Reductive P-adic Groups Research Articles

Bruhat-Tits buildings, representations of p-adic groups and Langlands correspondence

The Bruhat-Tits theory is a key ingredient in the construction of irreducible smooth representations of p-adic reductive groups. We describe generalizations to arbitrary such representations of several results recently obtained in the case of supercuspidal representations, in particular regarding the local Langlands correspondence and the internal structure of the L-packets.We prove that the enhanced L-parameters with semisimple cuspidal support are those which are obtained via the (ordinary) Springer correspondence.Let G be a connected reductive group over a non-archimedean field F of residual characteristic p. In the case where G splits over a tamely ramified extension of F and p does not divide the order of the Weyl group of G, we show that the enhanced L-parameters with semisimple cuspidal support correspond to irreducible smooth representations of G(F) with non-singular supercuspidal support via the local Langlands correspondence constructed by Kaletha, under the assumption that the latter satisfies certain expected properties of functorial nature. As a consequence, we obtain that every compound L-packet of G(F) contains at least one representation with non-singular supercuspidal support.

Hecke algebras for p-adic reductive groups and Local Langlands Correspondences for Bernstein blocks

We study the endomorphism algebras attached to Bernstein components of reductive p-adic groups and construct a local Langlands correspondence with the appropriate set of enhanced L-parameters, using certain “desiderata” properties for the LLC for supercuspidal representations of proper Levi subgroups. We give several applications of our LLC to various reductive groups with Bernstein blocks cuspidally supported on general linear groups.In particular, for Levi subgroups of maximal parabolic subgroups of the split exceptional group G2, we compute the explicit weight functions for the corresponding Hecke algebras, and show that they satisfy a conjecture of Lusztig's. Some results from §4 are used by the same authors to construct a full local Langlands correspondence in [9]. Moreover, we prove a “reduction to depth-zero” result for regular Bernstein blocks (i.e., blocks for which the supercuspidal support of each irreducible representation is regular).

Some Unipotent Arthur Packets for Reductive p-adic Groups

Abstract Let ${\textsf k}$ be a $p$-adic field, and let $\textbf {G}({\textsf k})$ be the ${\textsf k}$-points of a connected reductive group, inner to split. The set of Aubert–Zelevinsky duals of the constituents of a tempered L-packet forms an Arthur packet for $\textbf {G}({\textsf k})$. In this paper, we give an alternative characterization of such Arthur packets in terms of the wavefront set, proving in some instances a conjecture of Jiang–Liu and Shahidi. Pursuing an analogy with real and complex groups, we define some special unions of Arthur packets, which we call weak Arthur packets and describe their constituents in terms of their Langlands parameters.

The left adjoint of derived parabolic induction

We prove that the derived parabolic induction functor, defined on the unbounded derived category of smooth mod p representations of a p-adic reductive group, admits a left adjoint textrm{L}(U,-). We study the cohomology functors textrm{H}^icirc textrm{L}(U,-) in some detail and deduce that textrm{L}(U,-) preserves bounded complexes and global admissibility in the sense of Schneider–Sorensen. Using textrm{L}(U,-) we define a derived Satake homomorphism and prove that it encodes the mod p Satake homomorphisms defined explicitly by Herzig.

Muller–Speh–Vogan type irreducibility criterion for generalized principal series

We provide an elementary way to obtain an irreducibility criterion for generalized principal series of p-adic reductive groups and the finite central covering groups, extending known and frequently employed results for principal series. Basically our approach is a counting argument which rests on a semi-direct product decomposition of the relative Weyl group and its action on generalized principal series, in conjunction with the theory of Jacquet module. We thus circumvent the obstacle that the relative Weyl group of a general Levi subgroup is not a Coxeter group. Along the way, we observe a structural irreducibility criterion with respect to Levi subgroups. Indeed, following Speh–Vogan's argument for real groups which is essentially a corollary of intertwining operator theory, the former result can be generalized to parabolic inductions inducing from essentially tempered representations, but the latter one can't be obtained in this way. In view of the lack of intertwining operator theory for mod ℓ representations, our counting argument may play a role there, as opposed to the failure of Speh–Vogan's argument.

Gelfand–Kirillov dimension and the p-adic Jacquet–Langlands correspondence

Abstract We bound the Gelfand–Kirillov dimension of unitary Banach space representations of p-adic reductive groups, whose locally analytic vectors afford an infinitesimal character. We use the bound to study Hecke eigenspaces in completed cohomology of Shimura curves and p-adic Banach space representations of the group of units of a quaternion algebra over ℚ p {\mathbb{Q}_{p}} appearing in the p-adic Jacquet–Langlands correspondence, deducing finiteness results in favorable cases.

A note on the representation theory of central extensions of reductive p-adic groups

In this note, we verify that several fundamental results from the theory of representations of reductive p-adic groups, extend to finite central extensions of these groups.

Representations of a reductive p-adic group in characteristic distinct from p

We investigate the irreducible cuspidal $C$-representations of a reductive $p$-adic group $G$ over a field $C$ of characteristic different from $p$. When $C$ is algebraically closed, for many groups $G$, a list of cuspidal $C$-types $(J,\lambda)$ has been produced satisfying exhaustion, sometimes for a restricted kind of cuspidal representations, and often unicity. We verify that those lists verify Aut($C$)-stability and we produce similar lists when $C$ is no longer assumed algebraically closed. Our other main results concern supercuspidality. This notion makes sense for the representations $\lambda$ in the cuspidal $C$-types $(J,\lambda)$ as above, which involve finite reductive groups. We check that an irreducible cuspidal representation of $G$ induced from $\lambda$ is supercuspidal if and only $\lambda$ is supercuspidal.

Endomorphism algebras and Hecke algebras for reductive p-adic groups

Let G be a reductive p-adic group and let Rep(G)s be a Bernstein block in the category of smooth complex G-representations. We investigate the structure of Rep(G)s, by analysing the algebra of G-endomorphisms of a progenerator Π of that category.We show that Rep(G)s is “almost” Morita equivalent with a (twisted) affine Hecke algebra. This statement is made precise in several ways, most importantly with a family of (twisted) graded algebras. It entails that, as far as finite length representations are concerned, Rep(G)s and EndG(Π)-Mod can be treated as the module category of a twisted affine Hecke algebra.We draw two major consequences. Firstly, we show that the equivalence of categories between Rep(G)s and EndG(Π)-Mod preserves temperedness of finite length representations. Secondly, we provide a classification of the irreducible representations in Rep(G)s, in terms of the complex torus and the finite group canonically associated to Rep(G)s. This proves a version of the ABPS conjecture and enables us to express the set of irreducible G-representations in terms of the supercuspidal representations of the Levi subgroups of G.Our methods are independent of the existence of types, and apply in complete generality.

From category [formula omitted] to locally analytic representations

Let G be a p-adic reductive group and g its Lie algebra. We construct a functor from the extension closure of the Bernstein-Gelfand-Gelfand category O associated to g into the category of locally analytic representations of G, thereby expanding on an earlier construction of Orlik-Strauch. A key role in this new construction is played by p-adic logarithms on tori. This functor is shown to be exact with image in the subcategory of admissible representations in the sense of Schneider and Teitelbaum. En route, we establish some basic results in the theory of modules over distribution algebras and related subalgebras, such as a tensor-hom adjunction formula. We also relate our constructions to certain representations constructed by Breuil and Schraen in the context of the p-adic Langlands program.

Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

We construct a family of representations of affine Hecke algebras, which depend on a number of auxiliary parameters g_i, and which we refer to as metaplectic representations. We realize these representations as quotients of certain parabolically induced modules, and we apply the method of Baxterization (localization) to obtain actions of corresponding Weyl groups on rational functions on the torus. Our construction both generalizes and provides a conceptual proof of earlier results of Chinta, Gunnells, and Puskas, which had depended on a crucial computer verification. A key motivation is that when the parameters g_i are specialized to certain Gauss sums, the resulting representation and its localization arise naturally in the consideration of p-parts of Weyl group multiple Dirichlet series. In this special case, similar results have been previously obtained in the literature by the study of Iwahori Whittaker functions for principal series of metaplectic covers of reductive p-adic groups. However this technique is not available for generic parameters g_i. It turns out that the metaplectic representations can be extended to the double affine Hecke algebra, where they share many important properties with Cherednik’s basic polynomial representation, which they generalize. This allows us to introduce families of metaplectic polynomials, which depend on the g_i, and which generalize Macdonald polynomials. In this paper we discuss in some detail the situation for type A, which is of considerable interest in algebraic combinatorics. We postpone some of the proofs, as well as a discussion of other types, to the sequel.

A note on sign of a self-dual representation

D. Prasad showed that the sign of a self-dual representation of a finite or p-adic reductive group is often detected by a central element. We study the extension of his results to some more general situations and make some observations about the consequences of his results.

ON FORMAL DEGREES OF UNIPOTENT REPRESENTATIONS

AbstractLet G be a reductive p-adic group which splits over an unramified extension of the ground field. Hiraga, Ichino and Ikeda [24] conjectured that the formal degree of a square-integrable G-representation $\pi $ can be expressed in terms of the adjoint $\gamma $ -factor of the enhanced L-parameter of $\pi $ . A similar conjecture was posed for the Plancherel densities of tempered irreducible G-representations. We prove these conjectures for unipotent G-representations. We also derive explicit formulas for the involved adjoint $\gamma $ -factors.

On representations of reductive p-adic groups over ℚ-algebras

In this paper we study certain category of smooth modules for reductive p-adic groups analogous to the usual smooth complex representations but with the field of complex numbers replaced by a ℚ-algebra. We prove some fundamental results in these settings, and as an example we give a classification of admissible unramified irreducible representations using the reduction to the complex case.

Cocenters of p-adic Groups, III: Elliptic and Rigid Cocenters

In this paper, we show that the elliptic cocenter of the Hecke algebra of a connected reductive p-adic group is contained in the rigid cocenter. As applications, we prove the trace Paley–Wiener theorem and the abstract Selberg principle for mod-l representations.

Distinguished regular supercuspidal representations

Based on recent work of Kaletha, we apply Hakim–Murnaghan’s result to study distinguished regular supercuspidal representations of tamely ramified reductive p-adic groups. Assuming p is sufficiently large, we obtain a necessary and sufficient condition for regular supercuspidal representations to be distinguished. We also investigate the relation between the distinction problem and the Langlands functoriality, and confirm a conjecture of Lapid for regular depth-zero or epipelagic supercuspidal representations.

Asymptotic behavior of supercuspidal representations and Sato-Tate equidistribution for families

We establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes. For the tame supercuspidals constructed by J.-K. Yu we prove the limit multiplicity property with error terms. Thereby we obtain a Sato-Tate equidistribution for the Hecke eigenvalues of these families. The main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive p-adic group tend to zero uniformly for every noncentral semisimple element.

A geometric interpretation of Newton strata

The Newton strata of a reductive p-adic group are introduced in He (Forum Math Pi 6:e2, 2018) and play some role in the representation theory of p-adic groups. In this paper, we give a geometric interpretation of the Newton strata.

Harish-Chandra's orthogonality relations for admissible representations

We prove a generalization of Harish-Chandra’s character orthogonality relations for discrete series to arbitrary admissible representations for real reductive Lie groups. This result is an analogue of a conjecture by Kazhdan for p-adic reductive groups proved by Bezrukavnikov and by Schneider and Stuhler.

A vanishing result for higher smooth duals

In this paper we prove a general vanishing result for Kohlhaase’s higher smooth duality functors Si. If G is any unramified connected reductive p-adic group, K is a hyperspecial subgroup, and V is a Serre weight, we show that Si(indKGV)=0 for i> dim(G∕B), where B is a Borel subgroup and the dimension is over ℚp. This is due to Kohlhaase for GL2(ℚp), in which case it has applications to the calculation of Si for supersingular representations. Our proof avoids explicit matrix computations by making use of Lazard theory, and we deduce our result from an analogous statement for graded algebras via a spectral sequence argument. The graded case essentially follows from Koszul duality between symmetric and exterior algebras.