid:31) O O a mod p m ✤ / / T a ❴ (cid:15) (cid:15) ✤ / / T a ❴ (cid:15) (cid:15) ❴ O O ( Z /p m Z ) × (cid:31) (cid:127) / / Z /p m Z ✤ / / ψ m : U p,m → A × m as the composite U p,m λ m −→ Z /p m Z ֒ → A × m , and we let u ∈ U p,m act on A m by multiplication by ψ m ( u ). Since the resulting action of U p,m on A m / ( T −
1) is trivial, we have canonical isomorphisms M ( U p U p,m , Z Np / ( p m )) ≃ M ( U p U p,m , Z p / ( p m )) ⊕ N ≃ ( M ( U p U p,m , A m ) / ( T − ⊕ N (3.1.6)as modules over Z p / ( p m ) ≃ A m / ( T − T S -equivariant,and observing that the action of T S on ( M ( U p U p,m , A m ) / ( T − ⊕ N is given via the di-agonal action of T S ( U p U p,m , A m / ( T − T S ( U p U p,m , Z Np / ( p m )) ≃ T S ( U p U p,m , A m / ( T − . Note that M ( U p U p,m , A m ) ⊗ Z p Q p ≃ M χ : Z /pm Z → Q × pχ =1 M ( U p U p,m , ( Q p ) χ ◦ λ m ) , (3.1.7)where ( Q p ) χ ◦ λ m denotes the free rank-1 Q p -module on which u ∈ U p,m acts by multiplicationby χ ◦ λ m . Since ( U p,m , λ m ) is omni-supercuspidal, every automorphic representation of G ( A F ) that contributes to ( M ( U p U p,m , A m )) ⊕ N ⊗ Z p Q p is supercuspidal at all places above p . (cid:3) ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 31
Non-constant coefficient.
Let L be a finite extension of F such that L/ Q is Galoisand G × F L is split. Then (Res F/ Q G ) × Q L is split and we denote by ˜ G a split reductivegroup over O L such that ˜ G L ≃ (Res F/ Q G ) × Q L . Let V be a linear algebraic representationof ˜ G over O L . This yields an algebraic representation of Res O L / Z ˜ G on V Z (i.e. V viewed asa Z -module) over Z . In particular, (Res O L / Z ˜ G )( Z p ) acts continuously on V Z p = V Z ⊗ Z Z p for the p -adic topology. Observe that(Res O L / Z ˜ G )( Q p ) = ˜ G ( L ⊗ Q Q p ) ≃ (Res F/ Q G )( L ⊗ Q Q p ) = G p ( L ⊗ Q Q p ) , and the latter naturally contains G p ( Q p ) as a closed subgroup. Therefore every sufficientlysmall open subgroup U p of G p ( Q p ) (as long as it maps into (Res O L / Z ˜ G )( Z p ) under theabove isomorphism) acts on V Z p . In that case, using this action, we obtain a space ofautomorphic forms M ( U p U p , V Z p / ( p m )) as defined earlier. Theorem 3.2.1.
Assume p >
Cox( G ) . Then there exists a basis of compact open neighbor-hoods { U p,m } m ≥ of ∈ G p ( Q p ) = Q w | p G ( F w ) with U p,m ′ normal in U p,m for m ′ ≥ m as wellas smooth actions of U p,m on A m arising from multiplication by a character ψ m : U p,m → A × m for m ≥ (factoring through Z /p m Z ), such that we have isomorphisms of Z p / ( p m ) -modules M ( U p U p,m , V Z p / ( p m )) ≃ ( M ( U p U p,m , A m / ( T − ⊕ dim Z p V (3.2.2) that are compatible with the action of T S ( U p U p,m , V / ( p m )) on the left hand side and thediagonal action of T S ( U p U p,m , A m / ( T − on the right hand side via the Z p -algebra iso-morphism T S ( U p U p,m , V Z p / ( p m )) ≃ T S ( U p U p,m , A m / ( T − . (3.2.3) Moreover, every automorphic representation of G ( A F ) that contributes to ( M ( U p U p,m , A m )) ⊕ dim Z p V Z p ⊗ Z p Q p is supercuspidal at all places above p .Remark . If V ′ C is an algebraic representation of (Res F/ Q G ) × Q C over C , then we canchoose a model of the representation V ′ C over O L (e.g. the sum of the Weyl modules asin [Jan03, p. 183] corresponding to the irreducible components of V ′ C ), i.e. a representation V of ˜ G over O L whose base change to C is the representation V ′ C of (Res F/ Q G ) × Q C . Thisway we obtain the setup needed at the start of this subsection. Remark . The space M ( U p U p,m , V Q p ), where V Q p = V Z p ⊗ Z p Q p , can be describedin terms of classical automorphic forms on G . Here we fix field embeddings Q ֒ → Q p and Q ֒ → C and a compatible isomorphism ι : Q p ≃ C , and extend scalars using theseembeddings to define V Q p and V C . Thus V C is an algebraic representation of (Res L/ Q ˜ G L ) × Q C = (Res F/ Q G ) × Q ( L ⊗ Q C ), which we can restrict to a representation of (Res F/ Q G ) × Q C via the obvious embedding C ֒ → L ⊗ Q C . The resulting representation can be viewedeither as an algebraic representation of (Res F/ Q G ) × Q C , or as a continuous representationof G ∞ ( C ) ⊃ G ∞ ( R ). For simplicity, suppose that the center of G is anisotropic over F andwrite A G for the space of L -automorphic forms on G ( F ) \ G ( A F ) (as in 2.1.2 of [Sor13] but without the need to fix a central character; note that his G is our Res F/ Q G ). Then Lemma2 of loc. cit. gives a T S C -equivariant isomorphism ιM ( U p U p,m , V Q p ) ≃ Hom G ∞ ( R ) ( V ∨ C , A G ) U p U p,m . Proof of Theorem 3.2.1.
Recall that an open subgroup of G p ( Q p ) acts continuously on V Z p for the p -adic topology. Hence, for every integer m ≥
1, there exists an open subgroup of G p ( Q p ) that acts trivially on (the finite set) V Z p /p m V Z p .Now let { ( U p,n , λ n ) } n ≥ be a sequence of omni-supercuspidal types for G p ( Q p ) = G ( F ⊗ Q Q p ) as in Theorem 2.2.15. By the preceding paragraph, there is an increasing sequence n < n < · · · such that U p,n m stabilizes V Z p and acts trivially on V Z p /p m V Z p for every m .Let pr i,j : Z /p i Z ։ Z /p j Z denote the canonical surjection when i ≥ j ≥
1. Then( U ′ p,m , λ ′ p,m ) := ( U p,n m , pr n m ,m ◦ λ n m ) , m ≥ , is an omni-supercuspidal type of level p m . Moreover, by construction, we have M ( U p U ′ p,m , V Z p / ( p m )) ≃ M ( U p U ′ p,m , Z p / ( p m )) ⊕ dim Z p V Z p , where the action of U ′ p,m is trivial on the right hand side and induced by that on V Z p on theleft hand side. Now we can proceed as in the proof of Theorem 3.1.1 : We let U ′ p,m act on A m via the character ψ ′ m : U ′ p,m λ ′ p,m ։ Z /p m Z ֒ → A × m to obtain a Z p /p m -linear isomorphism M ( U p U ′ p,m , V Z p / ( p m )) ≃ M ( U p U ′ p,m , A m / ( T − ⊕ dim Z p V Z p . (3.2.6)As at the start of §
3, we obtain a Z p /p m -algebra isomorphism T S ( U p U ′ p,m , V Z p / ( p m )) ≃ T S ( U p U ′ p,m , A m / ( T − , m ≥ , (3.2.7)which is compatible with (3.2.2) via the respective Hecke algebra actions on both sides. (cid:3) Remark . In fact the argument of this section still goes through and produces somecongruence without the assumption that ( U p,m , λ p,m ) is omni-supercuspidal, which plays arole only in applications. However the outcome is less interesting without a careful choiceof the pair ( U p,m , λ p,m ).3.3. An application to Galois representations.
We illustrate how to employ Theorems3.1.1 and 3.2.1 to construct Galois representations from automorphic representations in asuitable context. The idea is to reduce to the case when automorphic representations havesupercuspidal components. In fact, Remark 7.4 of [Sch18] reads “...and it seems reasonableto expect that one could do a similar argument in the compact unitary case, providing analternative to the construction of Galois representations of Shin [Shi11] and Chenevier–Harris [CH13], by reducing directly to the representations constructed by Harris–Taylor.” Confirmation of this is the goal of this section. We copied the sentence except that the bibliographic items have been adapted. Remark 7.4 of [Sch18]also mentions the case of Hilbert modular forms but we chose to concentrate on the more complicated casetreated here.
ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 33
Let n ∈ Z ≥ . Recall that F is a totally real field, and let E be a totally imaginary quadraticextension of F with complex conjugation c ∈ Gal(
E/F ). For a finite place w of E , wewrite LL w for the unramified local Langlands correspondence for GL n ( E w ), from irreducibleunramified representations of GL n ( E w ) to continuous semisimple unramified n -dimensionalrepresentations of the Weil group W E w of E w (with coefficients in C , up to isomorphisms).Let Π be a regular C-algebraic cuspidal automorphic representation of GL n ( A E ) that isconjugate self-dual, i.e. its contragredient Π ∨ is isomorphic to the complex conjugate Π c .Let S E be the set of all infinite places and the finite places of E where Π is ramified, and S all the places of F below S E . Fix a prime p and an isomorphism ι : Q p ≃ C . A well knowntheorem by Clozel ([Clo91, Thm 1.1], based on [Kot92]) states the following. At each place w of E , write | det w | for the determinant map on GL n ( E w ) composed with the absolutevalue on E w which is normalized to send a uniformizer to the inverse of the residue fieldcardinality. Theorem 3.3.1 ([Clo91]) . Suppose that there exists a finite place v of E where Π v is adiscrete series representation. Then there exists a continuous representation ρ Π ,ι : Gal( E/E ) → GL n ( Q p ) which is unramified outside S E such that LL w (Π w ) ⊗ | det w | (1 − n ) / is isomorphic to thesemisimplification of ιρ Π ,ι | W Ew for all w / ∈ S E .Remark . Harris and Taylor showed in [HT01] (see Thm. VII.1.9 therein) that Π w and ρ Π ,ι | W Ew still correspond under the local Langlands correspondence at ramified primes notabove p . The latter result was later also obtained in [Sch13] by a different approach.We remove the assumption that Π v is a discrete series representation from the abovetheorem when p is not too small using congruences, to obtain Theorem 3.3.3. The conditionon p is due to Proposition 2.1.2. Theorem 3.3.3.
Suppose that p > n . Then for every regular C-algebraic conjugate self-dual cuspidal automorphic representation Π of GL n ( A E ) , there exists a continuous repre-sentation ρ Π ,ι : Gal( E/E ) → GL n ( Q p ) which is unramified outside S E such that LL w (Π w ) ⊗ | det w | (1 − n ) / is isomorphic to thesemisimplification of ιρ Π ,ι | W Ew for all w / ∈ S E .Remark . This theorem is not new. A stronger statement on the existence and local-global compatibility for ρ Π ,ι has been known by [CHL11, Shi11, CH13] without restrictionon p . The local-global compatibility was further strengthened by [Car12,Car14,BLGGT14].Much of [Shi11] was reproved in [SS13] by a simpler method. The point is that the proofhere is still simpler as there is no eigenvariety as in [CH13] and no elaborate geometricand endoscopic arguments as in [Shi11], as far as Theorem 3.3.1 is taken for granted. (Thegeometry and harmonic analysis involved in Theorem 3.3.1 are less complicated than thoseof [Shi11, SS13].) Proof.
We will freely use the notions and base change theorems of [Lab11] for unitarygroups to go between automorphic representations on unitary groups and general lineargroups.Let Spl SE (resp. Spl SF ) be the set of places of E (resp. F ) outside S E (resp. S ) that aresplit in E/F . By a standard reduction step using automorphic base change over quadraticextensions as in the proof of [HT01, Thm. VII.1.9] (also see the proof of [Shi11, Prop. 7.4]),it suffices to show the compatibility for ρ Π ,ι only at w ∈ Spl SE .By a patching argument (executed as in the proof of [CH13, Thm. 3.1.2]), we further reduceto the case where • [ F : Q ] is even, • every place of F above p is split in E , • every finite place of F is unramified in E .Since [ F : Q ] is even, there exists a unitary group G over F which is an outer formof GL n with respect to the quadratic extension E/F such that G is quasi-split at allfinite places and anisotropic at all infinite places. We may view Π as a representation of(Res E/F ( G ⊗ F E ))( A F ), which is isomorphic to GL n ( A E ); in particular Π x stands for thecomponent of Π at a place x of F . Since Π is conjugate self-dual, by [Lab11, Thm. 5.4, 5.9],there exists an automorphic representation π on G ( A F ) such that Π x is the unramified basechange of π x at all places x of F outside S .We set up some more notation. For w ∈ Spl SE , we denote by w ∈ Spl SF the restriction of w to F . Recall that we denote by U p = Q x ∤ p ∞ U x a (fixed) compact open subgroup of G ( A ∞ ,pF ) such that U x is hyperspecial for x / ∈ S . For x ∈ Spl SF and for each w above x , wefix isomorphisms i w : G ( F x ) ≃ GL n ( E w ) carrying U x onto GL n ( O E w ) such that π x ≃ Π w via i w . We write ̟ w for a uniformizer of E w and write T ( i ) w for the following double coset T ( i ) w := (cid:20) GL n ( O E w ) (cid:18) ̟ w I i
00 I n − i (cid:19) GL n ( O E w ) (cid:21) , which we might also view as a double coset of U S \ G ( A SF ) /U S by requiring all other factorsto be the trivial double coset. We also denote by T ( i ) w the double coset operator corre-sponding to this double coset acting on appropriate spaces that can be deduced from thecontext. Given an irreducible unramified representation σ w of GL n ( E w ) (or G ( F x )), wewrite T ( i ) w ( σ w ) for the eigenvalue of T ( i ) w on the one-dimensional space σ GL n ( O Ew ) w . For con-venience, we introduce the following variant of the big Hecke algebra : define T S Spl to bethe Z -subalgebra of Z [ U S \ G ( A SF ) /U S ] generated by T ( i ) w for w ∈ Spl SE (excluding w notsplit over F ) and 1 ≤ i ≤ n . Replacing T S with T S Spl , we define other Hecke algebras tobe the image of T S Spl in the endomorphism algebras of appropriate spaces of automorphicforms. Note that Theorems 3.1.1 and 3.2.1 are still valid with T S Spl in place of T S : indeedwe retain the same isomorphism between the same spaces of automorphic forms, and the T S Spl -equivariance is simply weaker than the T S -equivariance. ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 35
We choose U = U p U p ⊂ G ( A ∞ F ) sufficiently small such that π ∞ has nonzero U -fixed vectors,while keeping U x hyperspecial for x / ∈ S . Since G ∞ ( R ) is compact, we see that π ∨∞ comesfrom an irreducible algebraic representation V ′ C of G ∞ × R C = (Res F/ Q G ) × Q C . As inRemark 3.2.4, there exists a finite Galois extension L/ Q in C containing F such that(Res F/ Q G ) × Q L is split, thus (Res F/ Q G ) × Q L ≃ ˜ G L for a split group ˜ G over O L , and suchthat there is an algebraic representation V of ˜ G over O L giving a model for V ′ C . Accordingto § U p on the corresponding free Z p -module V Z p such that π contributes to M ( U, V Z p ), see Remark 3.2.5. (Note that π ∞ is a direct summand of( V ⊗ Z C ) ∨ = ( V Z p ⊗ Z p C ) ∨ as a G ∞ ( R )-representation.) We let { U p,m } m ≥ be compactopen subgroups of U ⊂ G p ( Q p ) with an action of U p,m on A m arising from multiplicationby a character ψ m : U p,m → A × m for m ≥
1, factoring through λ m : U p,m → Z /p m Z , as inTheorem 3.2.1. This means we have T S Spl ( U p U p,m , V Z p / ( p m )) ≃ T S Spl ( U p U p,m , A m / ( T − , m ≥ , (3.3.5)Let A S ( U p U p,m , A m ) be the set of irreducible T S Spl , Q p -modules appearing as a constituent of M ( U p U p,m , A m ) ⊗ Z p Q p . This set is identified with the set of σ S = { σ x } x ∈ Spl SF , where σ x isan irreducible unramified representation of G ( F x ), such that there exists an automorphicrepresentation σ of G ( A F ) satisfying • σ x ≃ σ x for x ∈ Spl SF , • ( σ ∞ ,p ) U p = { } , • σ ∞ = , i.e. the archimedean components of σ are trivial, and • Hom U p,m ( ψ ◦ λ m , σ p ) = 0 for some nontrivial character ψ : Z /p m Z → C ∗ .The last condition implies that σ p is a supercuspidal representation of G p ( Q p ). For each σ S ∈ A S ( U p U p,m , A m ), we choose a σ as two sentences above. By base change theo-rems [Lab11, Cor. 5.3, Thm. 5.9] we obtain a cuspidal conjugate self-dual automorphicrepresentation Σ of GL n ( A E ) that is regular and C-algebraic such that Σ x is the unrami-fied base change of σ x at each place x of F outside S and also that Σ p is supercuspidal. Itfollows from Theorem 3.3.1 that there exists a continuous semisimple representation ρ Σ ,ι : Gal( E/E ) → GL n ( Q p )which is unramified outside S E such that LL w (Σ w ) is isomorphic to the semisimplificationof ιρ Σ ,ι | W Ew for all w / ∈ S E . By the Chebotarev density and Brauer–Nesbitt theorems, ρ Σ ,ι is independent of the choice of σ up to isomorphism. Thus we write ρ σ S ,ι := ρ Σ ,ι . Let E S denote the maximal extension of E in E unramified outside S E . Then ρ σ S ,ι factors throughGal( E S /E ). At w ∈ Spl SE , let N ( w ) ∈ Z ≥ denote the absolute norm of the finite prime w . Using t as an auxiliary variable, the compatibility at w of Theorem 3.3.1 means that(cf. [CHT08, Prop. 3.4.2. part 2])det(1 + tρ σ S ,ι (Frob w )) = n X i =0 t i N ( w ) i ( i − / T ( i ) w ( σ w ) . (3.3.6) On the other hand, T S Spl ( U p U p,m , A m ) ֒ → T S Spl ( U p U p,m , A m ) ⊗ Z p Q p ≃ Y σ S ∈A S ( U p U p,m ,A m ) Q p , where for each w ∈ Spl SE and each 1 ≤ i ≤ n , the image of T ( i ) w ∈ T S Spl ( U p U p,m , A m ) in the σ S -component is the scalar in Q p by which T ( i ) w acts on the x -component of σ S (viewed asa representation of GL n ( E w ) via i w ). Let Mat n × n ( · ) denote the n × n -matrix algebra overthe specified coefficient ring. For m ≥
1, write ρ m : Gal( E S /E ) → Y σ S ∈A S ( U p U p,m ,A m ) GL n (cid:0) Q p (cid:1) ֒ → Mat n × n Y σ S ∈A S ( U p U p,m ,A m ) Q p , where the σ S -part of the first map is ρ σ S ,ι . We can extend this map linearly to a map ρ Bm : B [Gal( E S /E )] → Mat n × n ( B ) , for every ( Q σ S ∈A S ( U p U p,m ,A m ) Q p )-algebra B . By composing ρ Bm with the determinant, weproduce a continuous n -dimensional determinant map in the sense of Chenevier [Che14] d m : Q p [Gal( E S /E )] → Y σ S ∈A S ( U p U p,m ,A m ) Q p (which consists of maps d Bm for all ( Q σ S ∈A S ( U p U p,m ,A m ) Q p )-algebra B , but we usually omitthe index B from the notation) such thatdet(1 + tρ m ( γ )) = d m (1 + tγ ) , γ ∈ Gal( E S /E ) (3.3.7)We deduce from (3.3.6) that all the coefficients of the characteristic polynomial χ (Frob w , t ) := d m ( t − Frob w ) in the sense of Chenevier are contained in T S Spl ( U p U p,m , A m ) for all w ∈ Spl SE .The same holds for all elements of Gal( E S /E ) because the union of Frobenius conjugacyclasses over Spl SE is dense in Gal( E S /E ) by the Chebotarev density theorem. Hence itfollows from [Che14, Corollary 1.14] that d m is the scalar extension of a T S Spl ( U p U p,m , A m )-valued n -dimensional continuous determinant (in Chenevier’s sense): Z p [Gal( E S /E )] → T S Spl ( U p U p,m , A m )satisfying (3.3.7). To save notation, we still write d m for the latter. Thus d m (1 + t Frob w ) = n X i =0 t i N ( w ) i ( i − / T ( i ) w , w ∈ Spl SE . Via (3.3.5) we obtain a continuous n -dimensional T S Spl ( U p U p,m , V Z p / ( p m ))-valued determi-nant D m : ( Z p / ( p m ))[Gal( E S /E )] → T S Spl ( U p U p,m , V Z p / ( p m )) ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 37 such that D m (1 + t Frob w ) = n X i =0 t i N ( w ) i ( i − / T ( i ) w (3.3.8)(equality taken inside T S Spl ( U p U p,m , V Z p / ( p m ))) for w ∈ Spl SE . We have the restriction mapres m, : T S Spl ( U p U p,m , V Z p / ( p m )) → T S Spl ( U p U p, , V Z p / ( p m ))and for m ′ ≤ m the projection mappr m,m ′ : T S Spl ( U p U p, , V Z p / ( p m )) → T S Spl ( U p U p, , V Z p / ( p m ′ )) . From (3.3.8) and the density of Frobenii, we deduce that for m ≥ m ′ ≥
1, we havepr m,m ′ ◦ res m, ◦ D m (1 + tγ ) = res m ′ , ◦ D m ′ (1 + tγ ) γ ∈ Gal( E S /E ) . By Amitsur’s formula [Che14, (1.5)] and the properties of the determinant, we see thatthe n -dimensional determinant res m, ◦ D m is uniquely determined by its values on 1 + t Gal( E S /E ). Hence we obtain an equality of T S Spl ( U p U p, , V Z p / ( p m ′ ))-valued n -dimensionalcontinuous determinants pr m,m ′ ◦ res m, ◦ D m = res m ′ , ◦ D m ′ Taking the inverse limit over res m, ◦ D m for m ≥ T S Spl ( U p U p, , V Z p ) = lim ←− m T S Spl ( U p U p, , V Z p / ( p m )), we obtain a T S Spl ( U p U p, , V Z p )-valued n -dimensional continuous determinant D : Z p [Gal( E S /E )] → T S Spl ( U p U p, , V Z p )with D (1 + t Frob w ) = P ni =0 t i N ( w ) i ( i − / T ( i ) w . Since π contributes to M ( U p U p, , V Z p ) (as π contributes to M ( U, V Z p ) and U p, ⊂ U p ), it gives rise to a Z p -algebra morphism c π : T S Spl ( U p U p, , V Z p ) → Q p , T ( i ) w T ( i ) w ( π w ) , ∀ w ∈ Spl SE . The composition c π ◦ D yields a continuous n -dimensional Q p -valued determinant. Itfollows from [Che14, Thm. A, Ex. 2.34] that c π ◦ D arises from a continuous representation ρ π : Gal( E S /E ) → GL n ( Q p ) in the sense that c π ( D (1 + tγ )) = det(1 + tρ π ( γ )) , γ ∈ Gal( E S /E ) . Therefore we conclude that for w ∈ Spl SE ,det(1 + tρ π (Frob w )) = c π ( D (1 + t Frob w )) = n X i =0 t i N ( w ) i ( i − / T ( i ) w ( π w ) . That is, LL w ( π w ) ⊗ | det w | (1 − n ) / is isomorphic to the semisimplification of ιρ π | W Ew . Theproof of the theorem is complete by setting ρ Π ,ι := ρ π . (cid:3) Density of supercuspidal points in the Hecke algebra.
As another applicationof our main local theorem, we show that the supercuspidal locus is Zariski dense in thespectrum of the Hecke algebra of p -adically completed (co)homology following Emerton–Paˇsk¯unas [EP20]. Using Bushnell–Kutzko’s study of types for GL n , they proved the resultfor a global definite unitary group which is isomorphic to a general linear group at p . Theirmachinery is quite general, enabling us to extend their result to general reductive groupswhich are compact modulo center at ∞ once we combine it with our local construction.We retain the same notation as at the start of §
3. We may and will assume that F = Q by replacing G with Res F/ Q G as this does not sacrifice the quality of the theorem. As in loc. cit. we assume that the central torus Z ( G ) has the same Q -rank and R -rank. Let L be a finite extension of Q p with ring of integers O . (We have renewed the use of L hereto be consistent with [EP20]. In § L denoted a certain number field which wewill denote by L below.) This will be our coefficient field for the involved representations.Fix an algebraic closure L of L and a uniformizer ̟ ∈ O . So far in this section, weworked with Hecke algebras as Z p -algebras acting on the space of automorphic forms as Z p -modules, but everything carries over verbatim with O and ̟ in place of Z p and p .This extension is not strictly necessary but sometimes convenient as irreducible algebraicrepresentations of G Q p need not be defined over Q p . If U p is a compact open subgroupof G ( Q p ), then we define the completed group algebra of U p over O to be O [[ U p ]] :=lim ←− U ′ p O [ U p /U ′ p ], where the limit is taken over open normal subgroups U ′ p ⊂ U p . Thetopology on O [[ U p ]] is given by the projective limit (with the usual topology on O [ U p /U ′ p ]as a finite free O -module). Whenever we work with O [[ U p ]]-modules, we work in thecategory of compact linear-topological O [[ U p ]]-modules (resp. O -modules) and denote theHom space by Hom cont O [[ U p ]] ( · , · ) (resp. Hom cont O ( · , · )).Let U p = Q w ∤ p, ∞ U w be an open compact subgroup of G ( A ∞ ,p ) such that U w is hyperspecialfor all w away from a finite set of places S . Define Y ( U p U p ) := G ( Q ) \ G ( A ) /U p U p G ( R ) ◦ .Consider the completed homology˜ H ( U p ) := lim ←− U p H ( Y ( U p U p ) , O ) , where U p runs over open compact subgroups of G ( Q p ). Then ˜ H ( U p ) is a finitely generated O [[ U p ]]-module that is O -torsion free for any compact open subgroup U p of G ( Q p ). If U p or U p is sufficiently small, for instance if U p U p is a neat subgroup (in the sense of [Pin90, § U p acts on points of G ( Q ) \ G ( A ) /U p G ( R ) ◦ with trivial stabilizers, and ˜ H ( U p ) isfree over O [[ U p ]]. We topologize ˜ H ( U p ) as a finitely generated O [[ U p ]]-module, using thetopology of O [[ U p ]]. This is equivalent to the inverse limit topology where the topology on H ( Y ( U p U p ) , O ) arises from the topology of O .We also define the completed cohomology˜ H ( U p ) := lim ←− s ≥ lim −→ U p H ( Y ( U p U p ) , O /̟ s ) , (3.4.1) ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 39 which is complete for the ̟ -adic topology, or equivalently the inverse limit topology over s of the discrete topology on the direct limit. In our earlier notation, H ( Y ( U p U p ) , O /̟ s ) = M ( U p U p , O /̟ s ). We have a canonical isomorphism ˜ H ( U p ) = Hom cont O ( ˜ H ( U p ) , O ) astopological O -modules, where the topology on the latter is given by the supremum norm.We define the “big” Hecke algebra T S ( U p ) := lim ←− U p , s T S ( U p U p , O /̟ s ) , (3.4.2)recalling that T S ( U p U p , O /̟ s ) was introduced around (3.0.1), where the Z p -algebra setupearlier extends to the O -algebra setup in the evident manner. We equip T S ( U p ) with theprofinite topology via (3.4.2), where the finite set T S ( U p U p , O /̟ s ) is equipped with thediscrete topology. For each compact open subgroup U p of G ( Q p ) and a locally algebraicrepresentation V of U p over L , we have T S ( U p )-equivariant isomorphismsHom O [[ U p ]] ( ˜ H ( U p ) , V ∗ ) = Hom U p ( V, ˜ H ( U p ) L ) = M ( U p U p , V ∗ ) , as explained in [EP20, 5.1]. The T S ( U p )-module structure is semisimple as it is the caseon the space of algebraic automorphic forms.Let { V i } i ∈ I be a family of continuous representations of U p on finite dimensional L -vectorspaces. We recall from [CDP14, Def. 2.6] (see Lemmas 2.7 and 2.10 therein for equivalentcharacterizations): Definition 3.4.3.
Let M be a compact linear-topological O [[ U p ]]-module. We say that { V i } captures M if there is no nontrivial (i.e. other than M = Q ) quotient M ։ Q inducing an isomorphismHom cont O [[ U p ]] ( Q, V ∗ i ) ≃ Hom cont O [[ U p ]] ( M, V ∗ i ) , ∀ i ∈ I. Proposition 3.4.4.
Assume p >
Cox( G ) . Then there exist • an open compact pro- p subgroup U p of G ( Q p ) and • a countable family of smooth representations { V i } i ∈ I of U p on finite dimensional L -vector spacessuch that the following hold: • ( U p , V i ⊗ L L ) is a supercuspidal type for every i , • { V i } i ∈ I captures O [[ U p ]] .Moreover U p can be chosen to be arbitrarily small.Remark . This proves a result similar to [EP20, Cor. 4.2] but slightly stronger in thatour proposition implies the conclusion there via Lemmas 2.8 and 2.10 of loc. cit.
Note thatthe proof below still produces { V i } i ∈ I capturing O [[ U p ]] as far as U p,m is a sequence of pro- p groups forming a neighborhood basis of 1. Indeed the proof readily adapts to other typessuch as principal series types. Remark . As can be seen in the proof, we construct V i to be irreducible representationsof U p over L which may become reducible over L . Proof.
Let { ( U p,m , λ m ) } m ≥ be a sequence for G ( Q p ) as in Theorem 2.2.15, where we mayassume that U p := U p, is a pro- p group and arbitrarily small (see the first paragraph inthe proof of Theorem 2.2.15).Let L m ⊂ L denote the totally ramified extension of L generated by p m -th roots of unity,with O m denoting its ring of integers and ̟ m a uniformizer in O m . We compose λ m : U p,m → Z /p m Z with a fixed character Z /p m Z ֒ → O m to define a smooth character ψ ◦ m : U p,m → O × m . Put ψ m := ψ ◦ m ⊗ O m L m . The dual representations, i.e., inverse characters,corresponding to ψ ◦ m and ψ m are denoted by ψ ◦∗ m and ψ ∗ m . For an embedding σ : L m ֒ → L over L , write ψ ∗ m,σ := ψ ∗ m ⊗ L m ,σ L (viewed as a one-dimensional representation over L ).We will think of ψ ◦ m , ψ ◦∗ m (resp. ψ m , ψ ∗ m ) as representations on a free O -module (resp. L -module) of rank [ L m : L ]. The notation is consistent as ψ ∗ m is then the dual representationof ψ m over L . The multiplication by ̟ m (on the underlying O m -module) yields a U p,m -equivariant map [ ̟ m ] : ψ ◦∗ m → ψ ◦∗ m . Since p m -th roots of unity become trivial in O m / ( ̟ m ),the U p,m -action on the cokernel ψ ◦∗ m / ( ̟ m ) is trivial.Take { V i } to be the family of irreducible L -subrepresentations of Ind U p U p,m ψ m (which issemisimple) for all m ≥
1. Since ( U p,m , λ m ) is omni-supercuspidal, and since ψ m ⊗ L L ≃⊕ σ ∈ Hom L ( L m ,L ) ψ m ⊗ L m ,σ L as a U p,m -representation, the pair ( U p,m , ψ m ⊗ L L ) is a supercus-pidal type. Via Frobenius reciprocity, it follows that ( U p , V i ⊗ L L ) is also a supercuspidaltype.It remains to prove that { V i } captures O [[ U p ]]. Put M := O [[ U p ]] and let Q be the smallestquotient of M such that Hom cont O [[ U p ]] ( Q, V ∗ i ) ≃ Hom cont O [[ U p ]] ( M, V ∗ i ) , ∀ i ∈ I. This implies that Q captures { V i } . We need to show that M ∼ → Q .For each m , we have Ind U p U p,m ψ ∗ m isomorphic to the direct sum of V ∗ i ’s for suitable indices i .So M ։ Q induces an isomorphismHom cont O [[ U p ]] ( Q, Ind U p U p,m ψ ∗ m ) ∼ → Hom cont O [[ U p ]] ( M, Ind U p U p,m ψ ∗ m ) , m ≥ . Via Frobenius reciprocity,Hom cont O [[ U p,m ]] ( Q, ψ ∗ m ) ∼ → Hom cont O [[ U p,m ]] ( M, ψ ∗ m ) , m ≥ . The isomorphism continues to hold when ψ ∗ m is replaced with ψ ◦∗ m . Indeed, the injectivityis obvious as the map is induced by the surjection M ։ Q . For the surjectivity, noticethat the cokernel is a finitely generated O -module that is torsion free and vanishes aftertaking ⊗ O L . Now we consider the following commutative diagram with exact rows and ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 41 vertical maps induced by M ։ Q .Hom cont O [[ U p,m ]] ( Q, ψ ◦∗ m ) [ ̟ m ] / / ∼ (cid:15) (cid:15) Hom cont O [[ U p,m ]] ( Q, ψ ◦∗ m ) / / ∼ (cid:15) (cid:15) Hom cont O [[ U p,m ]] ( Q, ψ ◦∗ m / ( ̟ m )) α (cid:15) (cid:15) Hom cont O [[ U p,m ]] ( M, ψ ◦∗ m ) [ ̟ m ] / / Hom cont O [[ U p,m ]] ( M, ψ ◦∗ m ) β / / Hom cont O [[ U p,m ]] ( M, ψ ◦∗ m / ( ̟ m ))The map α is injective by surjectivity of M ։ Q . To see that α is surjective, it is enough tocheck that β is surjective; this is true since the next term in the bottom row exact sequenceis Ext O [[ U p,m ]] ( M, ψ ◦∗ m ), which vanishes by projectivity of M (as a compact linear-topological O [[ U p,m ]]-module). Hence α gives an isomorphismHom cont O [[ U p,m ]] ( Q, ψ ◦∗ m / ( ̟ m )) ∼ → Hom cont O [[ U p,m ]] ( M, ψ ◦∗ m / ( ̟ m ))as vector spaces over O / ( ̟ ). The Hom spaces do not change if Q and M are replaced with Q/ ( ̟ ) and M/ ( ̟ ). Since U p,m acts trivially on ψ ◦∗ m / ( ̟ m ), we deduce that( M/ ( ̟ )) U p,m ∼ → ( Q/ ( ̟ )) U p,m , m ≥ , where the subscripts signify the U p,m -coinvariants. Since { U p,m } is a neighborhood basisof 1, we deduce that M/ ( ̟ ) ∼ → Q/ ( ̟ ). As Q is O -torsion free by [EP20, Lem. 2.4], thisimplies that (ker( M ։ Q )) / ( ̟ ) = { } . The topological Nakayama’s lemma for compact O [[ U p,m ]]-modules [NSW08, Lem. 5.2.18] implies that ker( M ։ Q ) = { } . Hence M ∼ → Q ,and { V i } captures M . (cid:3) Corollary 3.4.7.
Assume p >
Cox( G ) , and let U p and { V i } be as in Proposition 3.4.4.If U p or U p is sufficiently small (e.g., if U p U p is neat), then for each continuous finitedimensional representation W of G ( Q p ) over L , the family { V i ⊗ L W } captures ˜ H ( U p ) and the evaluation morphism M i Hom U p ( V i ⊗ W, ˜ H ( U p ) L ) ⊗ ( V i ⊗ W ) −→ ˜ H ( U p ) L has dense image.Proof. As remarked earlier, if U p U p is neat, then ˜ H ( U p ) is a free O [[ U p ]]-module. ThereforeLemmas 2.8 and 2.9 of [EP20] imply that { V i ⊗ L W } captures ˜ H ( U p ). The assertionabout density is simply an equivalent characterization of capture in [CDP14, Lem. 2.10](or [EP20, Lem. 2.3]), noting that ˜ H ( U p ) L = Hom cont O ( ˜ H ( U p ) , L ). (cid:3) Remark . To apply the corollary when U p is fixed (and p > Cox( G )), we choose smallenough U p such that U p U p is neat. This is possible as U p can be made arbitrarily small inProposition 3.4.4.Here is an informal discussion of the corollary. Recall that Hom U p ( V i ⊗ W, ˜ H ( U p ) L ) = M ( U p U p , ( V i ⊗ W ) ∗ ). If W is the restriction of an irreducible algebraic representation of G L (we are writing L for the number field L in § forms such that every automorphic representation π that contributes to it (as in Remark3.2.5) has the properties that π p is supercuspidal (since π p contains the type ( U p , V i )) andthat π | G ( R ) = W . Thus the corollary roughly asserts that automorphic forms which aresupercuspidal at p and have “weight” W at ∞ form a dense subspace in the completedcohomology.Now we formulate a density statement in terms of Hecke algebras. Fix an open maximalideal m ⊂ T S ( U p ) and consider the m -adic localization T S ( U p ) m , which is a direct factorof T S ( U p ) as a topological ring. (See [EP20, 5.1], cf. (C.4) and Remark C.5.) Fix arepresentation W of G ( Q p ) coming from an irreducible algebraic representation as in thepreceding paragraph. Let Σ( W ) sc ⊂ Spec T S ( U p ) m [1 /p ]denote the subset of closed points such that the corresponding morphism T S ( U p ) m [1 /p ] → L (up to the Gal( L/L )-action) comes from an eigen-character of T S ( U p ) in M ( U p U p , W ∗ ) forsome U p , and such that the eigenspace gives rise to an automorphic representation of G ( A )whose component at p is supercuspidal. Theorem 3.4.9. If p > Cox( G ) , the subset Σ( W ) sc ⊂ Spec T S ( U p ) m is Zariski dense.Proof. Essentially the same argument as in the proof of [EP20, Thm. 5.1] works, so weonly sketch the proof. Choose a sufficiently small U p and { V i } as in Proposition 3.4.4.By Corollary 3.4.7, { W ⊗ V i } captures the m -adic localization ˜ H ( U p ) m , which is a directsummand of ˜ H ( U p ). As in loc. cit. we obtain thatHom cont O [[ U p ]] ( ˜ H ( U p ) m , ( W ⊗ V i ) ∗ ) ≃ Hom cont O [[ U p ]] ( W ⊗ V i , ˜ H ( U p ) m ⊗ L ) ≃ Hom cont O [[ U p ]] ( W ⊗ V i , ( ˜ H ( U p ) m ⊗ L ) alg ) , where ( · ) alg designates the subspace of locally algebraic vectors for the U p -action. Thus T S ( U p ) m [1 /p ] acts semisimply on ( ˜ H ( U p ) m ⊗ L ) alg as it is the case on the space of algebraicautomorphic forms. Moreover the support of Hom cont O [[ U p ]] ( W ⊗ V i , ( ˜ H ( U p ) m ⊗ L ) alg ) in themaximal spectrum of T S ( U p ) m [1 /p ] is contained in Σ( W ) sc as explained in loc. cit. (Thesupercuspidality at p comes from the fact that ( U p , V i ) is a supercuspidal type.) Finally theZariski density of Σ( W ) sc follows from Proposition 2.11 of [EP20] based on Remarks 2.12and 2.13 therein with R = T S ( U p ) m . (We need Remark 2.13 as T S ( U p ) m is not known to beNoetherian in general; refer to the discussion in § loc. cit. Sometimes the Noetherianproperty can be proved, as in Appendix C.) (cid:3)
Appendix A. Calculations for D N +1 The purpose of this appendix is to prove Proposition 2.1.2 for split groups G of type D N +1 .In this appendix, by “the proof” we will always refer to the proof of Proposition 2.1.2. Wemaintain the notation from there. Recall that it suffices to exhibit an elliptic maximaltorus T ⊂ G such that for every n ∈ Z ≥ , there exists a G -generic element X ∈ t ∗ of depth − r with n < r ≤ n + 1. Put s := 2 N + 1 in favor of simpler notation. ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 43
Our strategy is similar to the other cases of the proof. We take
E/F to be the unramifiedextension of degree 2 s − N with Gal( E/F ) = h σ i , where σ denotes the (arithmetic)Frobenius automorphism. Recall that T sp is a split maximal torus in G .Let { e , ..., e s } be a basis for X ∗ ( T sp ) ⊗ Z R . Without loss of generality, i.e. by changingthe basis if necessary, we let the simple coroots beˇ α i = e i − e i +1 , ≤ i ≤ s − , ˇ α s = e s − + e s , and the coroots ˇΦ = {± ( e i ± e j ) : 1 ≤ i < j ≤ s } . Take the Coxeter element w = s ˇ α s ˇ α . . . s ˇ α s , where s ˇ α i denotes the reflection on X ∗ ( T sp ) ⊗ Z R corresponding to ˇ α i . Theorder of w is equal to Cox( G ) = 2 s − N . An easy computation shows: w ( ˇ α i ) = ˇ α i +1 , ≤ i ≤ s − ,w ( ˇ α s − ) = ˇ α + ˇ α + · · · + ˇ α s w ( ˇ α s − ) = − ( ˇ α + ˇ α + · · · + ˇ α s − ) w ( ˇ α s ) = − ( ˇ α + ˇ α + · · · + ˇ α s − + ˇ α s )As in the earlier proof, we define T over F from the cocycle f : Gal( E/F ) → W sending σ to w . (A cocycle here is a homomorphism as Gal( E/F ) acts trivially on W .) We mayand will identify T E with T sp E and fix T henceforth. Let r ∈ Z . Then giving X ∈ t ∗ ( E ) − r is equivalent to assigning a , ..., a s ∈ O E such that X ( H ̟ rF ˇ α ) = a , ..., X ( H ̟ rF ˇ α s ) = a s since { ̟ rF ˇ α i } ≤ i ≤ s is an O E -basis of t ( E ) r . (To see this, it is enough to check it for r = 0. Inthis case, { ˇ α i } ≤ i ≤ s generates a subgroup of the free Z -module X ∗ ( T E ) with index coprimeto p , since p > Cox( G ) and thus p does not divide the order of the Weyl group. It followsthat { ˇ α i } ≤ i ≤ s indeed generates t ( E ) over O E . Since t ( E ) is a free O E -module of rank s ,linear independence follows.)The G -genericity means that we need v ( X ( H ˇ β )) = − r, ∀ ˇ β ∈ ˇΦ . (A.1)On the other hand, X descends to an O F -linear functional on t r if X ( σ ( H ̟ rF ˇ α i )) = σ ( a i ) , ∀ ≤ i ≤ s. (A.2)Hence in order to prove Proposition 2.1.2 for split groups G of type D N +1 = D s , it sufficesto find X satisfying (A.1) and (A.2). Moreover, it suffices to find such an X only when r = 0, since the case n < r ≤ n + 1 follows by multiplying X with ̟ − n − F . Thus we set r = 0 from now on. Then (A.2) can be rewritten as σ ( a i ) = a i +1 , ≤ i ≤ s − ,σ ( a s − ) = a + a + · · · + a s ,σ ( a s − ) = − ( a + a + · · · + a s − ) ,σ ( a s ) = − ( a + a + · · · + a s − + a s +1 ) . (A.3) Take ζ to be a primitive ( q s − − O E , where q is the residue fieldcardinality of F (so k E = F q s − ). Set a = ζ qs − , b = ζ q s − − q − . We would like to verify that the following solution to the system of equations (A.3) works: a i = σ i − ( a ) , ≤ i ≤ s − ,a s − = 12 ( b − ( a + σ ( a ) + · · · + σ s − ( a )) + σ s − ( a )) ,a s = 12 ( − b − ( a + σ ( a ) + · · · + σ s − ( a )) + σ s − ( a )) . A simple computation shows σ s − ( a ) = − a, σ ( b ) = − b, (A.4) a s − − a s = b, a s − + a s = − ( a + σ ( a ) + · · · + σ s − ( a )) + σ s − ( a ) . Using this, it is elementary to check that (A.3) is satisfied.It remains to prove that (A.1) holds with r = 0. As a preparation, observe that thereduction a ∈ k E of a generates k E , namely σ j ( a ) = a, j ∈ Z ≥ ⇒ (2 s − | j. (A.5)When each coroot ˇ β is written (uniquely) as ˇ β = P si =1 λ i ˇ α i with λ i ∈ Z , the conditionimposed by (A.1) is that v ( P si =1 λ i X ( H ˇ α i )) = 0. As ˇ β runs through all coroots (enoughto consider positive coroots), the following are the conditions to check:(1) v ( a i +1 + · · · + a j ) = 0 for 0 ≤ i < j ≤ s .(2) v ( a i +1 + · · · + a s + ( a j +1 + · · · + a s − )) = 0 for 0 ≤ i ≤ j ≤ s − v ( a i +1 + · · · + a s − + a s ) = 0 for 0 ≤ i < s − j ≤ s −
2, (1b) j = s −
1, and (1c) j = s . To check an elementof O E has valuation zero, it suffices to show that the reduction is nonzero in k E . Cases (1a), (1c), and (2) . In these cases, the condition to be checked has the form σ i ′ ( a ) + · · · + σ j ′ ( a ) = 0 , ≤ j ′ − i ′ ≤ s − . (A.6)Indeed this is clear in Case (1a) with i ′ = i and j ′ = j −
1. In Case (1c), it suffices to showthat − σ ( a i +1 + · · · + a s ) = 0 but the left-hand side equals, via (A.4), − σ (cid:0) σ s − ( a ) − ( a + σ ( a ) + · · · + σ i − ( a )) (cid:1) = a + σ ( a ) + · · · + σ i ( a ) . So this case corresponds to showing (A.6) with i ′ = 0 and j ′ = i . In Case (2), a i +1 + · · · + a s + ( a j + · · · + a s − ) = − a − σ ( a ) − · · · − σ i − ( a ) + σ j ( a ) + σ j +1 ( a ) + · · · + σ s − ( a )= σ j ( a ) + σ j +1 ( a ) + · · · + σ s − i ( a )so the condition that this expressions is non-zero amounts to (A.6) with i ′ = j and j ′ = s − i . Note that j ′ − i ′ = s − i − j ≤ s − ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 45
Now the verification of (A.6) is the same as for type A in the proof of Proposition 2.1.2,based on (A.5).
Case (1b) . To prove the claim by contradiction, we suppose that2( a i +1 + a i +2 + · · · + a s − ) = − ( a + σ ( a ) + · · · + σ i − ( a )) + ( σ i ( a ) + · · · + σ s − ( a )) + b = 0 . We apply σ to the equation and subtract it from the original equation, recalling that σ ( b ) = b . Then − a − σ ( a ) + 2( σ i ( a ) + σ i +1 ( a )) − σ s − ( a ) − σ s ( a ) = 0 . This is simplified via (A.4) as σ i ( a )+ σ i +1 ( a ) = 0, also using that q is odd. Hence σ ( a ) = − a and therefore σ ( a ) = a , which contradicts (A.5) (as s ≥ Case (3) . Again suppose that2( a i +1 + · · · + a s − + a s ) = − ( a + σ ( a ) + · · · + σ i − ( a )) + ( σ i ( a ) + · · · + σ s − ( a )) − b = 0 . The equation is the same as in Case (1b) except that the coefficient of b has opposite sign,which does not affect the argument, so we reach contradiction in the same way as before. Appendix B. Calculations for E Here we prove Proposition 2.1.2 for a split group G of type E . That is, we exhibit anelliptic maximal torus T ⊂ G , and a G -generic element X ∈ t ∗ of depth − r for the same T (as r varies). The notation in the proof of the proposition is maintained.We denote the simple coroots of E by ˇ α , ˇ α , . . . , ˇ α as shown in Figure 1.ˇ α ˇ α ˇ α ˇ α ˇ α ˇ α Figure 1.
Dynkin diagram for (the dual of) E Then the positive coroots of E are (1) ˇ α + . . . + ˇ α j − ˇ α for 1 < j ≤ α i + . . . + ˇ α j for 3 ≤ i ≤ j ≤ α + . . . + ˇ α j − ˇ α for 3 ≤ j ≤ α i + . . . + ˇ α for 1 ≤ i ≤ α i + . . . + ˇ α j for 1 ≤ i ≤ ≤ ≤ j ≤ α i + . . . + ˇ α j + ˇ α for 1 ≤ i ≤ ≤ ≤ j ≤ The list coincides with the one in [Bou02, Planche V] up to identifying roots with coroots. This is fineas the root system of type E is self-dual. (7) ˇ α + ˇ α + 2 ˇ α + 2 ˇ α + ˇ α (8) ˇ α + ˇ α + ˇ α + 2 ˇ α + 2 ˇ α + ˇ α (9) ˇ α + ˇ α + 2 ˇ α + 2 ˇ α + ˇ α (10) ˇ α + ˇ α + 2 ˇ α + 2 ˇ α + ˇ α + ˇ α (11) ˇ α + ˇ α + 2 ˇ α + 2 ˇ α + 2 ˇ α + ˇ α (12) ˇ α + ˇ α + 2 ˇ α + 3 ˇ α + 2 ˇ α + ˇ α (13) ˇ α + 2 ˇ α + 2 ˇ α + 3 ˇ α + 2 ˇ α + ˇ α (Note that (1)–(4) correspond to coroots of subroot systems of type A .)Let w h = s s s s s s , where s i denotes the reflection corresponding to ˇ α i . Then w h is aCoxeter element, hence it has order Cox( G ) = 12 and its eigenvalues when acting on thecomplex vector space spanned by the coroots are ζ , ζ , ζ , ζ , ζ , ζ , where ζ denotesa primitive (complex) twelfth root of unity (see [Hum90, 3.7. Table 1 and 3.19. Theorem]).Hence w := w h is an elliptic element, i.e. does not have any nonzero fixed vector whenacting on the above complex vector space. One easily calculates that w ( ˇ α ) = − ˇ α − ˇ α − ˇ α − ˇ α w ( ˇ α ) = ˇ α + ˇ α + ˇ α + ˇ α + ˇ α w ( ˇ α ) = ˇ α + ˇ α + ˇ α + 2 ˇ α + ˇ α w ( ˇ α ) = − ˇ α − ˇ α − α − α − α − ˇ α w ( ˇ α ) = ˇ α + ˇ α + 2 ˇ α + ˇ α + ˇ α w ( ˇ α ) = − ˇ α − ˇ α − ˇ α − ˇ α Let E be a cubic Galois extension of F . Fix a generator σ of Gal( E/F ) and define f : Gal( F sep /F ) ։ Gal(
E/F ) → W by sending σ ∈ Gal(
E/F ) to w . As in the proof ofProposition 2.1.2, f gives rise to (the conjugacy class of) a maximal torus T of G . Since w is elliptic, the torus T is elliptic. We divide into two cases. Case (1) : when F does not contain a nontrivial third root of unity. In this case we let E be the unramified cubic extension of F . Let a ∈ E be an element of valuation zero suchthat a + σ ( a ) + σ ( a ) = 0 and such that the image ¯ a of a in the residue field k E of E is agenerator for the field extension k E /k F (see Lemma 2.1.1), and define a = a = a = σ ( a ) , a = a − σ ( a ) , a = a + σ ( a ) , a = − a − σ ( a ) . Then σ ( a ) = − a − a − a − a ,σ ( a ) = a + a + a + a + a ,σ ( a ) = a + a + a + 2 a + a ,σ ( a ) = − a − a − a − a − a − a ,σ ( a ) = a + a + 2 a + a + a ,σ ( a ) = − a − a − a − a . ONGRUENCES AND SUPERCUSPIDAL REPRESENTATIONS 47
Hence the linear functional X on t ( E ) y,n +1 defined by X ( ̟ n +1 F H ˇ α i ) = a i descends to alinear functional on t y,n +1 .We claim that { , ¯ a, σ (¯ a ) } is a k F -basis for k E . To this end, suppose that there exist c , c , c ∈ k F such that c + c ¯ a + c σ (¯ a ) = 0. Applying σ , we have c + c σ (¯ a ) + c ( − ¯ a − σ (¯ a )) = 0. Taking the difference of the two equations, we obtain( c − c ) σ (¯ a ) = ( c + c )¯ a. If c − c = 0 then σ (¯ a ) = c ¯ a with c = ( c + c ) / ( c − c ), thus ¯ a = σ (¯ a ) = c ¯ a . Since F does not contain any nontrivial third root of unity, neither does k F , and hence c = 1.But then σ (¯ a ) = ¯ a , contradicting k E = k F (¯ a ) = k F . Therefore c − c = 0, thus also c + c = 0 as ¯ a = 0. Since p > Cox( G ) >
3, it follows that c = c = 0. Hence c = 0 aswell, proving the desired linear independence of { , ¯ a, σ (¯ a ) } over k F .Now using the linear independence of ¯ a and σ (¯ a ) together with the explicit formulas forthe (positive) coroots of E above, it is easy to check that v ( X ( H ˇ α )) = − ( n + 1) for allcoroots ˇ α of G E with respect to T E . Hence X is G -generic of depth n + 1. Case (2) : when F contains a nontrivial third root of unity ζ . In this case let E be thetotally ramified extension F ( ̟ E ) for a root ̟ E of the equation x − ̟ F = 0. As our choiceof ζ and ̟ E is flexible, we may assume that σ ( ̟ E ) = ζ ̟ E . We set a = 2 , a = a = a = 1 , a = − − ζ , a = 3 ζ . Then ζ a = − a − a − a − a ,ζ a = a + a + a + a + a ,ζ a = a + a + a + 2 a + a ,ζ a = − a − a − a − a − a − a ,ζ a = a + a + 2 a + a + a ,ζ a = − a − a − a − a . Thus the linear functional X on t ( E ) y,n +1 / defined by X ( ̟ n +1 E H ˇ α i ) = a i descends to alinear functional on t y,n +1 / . It remains to check that v ( X ( H ˇ α )) = − ( n + ) for all corootsˇ α of G E with respect to T E . Note that it suffices to consider the positive coroots. Usingthe explicit formulas above, we obtain that for all positive coroots ˇ α , X ( ̟ n +1 E H ˇ α ) ∈ { , , , − − ζ } ∪ { i − ζ | − ≤ i ≤ } ∪ { i − ζ | − ≤ i ≤ }∪{ i + ζ | − ≤ i ≤ } ∪ { i + 3 ζ | ≤ i ≤ } . If ζ / ∈ F p , then p > Cox( G ) = 12 > X ( ̟ n +1 E H ˇ α ) of X ( ̟ n +1 E H ˇ α ) ∈O E in the residue field k E is non-zero, hence v ( X ( ̟ n +1 E H ˇ α )) = 0 as desired. If ζ ∈ F p ,then one can treat each case separately and show that X ( ̟ n +1 E H ˇ α ) = 0 using that p > X ( ̟ n +1 E H ˇ α ) = 0 when the value of X ( ̟ n +1 E H ˇ α ) is aninteger, a multiple of ζ , or of the form ± ± ζ . In the remaining cases, we have the form X ( ̟ n +1 E H ˇ α ) = c + c ζ , and one verifies that c
6≡ − c mod p in each case so that c + c ζ = 0.We conclude that X is G -generic of depth n + . Appendix C. A note on Galois representations associated to automorphicforms (by Vytautas Paˇsk¯unas)
The aim of this note is to explain how to deduce Theorem 3.3.3 (in the main article) byreplacing the use of Theorem 3.2.1 in its proof by the density results proved in [4]. Weput ourselves in the setting of the proof of Theorem 3.3.3. In particular, F is a totallyreal field, E is a totally imaginary quadratic extension of F , S is a finite set of places of F containing all the places above p and ∞ , S E is the set of places of E above S , E S isthe maximal extension of E unramified outside S , G is a unitary group over F which is anouter form of GL n with respect to the quadratic extension E/F such that G is quasi-splitat all finite places and anisotropic at all infinite places. We assume that all the places of F above p split in E . This implies that G ( F ⊗ Q Q p ) is isomorphic to a product of GL n ( F v ) for v | p . Let U p be a compact open subgroup of G ( A ∞ ,pF ). If U p is a compact open subgroupof G ( F ⊗ Q Q p ) then the double coset Y ( U p U p ) := G ( F ) \ G ( A F ) /U p U p G ( F ⊗ Q R ) ◦ is a finite set. From our point of view the key objects are the completed cohomology e H ( U p ) := lim ←− m lim −→ U p H ( Y ( U p U p ) , Z /p m Z ) , where the inner limit is taken over all open compact subgroups U p and the big Heckealgebra: T S Spl ( U p ) := lim ←− m,U p T S Spl ( U p U p , Z /p m Z ) , (C.1)where T S Spl ( U p U p , Z /p m Z ) is the image the algebra T S Spl , defined in the proof of Theorem3.3.3, in End Z p ( H ( Y ( U p U p ) , Z /p m Z )). This algebra is denoted by T ′ in [4, § T S Spl ( U p ), which makes it into a profinitering. We also note that H ( Y ( U p U p ) , Z /p m Z ) is just a space of Z /p m Z -valued functionson Y ( U p U p ) and so coincides with M ( U p U p , Z /p m Z ) in § T S Spl ( U p ) is a noetherian semi-local ring and will attach a Galois rep-resentation of Gal( E S /E ) to each maximal ideal of T S Spl ( U p )[1 /p ], assuming the result ofClozel recalled in Theorem 3.3.1. This is slightly more general than Theorem 3.3.3, as weallow maximal ideals, which do not correspond to the classical automorphic forms, and wedo not have a restriction on the prime p , as we work with Bushnell–Kutzko types.We may identify e H ( U p ) (resp. e H ( U p ) Q p ) with the space of continuous Z p -valued (resp. Q p -valued) functions on the profinite set Y ( U p ) := G ( F ) \ G ( A F ) /U p G ( F ⊗ Q R ) ◦ ∼ = lim ←− U p Y ( U p U p ) . ppendix C. A note on Galois representations associated to automorphic forms The action of G ( F ⊗ Q Q p ) on Y ( U p ) makes e H ( U p ) Q p into an admissible unitary Q p -Banachspace representation of G ( F ⊗ Q Q p ) with unit ball equal to e H ( U p ). We fix an open pro- p subgroup K of G ( F ⊗ Q Q p ), which acts freely on Y ( U p ) with finitely many orbits. Thisenables us to identify e H ( U p ) as a representation of K with a direct sum of finitely manycopies of C ( K, Z p ), the space of continuous functions from K to Z p on which K acts viaright translations. Thus the Schikhof dual M := Hom cont Z p ( e H ( U p ) , Z p ) is a free Z p [[ K ]]-module of finite rank and we may apply the results of [4] to M . We point out that it followsfrom the Schikhof duality that e H ( U p ) Q p is isometric to the Banach space representationdenoted by Π( M ) in [4, Lem. 2.3].In [4, § { V i } i ∈ N of smooth absolutely irreducible repre-sentations of K defined over a finite extension of Q p such that if π is a smooth irreducible Q p -representation of G ( F ⊗ Q Q p ) and Hom K ( V i , π ) is non-zero then π is a supercuspidalrepresentation. (We actually work with GL n ( F v ) with v | p , but the argument readilyadapts to the product of such groups by taking tensor products.) Moreover, the evaluationmap ev : M i ≥ Hom K ( V i , e H ( U p ) Q p ) ⊗ V i → e H ( U p ) Q p (C.2)has dense image, see Proposition 3.26, together with Lemmas 2.3 and 2.17 in [4]. Thisdensity result is the key input in this note.The representations V i are obtained as direct summands of inductions of certain charactersof open subgroups of K , analogous to the characters ( U p,m , ψ ◦ λ m ) in the main text, butconstructed using Bushnell–Kutzko theory of types. This theory is not available for everyreductive group, but it does not impose any restrictions on the prime p . To orientatethe reader we point out that we may identify Hom K ( V i , e H ( U p ) Q p ) with M ( U p K, V ∗ i ) in § ϕ to the function f : Y ( U p K ) → V ∗ i , which satisfies f ( x )( v ) = ϕ ( v )( x ),for all x ∈ Y ( U p K ) and v ∈ V i . As explained in §
3, see also [3, Prop. 3.2.4], the spaceHom K ( V i , e H ( U p ) Q p ) is related to the space of automorphic forms on G , the action of T S Spl ( U p )[1 /p ] on this finite dimensional vector space is semi-simple and the maximal idealsin the support of Hom K ( V i , e H ( U p ) Q p ) correspond to certain classical automorphic forms,such that the associated automorphic representations are supercuspidal at places above p .For k ≥ A k be the image of ⊕ ki =1 Hom K ( V i , e H ( U p ) Q p ) ⊗ V i in e H ( U p ) Q p and let A ∞ be the image of (C.2). Let a k be the T S Spl ( U p )-annihilator of A n . Each Z p -algebra homo-morphism x : T S Spl ( U p ) / a k → Q p will correspond to a set of Hecke eigenvalues appearingin Hom K ( V i , e H ( U p ) Q p ) ⊗ Q p for some 1 ≤ i ≤ k , and hence will correspond to a classicalautomorphic form which, by construction of V i , will be supercuspidal at all places above p . Hence, to such x we may attach a Galois representation ρ x : Gal( E S /E ) → GL n ( Q p ),using the results of Clozel. Moreover, ( T S Spl ( U p ) / a k )[1 /p ] is semi-simple. Vytautas Paˇsk¯unas
Lemma C.3. If U p is an open pro- p subgroup of G ( F ⊗ Q Q p ) , then the open maximal idealsof T S Spl ( U p ) coincide with the maximal ideals of T S Spl ( U p U p , Z /p Z ) . In particular, T S Spl ( U p ) has only finitely many open maximal ideals.Proof. The transition maps in (C.1) are surjective and thus a maximal ideal m of T S Spl ( U p ) isopen if and only if it is equal to the preimage of a maximal ideal of T S Spl ( U p U ′ p , Z /p m Z ) underthe surjection T S Spl ( U p ) ։ T S Spl ( U p U ′ p , Z /p m Z ), for some open subgroup U ′ p and m ≥ U ′ p is contained in U p .Since the action of T S Spl ( U p U ′ p , Z /p m Z ) on H ( Y ( U p U ′ p ) , Z /p m Z ) is faithful by definition, weconclude that the localisation H ( Y ( U p U ′ p ) , Z /p m Z ) m is non-zero. Since the module is p -torsion and U p is pro- p the U p -invariants of its reduction modulo p are non-zero. Since theseoperations commute with localisation, we conclude that H ( Y ( U p U p ) , Z /p Z ) m is non-zeroand so m is a maximal ideal of T S Spl ( U p U p , Z /p Z ). (cid:3) If m is an open maximal ideal of T S Spl ( U p ) we let e H ( U p ) m and T S Spl ( U p ) m be the m -adiccompletions of e H ( U p ) and T S Spl ( U p ), respectively. It follows from the Chinese remaindertheorem applied at each finite level that e H ( U p ) ∼ = Y m e H ( U p ) m , T S Spl ( U p ) ∼ = Y m T S Spl ( U p ) m , (C.4)where the (finite) product is taken over all open maximal ideals of T S Spl ( U p ). Remark
C.5 . It follows from (C.4) that the completion of e H ( U p ) and T S Spl ( U p ) at an openmaximal ideal m coincides with the localisation, because inverting an element of T S Spl ( U p ),which maps to 1 in T S Spl ( U p ) m and to 0 in other completions, kills off the other factors.For k ≥ A k := A k ∩ e H ( U p ) and let A ∞ := A ∞ ∩ e H ( U p ). Then A k /p m injects into A ∞ /p m and, since A ∞ is dense in e H ( U p ) Q p , we have e H ( U p ) /p m ∼ = A ∞ /p m ∼ = lim −→ k ≥ A k /p m , ∀ m ≥ . (C.6)In particular, there exists k , such that A k /p contains H ( Y ( U p K ) , Z /p Z ). For such k , a k will also annihilate H ( Y ( U p K ) , Z /p Z ). Hence, there is a surjection T S Spl ( U p ) / a k ։ T S Spl ( U p K, Z /p Z ). It follows from Lemma C.3 that the maximal ideals T S Spl ( U p ) / a k coincidewith the open maximal ideals of T S Spl ( U p ). We note that, since A k is a finite free Z p -module,the same applies to T S Spl ( U p ) / a k and to its localisation ( T S Spl ( U p ) / a k ) m . In particular, thequotient topology on T S Spl ( U p ) / a k coincides with the p -adic one and every maximal idealof T S Spl ( U p ) / a k is open. Lemma C.7.
Let ρ x : Gal( E S /E ) → GL n ( Q p ) be the Galois representation correspond-ing to a Z p -algebra homomorphism x : ( T S Spl ( U p ) / a k ) m → Q p . Then the function D x : ppendix C. A note on Galois representations associated to automorphic forms Z p [Gal( E S /E )] → Q p , a det( ρ x ( a )) takes values in the image of x . Moreover, there isa semi-simple Galois representation ¯ ρ : Gal( E S /E ) → GL n ( F p ) such that the function D : Z p [Gal( E S /E )] → F p , a det( ¯ ρ ( a )) takes values in the residue field κ ( m ) and D ( a ) ≡ D x ( a ) (mod m ) , ∀ a ∈ Z p [Gal( E S /E )] and for all Z p -algebra homomorphisms x : ( T S Spl ( U p ) / a k ) m → Q p .Proof. Let ¯ ρ be the semisimplification of the reduction modulo p of a Gal( E S /E )-stablelattice in ρ x , for some x . As explained in the proof of Theorem 3.3.3 using density argumentsit is enough to check the assertions for a = 1+ t Frob w , w S E , split over F and t ∈ Z p . Theassertion then follows from equation (3.3.6), which expresses the characteristic polynomialof ρ x (Frob w ) in terms of Hecke operators. (cid:3) The function D : Z p [Gal( E S /E )] → κ ( m ) is a continuous n -dimensional determinant inthe sense of Chenevier [2]. The universal deformation ring R D of D is a complete localnoetherian algebra over the ring of Witt vectors of κ ( m ) by [2, Prop. 3.3, 3.7, Ex. 3.6]. Itfollows from Lemma C.7 that D x is a deformation of D and hence induces a map R D → Q p .By taking the product over all Z p -algebra homomorphisms x : ( T S Spl ( U p ) / a k ) m → Q p weobtain a continuous map R D → Y x Q p ∼ = ( T S Spl ( U p ) / a k ) m ⊗ Z p Q p , (C.8)where the last isomorphism follows since ( T S Spl ( U p ) / a k ) m [1 /p ] is semi-simple and finite over Q p . Lemma C.9.
The map (C.8) induces a surjection R D ։ ( T S Spl ( U p ) / a k ) m .Proof. This is proved in the course of the proof of Theorem 3.3.3 - let R ′ be the image of(C.8) and let D ′ be the tautological deformation of D to R ′ . Then R ′ is equal to the closureof the subring generated by the coefficients of D ′ (1 + t Frob w ) for all w S E and these arecontained in ( T S Spl ( U p ) / a k ) m . Since the coefficients of D ′ (1 + t Frob w ) can be expressed interms of Hecke operators, they are contained in R ′ . Since these generate ( T S Spl ( U p ) / a k ) m the map is surjective. We note that in the case of modular forms the analogous argumentappears in [1, § (cid:3) Theorem C.10.
The maps R D ։ ( T S Spl ( U p ) / a k ) m for k ≥ induce a surjection R D ։ T S Spl ( U p ) m . In particular, T S Spl ( U p ) m is noetherian and for every Z p -algebra homomorphism x : T S Spl ( U p ) m → Q p there is a continuous semi-simple representation ρ x : Gal( E S /E ) → GL n ( Q p ) such that det(1 + tρ x (Frob w )) = n X i =0 t i N ( w ) i ( i − / x ( T ( i ) w ) , ∀ w S E . (C.11) Vytautas Paˇsk¯unas
Proof.
Using Lemma C.9 we obtain a continuous action of R D on A k, m compatible with theinclusions A k, m ⊂ A k +1 , m , for k ≥
1. Thus for each m ≥ R D on lim −→ k ≥ ( A k /p m ) m ∼ = ( A ∞ /p m ) m ∼ = ( e H ( U p ) /p m ) m , where the last isomorphism follows from (C.6). By passing to the projective limit we obtaina continuous action of R D on e H ( U p ) m , which factors through the action of T S Spl ( U p ) m . Let R be the image of the map R D → T S Spl ( U p ) m . Since R D is a complete local noetherian ringwith a finite residue field it is compact. Since T S Spl ( U p ) m is profinite it is Hausdorff andhence R is closed in T S Spl ( U p ) m . But R is also dense, since by construction R surjects onto T S Spl ( U p U p , Z /p m Z ) m for all U p and m ≥
1. Hence, R = T S Spl ( U p ) m .Let D R be the tautological deformation of D to R . Then D R (1 + t Frob w ) = n X i =0 t i N ( w ) i ( i − / T ( i ) w , ∀ w S E , as this relation holds for all D x and hence in ( T S Spl ( U p ) / a k ) m by construction. If x : R → Q p is a homomorphism of Z p -algebras then by [2, Thm. 2.12] there is a unique semi-simplerepresentation ρ x : Gal( E S /E ) → GL n ( Q p ) such that x ( D R (1 + tg )) = det(1 + tρ x ( g )) , ∀ g ∈ Gal( E S /E ) . Since x ◦ D R is continuous, the representation ρ x is continuous by [2, Ex. 2.34]. (cid:3) Remark
C.12 . It follows from (C.4) and the theorem above that T S Spl ( U p ) is noetherian andwe may attach a Galois representation satisfying (C.11) to any Z p -algebra homomorphism x : T S Spl ( U p ) → Q p . Acknowledgements . I would like to thank the organisers Brandon Levin, RebeccaBellovin, Matthew Emerton and David Savitt for inviting me to the Banff Workshop
Mod-ularity and Moduli Spaces , which took place in Oaxaca in October 2019, where I first heardJessica Fintzen talk about her joint work with Sug Woo Shin. I would like to thank JessicaFintzen and Sug Woo Shin for kindly agreeing to include this note as an appendix to theirpaper and for their comments on earlier drafts.
References for Appendix C [1]
H. Carayol , Formes modulaires et repr´esentations galoisiennes ´a valeurs dans un anneau local com-plet, in p -adic monodromy and the Birch and Swinnerton-Dyer conjecture , Contemp. Math. 165, Amer.Math. Soc., 1994.[2] G. Chenevier , The p -adic analytic space of pseudocharacters of a profinite group, and pseudorepre-sentations over arbitrary rings, London Math. Soc. Lecture Notes Series 414, Proceedings of the LMSDurham Symposium 2011, Automorphic forms and Galois representations vol. 1, 221–285 (2014)[3]
M. Emerton , On the interpolation of eigenvalues attached to automorphic Hecke eigenforms,
Invent.Math.
164 (2006), no. 1, 1–84. ongruences and supercuspidal representations [4] M. Emerton and V. Paˇsk¯unas , On the density of supercuspidal points of fixed regular weight inlocal deformation rings and global Hecke algebras,
Journal de l’ ´Ecole polytechnique – Math´ematiques ,Volume 7 (2020) , pp. 337–371.
References [Adl98] Jeffrey D. Adler,
Refined anisotropic K -types and supercuspidal representations , Pacific J.Math. (1998), no. 1, 1–32. MR 1653184 (2000f:22019)[BLGGT14] Thomas Barnet-Lamb, Toby Gee, David Geraghty, and Richard Taylor, Local-global compat-ibility for l = p , II , Ann. Sci. ´Ec. Norm. Sup´er. (4) (2014), no. 1, 165–179. MR 3205603[Bor91] Armand Borel, Linear algebraic groups , second ed., Graduate Texts in Mathematics, vol. 126,Springer-Verlag, New York, 1991. MR 1102012[Bou02] Nicolas Bourbaki,
Lie groups and Lie algebras. Chapters 4–6 , Elements of Mathematics(Berlin), Springer-Verlag, Berlin, 2002, Translated from the 1968 French original by AndrewPressley. MR 1890629[Car12] Ana Caraiani,
Local-global compatibility and the action of monodromy on nearby cycles , DukeMath. J. (2012), no. 12, 2311–2413. MR 2972460[Car14] ,
Monodromy and local-global compatibility for l = p , Algebra Number Theory (2014),no. 7, 1597–1646. MR 3272276[CDP14] Pierre Colmez, Gabriel Dospinescu, and Vytautas Paˇsk¯unas, The p -adic local Langlands cor-respondence for GL ( Q p ), Camb. J. Math. (2014), no. 1, 1–47. MR 3272011[CH13] Ga¨etan Chenevier and Michael Harris, Construction of automorphic Galois representations,II , Camb. J. Math. (2013), no. 1, 53–73. MR 3272052[Che14] Ga¨etan Chenevier, The p -adic analytic space of pseudocharacters of a profinite group andpseudorepresentations over arbitrary rings , Automorphic forms and Galois representations.Vol. 1, London Math. Soc. Lecture Note Ser., vol. 414, 2014.[CHL11] Laurent Clozel, Michael Harris, and Jean-Pierre Labesse, Construction of automorphic Galoisrepresentations, I , On the stabilization of the trace formula, Stab. Trace Formula ShimuraVar. Arith. Appl., vol. 1, Int. Press, Somerville, MA, 2011, pp. 497–527. MR 2856383[CHT08] Laurent Clozel, Michael Harris, and Richard Taylor,
Automorphy for some l -adic lifts of au-tomorphic mod l Galois representations , Publ. Math. Inst. Hautes ´Etudes Sci. (2008), no. 108,1–181, With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B byMarie-France Vign´eras. MR 2470687 (2010j:11082)[Clo91] Laurent Clozel,
Repr´esentations galoisiennes associ´ees aux repr´esentations automorphes au-toduales de
GL( n ), Inst. Hautes ´Etudes Sci. Publ. Math. (1991), no. 73, 97–145. MR 1114211(92i:11055)[DS74] Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids
1, Ann. Sci. ´Ecole Norm.Sup. (4) (1974), 507–530 (1975). MR 379379[DS18] Stephen DeBacker and Loren Spice, Stability of character sums for positive-depth, supercus-pidal representations , J. Reine Angew. Math. (2018), 47–78. MR 3849622[Eme14] Matthew Emerton,
Completed cohomology and the p -adic Langlands program , Proceedings ofthe International Congress of Mathematicians—Seoul 2014. Vol. II, Kyung Moon Sa, Seoul,2014, pp. 319–342. MR 3728617[EP20] Matthew Emerton and Vytautas Paˇsk¯unas, On the density of supercuspidal points of fixedregular weight in local deformation rings and global Hecke algebras , J. de l’Ecole Poly. Math. (2020), 337–371.[Fina] Jessica Fintzen, On the construction of tame supercuspidal representations , https://arxiv.org/pdf/1908.09819.pdf . Jessica Fintzen and Sug Woo Shin [Finb] ,
Types for tame p -adic groups , version from Jan 24, 2020, available at .[Fin19] , Tame Tori in p -Adic Groups and Good Semisimple Elements , International Mathe-matics Research Notices (2019), rnz234.[Fla79] D. Flath, Decomposition of representations into tensor products , Automorphic forms, repre-sentations and L -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore.,1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979,pp. 179–183. MR 546596[Gro99] Benedict H. Gross, Algebraic modular forms , Israel J. Math. (1999), 61–93. MR 1729443(2001b:11037)[HM08] Jeffrey Hakim and Fiona Murnaghan,
Distinguished tame supercuspidal representations , Int.Math. Res. Pap. IMRP (2008), no. 2, Art. ID rpn005, 166. MR 2431732[HT01] Michael Harris and Richard Taylor,
The geometry and cohomology of some simple Shimuravarieties , Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton,NJ, 2001, With an appendix by Vladimir G. Berkovich. MR 1876802 (2002m:11050)[Hum90] James E. Humphreys,
Reflection groups and Coxeter groups , Cambridge Studies in AdvancedMathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460[Jan03] Jens Carsten Jantzen,
Representations of algebraic groups , second ed., Mathematical Sur-veys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003.MR 2015057[Kal19] Tasho Kaletha,
Regular supercuspidal representations , J. Amer. Math. Soc. (2019), no. 4,1071–1170. MR 4013740[Kot86] Robert E. Kottwitz, Stable trace formula: elliptic singular terms , Math. Ann. (1986),no. 3, 365–399. MR 858284 (88d:22027)[Kot92] ,
On the λ -adic representations associated to some simple Shimura varieties , Invent.Math. (1992), no. 3, 653–665. MR 1163241[Lab11] J.-P. Labesse, Changement de base CM et s´eries discr`etes , On the stabilization of the traceformula, Stab. Trace Formula Shimura Var. Arith. Appl., vol. 1, Int. Press, Somerville, MA,2011, pp. 429–470. MR 2856380[NSW08] J¨urgen Neukirch, Alexander Schmidt, and Kay Wingberg,
Cohomology of number fields , sec-ond ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Math-ematical Sciences], vol. 323, Springer-Verlag, Berlin, 2008. MR 2392026[Pin90] Richard Pink,
Arithmetical compactification of mixed Shimura varieties , Bonner Mathematis-che Schriften [Bonn Mathematical Publications], vol. 209, Universit¨at Bonn, MathematischesInstitut, Bonn, 1990, Dissertation, Rheinische Friedrich-Wilhelms-Universit¨at Bonn, Bonn,1989. MR 1128753[Rag04] M. S. Raghunathan,
Tori in quasi-split-groups , J. Ramanujan Math. Soc. (2004), no. 4,281–287. MR 2125504[Sch13] Peter Scholze, The local Langlands correspondence for GL n over p -adic fields , Invent. Math. (2013), no. 3, 663–715. MR 3049932[Sch18] , On the p -adic cohomology of the Lubin-Tate tower , Ann. Sci. ´Ec. Norm. Sup´er. (4) (2018), no. 4, 811–863, With an appendix by Michael Rapoport. MR 3861564[Shi11] Sug Woo Shin, Galois representations arising from some compact Shimura varieties , Ann. ofMath. (2) (2011), no. 3, 1645–1741. MR 2800722[Sor13] Claus M. Sorensen,
A proof of the Breuil-Schneider conjecture in the indecomposable case ,Ann. of Math. (2) (2013), no. 1, 367–382. MR 2999043[SS13] Peter Scholze and Sug Woo Shin,
On the cohomology of compact unitary group Shimuravarieties at ramified split places , J. Amer. Math. Soc. (2013), no. 1, 261–294. MR 2983012 ongruences and supercuspidal representations [Tay88] Richard Lawrence Taylor, On congruences between modular forms , ProQuestLLC, Ann Arbor, MI, 1988, Thesis (Ph.D.)–Princeton University, available at http://virtualmath1.stanford.edu/~rltaylor . MR 2636500[Tay91] Richard Taylor,
Galois representations associated to Siegel modular forms of low weight , DukeMath. J. (1991), no. 2, 281–332. MR 1115109 (92j:11044)[Xu16] Bin Xu, On a lifting problem of L-packets , Compos. Math. (2016), no. 9, 1800–1850.MR 3568940[Yu01] Jiu-Kang Yu,
Construction of tame supercuspidal representations , J. Amer. Math. Soc. (2001), no. 3, 579–622 (electronic). MR 1824988 (2002f:22033) Trinity College, Cambridge, CB2 1TQ, United Kingdom
E-mail address : [email protected] Department of Mathematics, UC Berkeley, Berkeley, CA 94720, USA // Korea Institutefor Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Republic of Korea
E-mail address : [email protected] Fakult¨at f¨ur Mathematik, Universit¨at Duisburg–Essen, 45117 Essen, Germany
E-mail address ::