Reductive P-adic Groups Research Articles

Restriction to compact subgroups in the cyclic homology of reductive p-adic groups

Restriction of functions from a reductive p-adic group G to its compact subgroups defines an operator on the Hochschild and cyclic homology of the Hecke algebra of G. We study the commutation relations between this operator and others coming from representation theory: Jacquet functors, idempotents in the Bernstein centre, and characters of admissible representations.

Distinction of depth-zero representations

Let π be a depth-zero irreducible admissible representation of a connected reductive p-adic group G. Let H be the group of fixed points of an involution θ of G. We relate H-distinction of π to existence of minimal K-types of π that exhibit particular symmetry properties relative to θ. In addition, we show that when π is H-distinguished, then (up to conjugacy) the support of π is of the form (M,τ) where M is a θ-stable Levi subgroup of G and τ is a depth-zero irreducible supercuspidal representation of M. Moreover, τ contains a minimal K-type (Mx,ρ) such that Mx is a θ-stable maximal parahoric subgroup of M and ρ is the inflation of a distinguished cuspidal representation of the quotient of Mx by its pro-unipotent radical.

Whittaker functions and Demazure operators

We show that elements of a natural basis of the Iwahori fixed vectors in a principal series representation of a reductive p-adic group satisfy certain recursive relations. The precise identities involve operators that are variants of the Demazure–Lusztig operators, with correction terms, which may be calculated by a combinatorial algorithm that is identical to the computation of the fibers of the Bott–Samelson resolution of a Schubert variety. This leads to an action of the affine Hecke algebra on functions on the maximal torus of the L-group. A closely related action was previously described by Lusztig using equivariant K-theory of the flag variety, leading to the proof of the Deligne–Langlands conjecture by Kazhdan and Lusztig. In the present paper, the action is applied to give a simple formula for the basis vectors of the Iwahori Whittaker functions. We also show that these Whittaker functions can be expressed as non-symmetric Macdonald polynomials.

Resolutions of tempered representations of reductive p-adic groups

Let G be a reductive group over a non-archimedean local field and let S(G) be its Schwartz algebra. We compare Ext-groups of tempered G-representations in several module categories: smooth G-representations, algebraic S(G)-modules, bornological S(G)-modules and an exact category of S(G)-modules on LF-spaces, which contains all admissible S(G)-modules. We simplify the proofs of known comparison theorems for these Ext-groups, due to Meyer and Schneider–Zink. Our method is based on the Bruhat–Tits building of G and on analytic properties of the Schneider–Stuhler resolutions.

Extensions of tempered representations

Let π, π′ be irreducible tempered representations of an affine Hecke algebra $${\mathcal{H}}$$ with positive parameters. We compute the higher extension groups Ext $${{}_\mathcal{H}^n (\pi,\pi')}$$ explicitly in terms of the representations of analytic R-groups corresponding to π and π′. The result has immediate applications to the computation of the Euler–Poincaré pairing EP (π, π′), the alternating sum of the dimensions of the Ext-groups. The resulting formula for EP(π, π′) is equal to Arthur’s formula for the elliptic pairing of tempered characters in the setting of reductive p-adic groups. Our proof applies equally well to affine Hecke algebras and to reductive groups over non-archimedean local fields of arbitrary characteristic. This sheds new light on the formula of Arthur and gives a new proof of Kazhdan’s orthogonality conjecture for the Euler–Poincaré pairing of admissible characters.

Abelian varieties and Weil representations

The main goal of this article is to construct and study a family of Weil representations over an arbitrary locally noetherian scheme without restriction on characteristic. The key point is to recast the classical theory in the scheme-theoretic setting. As in work of Mumford, Moret-Bailly and others, a Heisenberg group (scheme) and its representation can be naturally constructed from a pair of an abelian scheme and a nondegenerate line bundle, replacing the role of a symplectic vector space. Once enough is understood about the Heisenberg group and its representations (e.g., the analogue of the Stone–von Neumann theorem), it is not difficult to produce the Weil representation of a metaplectic group (functor) from them. As an interesting consequence (when the base scheme is SpecF¯p), we obtain the new notion of mod p Weil representations of p-adic metaplectic groups on F¯p-vector spaces. The mod p Weil representations admit an alternative construction starting from a p-divisible group with a symplectic pairing. We have been motivated by a few possible applications, including a conjectural mod p theta correspondence for p-adic reductive groups and a geometric approach to the (classical) theta correspondence.

Nonexistence of arithmetic fake compact Hermitian symmetric spaces of type other than An (n ≤ 4)

The quotient of a Hermitian symmetric space of non-compact type by a torsion-free cocompact arithmetic subgroup of the identity component of the group of isometries of the symmetric space is called an arithmetic fake compact Hermitian symmetric space if it has the same Betti numbers as the compact dual of the Hermitian symmetric space. This is a natural generalization of the notion of “fake projective planes” to higher dimensions. Study of arithmetic fake compact Hermitian symmetric spaces of type An with even n has been completed in [PY1], [PY2]. The results of this paper, combined with those of [PY2], imply that there does not exist any arithmetic fake compact Hermitian symmetric space of type other than An, n ≤ 4 (see Theorems 1 and 2 in the Introduction below and Theorem 2 of [PY2]). The proof involves the volume formula given in [P], the Bruhat-Tits theory of reductive p-adic groups, and delicate estimates of various number theoretic invariants.

Théorème de Paley—Wiener pour les fonctions de Whittaker sur un groupe réductif p-adique

RésuméSoit G un groupe réductif p-adique et U0 le radical unipotent d'un sous-groupe parabolique minimal de G. Nous introduisons une transformation de Fourier pour l'espace des fonctions de Whittaker lisses sur G et à support compact modulo U0. Nous en déterminons l'image. La preuve suit celle d'Heiermann pour les fonctions sur le groupe.Au cours de la preuve, une formule d'inversion est prouvée. Celle-ci permet de montrer qu'une représentation lisse irréductible de G, qui possède modèle de Whittaker dans les fonctions de Whittaker à support compact modulo U0, est cuspidale.Ce travail nous a donné l'opportunité de préparer un cadre pour l'analyse harmonique sur les espaces symétriques réductifs p-adiques: B-matrices et terme constant, propriétés des paquets d'ondes.

Murnaghan-Kirillov theory for depth-zero supercuspidal representations: reduction to Lusztig functions

The goal of Murnaghan-Kirillov theory is to associate to an irreducible smooth representation of a reductive p-adic group a family of regular semisimple orbital integrals in the Lie algebra with the following property: the character of π is given, on a well determined set, by an explicit combination of the Fourier transforms of these orbital integrals. Subject to certain restrictions, we adapt arguments of Waldspurger to show that, for depth-zero irreducible smooth supercuspidal representations, this problem may be reduced to a similar one for distributions associated to Lusztig functions.

Characters and growth of admissible representations of reductive p-adic groups

AbstractWe use coefficient systems on the affine Bruhat–Tits building to study admissible representations of reductive p-adic groups in characteristic not equal to p. We show that the character function is locally constant and provide explicit neighbourhoods of constancy. We estimate the growth of the subspaces of invariants for compact open subgroups.

A construction of elements in the Bernstein center for quasi-split groups

The Bernstein center of a reductive p-adic group is the algebra of conjugation invariant distributions on the group which are essentially compact, i.e., invariant distributions whose convolution against a locally constant compactly supported function is again locally constant compactly supported. The center acts naturally on any smooth representation, and if the representation is irreducible, each element of the center acts as a scalar. For a connected reductive quasi-split group, we show certain linear combinations of orbital integrals belong to the Bernstein center. Furthermore, when these combinations are projected to a Bernstein component, they form an ideal in the Bernstein center which can be explicitly described and is often a principal ideal. The elements constructed here should have applications to various questions in harmonic analysis.

The classification of irreducible admissible mod p representations of a p-adic GL n

Let F be a finite extension of ℚ p . Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over \(\overline{ \mathbb{F}}_{p}\) to be supersingular. We then give the classification of irreducible admissible smooth GL n (F)-representations over \(\overline{ \mathbb{F}}_{p}\) in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel–Livné for n=2. For general split reductive groups we obtain similar results under stronger hypotheses.

Exactness of the reduction on étale modules

We prove the exactness of the reduction map from étale ( φ , Γ ) -modules over completed localized group rings of compact open subgroups of unipotent p-adic algebraic groups to usual étale ( φ , Γ ) -modules over Fontaine's ring. This reduction map is a component of a functor from smooth p-power torsion representations of p-adic reductive groups (or more generally of Borel subgroups of these) to ( φ , Γ ) -modules. Therefore this gives evidence for this functor—which is intended as some kind of p-adic Langlands correspondence for reductive groups—to be exact. We also show that the corresponding higher Tor-functors vanish. Moreover, we give the example of the Steinberg representation as an illustration and show that it is acyclic for this functor to ( φ , Γ ) -modules whenever our reductive group is GL d + 1 ( Q p ) for some d ⩾ 1 .

A Satake isomorphism in characteristic p

AbstractSuppose that G is a connected reductive group over a p-adic field F, that K is a hyperspecial maximal compact subgroup of G(F), and that V is an irreducible representation of K over the algebraic closure of the residue field of F. We establish an analogue of the Satake isomorphism for the Hecke algebra of compactly supported,K-biequivariant functions f:G(F)→End V. These Hecke algebras were first considered by Barthel and Livné for GL 2. They play a role in the recent mod p andp-adic Langlands correspondences for GL 2 (ℚp) , in generalisations of Serre’s conjecture on the modularity of mod p Galois representations, and in the classification of irreducible mod p representations of unramified p-adic reductive groups.

Distinguished Tame Supercuspidal Representations

In these notes, we describe some examples of distinguished tame supercuspidal representations of reductive p-adic groups. We also discuss some general results about distinguished tame supercuspidal representations from a series of joint papers with Jeffrey Hakim ([HM2]– [HM4]). Finally, we discuss the application of results from [HM4] to obtain a parametrization of equivalence classes of tame supercuspidal representations.

On some generic very cuspidal representations

AbstractLet G be a reductive p-adic group. Given a compact-mod-center maximal torus S⊂G and sufficiently regular character χ of S, one can define, following Adler, Yu and others, a supercuspidal representation π(S,χ) of G. For S unramified, we determine when π(S,χ) is generic, and which generic characters it contains.

Resolutions for representations of reductive p-adic groups via their buildings

Schneider-Stuhler and Vigneras have used cosheaves on the affine Bruhat-Tits building to construct natural finite type projective resolutions for admissible representations of reductive p-adic groups in characteristic not equal to p. We use a system of idempotent endomorphisms of a representation with certain properties to construct a cosheaf and a sheaf on the building. We establish that these are acyclic and compute homology and cohomology with these coefficients. This implies Bernstein's result that certain subcategories of the category of representations are Serre subcategories. Furthermore, we also get results for convex subcomplexes of the building. Following work of Korman, this leads to trace formulas for admissible representations.

Constant term of smooth $H_\psi $-spherical functions on a reductive $p$-adic group

Let Ψ be a smooth character of a closed subgroup, H, of a reductive p-adic group G. If P is a parabolic subgroup of G such that PH is open in G, we define the constant term of every smooth function on G which transforms by Ψ under the right action of G. The example of mixed models is given: it includes symmetric spaces and Whittaker models. In this case a notion of cuspidal function is defined and studied. It leads to finiteness theorems.

The Generalized Injectivity Conjecture for Classical p -Adic Groups

We prove that the conjecture posed by Casselman and Shahidi, which says that a generic subquotient of a generic standard representation of a connected quasi-split reductive p-adic group is a subrepresentation (“generalized injectivity conjecture”), is valid in the case of classical groups with the generalized rank one standard representation. The generalized rank one case that we prove in this article is the key step in proving the claim in general, for every generic standard representation of a classical group.

On an algebraic approach to the Zelevinsky classification for classical p-adic groups

In this paper we classify irreducible, admissible representations of p-adic classical groups in a purely algebraic way. We present a new and simple algorithm for constructing the classifying data which is also useful in other contexts where the computation with Jacquet modules or constant terms of Eisenstein series appears. We construct and study non-standard intertwining operators. They are defined in a purely algebraic way although their rationality follows from [G. Muić, A geometric construction of intertwining operators for reductive p-adic groups, Manuscripta Math. 125 (2008) 241–272].