Supercuspidal Representations Research Articles

Depth zero supercuspidal representations of classical groups into l-packets: The typically almost symmetric case

We classify what we call “typically almost symmetric” depth zero supercuspidal representations of a classical group over a local field of odd residual characteristic into L-packets. Our main results resolve an ambiguity in the paper of Lust-Stevens [24] in this case, where they could only classify these representations in two or four, if not one, L-packets. By assuming the expected numbers of supercuspidal representations in the L-packets, we employ only simple properties of the representations to prove the main results. In particular, we do not require any deep calculations of character values. With the same method, we also compute the parity of a (conjugate-)self-dual depth zero supercuspidal representation of a general linear group.

The wavefront sets of unipotent supercuspidal representations

The wavefront sets of unipotent supercuspidal representations

Modulo ℓ distinction problems

Let $F$ be a non-archimedean local field of characteristic different from 2 and residual characteristic $p$ . This paper concerns the $\ell$ -modular representations of a connected reductive group $G$ distinguished by a Galois involution, with $\ell$ an odd prime different from $p$ . We start by proving a general theorem allowing to lift supercuspidal $\overline {\mathbf {F}}_{\ell }$ -representations of $\operatorname {GL}_n(F)$ distinguished by an arbitrary closed subgroup $H$ to a distinguished supercuspidal $\overline {\mathbf {Q}}_{\ell }$ -representation. Given a quadratic field extension $E/F$ and an irreducible $\overline {\mathbf {F}}_{\ell }$ -representation $\pi$ of $\operatorname {GL}_n(E)$ , we verify the Jacquet conjecture in the modular setting that if the Langlands parameter $\phi _\pi$ is irreducible and conjugate-selfdual, then $\pi$ is either $\operatorname {GL}_n(F)$ -distinguished or $(\operatorname {GL}_{n}(F),\omega _{E/F})$ -distinguished (where $\omega _{E/F}$ is the quadratic character of $F^\times$ associated to the quadratic field extension $E/F$ by the local class field theory), but not both, which extends one result of Sécherre to the case $p=2$ . We give another application of our lifting theorem for supercuspidal representations distinguished by a unitary involution, extending one result of Zou to $p=2$ . After that, we give a complete classification of the $\operatorname {GL}_2(F)$ -distinguished representations of $\operatorname {GL}_2(E)$ . Using this classification we discuss a modular version of the Prasad conjecture for $\operatorname {PGL}_2$ . We show that the ‘classical’ Prasad conjecture fails in the modular setting. We propose a solution using non-nilpotent Weil–Deligne representations. Finally, we apply the restriction method of Anandavardhanan and Prasad to classify the $\operatorname {SL}_2(F)$ -distinguished modular representations of $\operatorname {SL}_2(E)$ .

Simple Supercuspidal 𝐿-Packets of Quasi-Split Classical Groups

In this memoir, for quasi-split classical groups over p p -adic fields, we determine the L L -packets consisting of simple supercuspidal representations and their corresponding L L -parameters, under the assumption that p p is not equal to 2 2 . The key is an explicit computation of characters of simple supercuspidal representations and the endoscopic character relation, which is a characterization of the local Langlands correspondence for quasi-split classical groups.

On input and Langlands parameters for epipelagic representations

A paper of Reeder–Yu [J. Amer. Math. Soc. 27 (2014), pp. 437–477] gives a construction of epipelagic supercuspidal representations of p p -adic groups. The input for this construction is a pair ( λ , χ ) (\lambda , \chi ) where λ \lambda is a stable vector in a certain representation coming from a Moy–Prasad filtration, and χ \chi is a character of the additive group of the residue field. We say two such pairs are equivalent if the resulting supercuspidal representations are isomorphic. In this paper we describe the equivalence classes of such pairs. As an application, we give a classification of the simple supercuspidal representations for split adjoint groups. Finally, under an assumption about unramified base change, we describe properties of the Langlands parameters associated to these simple supercuspidals, showing that they have trivial L-functions and minimal Swan conductors, and showing that each of these simple supercuspidals lies in a singleton L-packet.

Functional equations and gamma factors of local zeta functions for the metaplectic cover of SL2

We introduce a local zeta-function for an irreducible admissible supercuspidal representation π of the metaplectic double cover of SL2 over a non-archimedean local field of characteristic zero. We prove a functional equation of the local zeta-functions showing that the gamma factor is given by a Mellin type transform of the Bessel function of π. We obtain an expression of the gamma factor, which shows its entireness on C. Moreover, we show that, through the local theta-correspondence, the local zeta-function on the covering group is essentially identified with the local zeta-integral for spherical functions on PGL2≅SO3 associated with the prehomogenous vector space of symmetric matrices of degree 2.

Formal degree of regular supercuspidals

Supercuspidal representations are usually infinite-dimensional, so the size of such a representation cannot be measured by its dimension; the formal degree is a better alternative. Hiraga, Ichino, and Ikeda conjectured a formula for the formal degree of a supercuspidal in terms of its L -parameter only. Our first main result is to compute the formal degrees of the supercuspidal representations constructed by Yu. Our second result, using the first, is to verify that Kaletha’s regular supercuspidal L -packets satisfy the conjecture.

Geometric $L$-packets of Howe-unramified toral supercuspidal representations

We show that L -packets of toral supercuspidal representations arising from unramified maximal tori of p -adic groups are realized by Deligne–Lusztig varieties for parahoric subgroups. We prove this by exhibiting a direct comparison between the cohomology of these varieties and algebraic constructions of supercuspidal representations. Our approach is to establish that toral irreducible representations are uniquely determined by the values of their characters on a domain of sufficiently regular elements. This is an analogue of Harish-Chandra’s characterization of real discrete series representations by their characters on regular elements of compact maximal tori, a characterization which Langlands relied on in his construction of L -packets of these representations. In parallel to the real case, we characterize the members of Kaletha’s toral L -packets by their characters on sufficiently regular elements of elliptic maximal tori.

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).

Classification of irreducible representations of metaplectic covers of the general linear group over a non-archimedean local field

Let F F be a non-archimedean local field and r r a non-negative integer. The classification of the irreducible representations of G L r ( F ) GL_r(F) in terms of supercuspidal representations is one of the highlights of the Bernstein–Zelevinsky theory. We give an analogous classification for metaplectic coverings of G L r ( F ) GL_r(F) .

Geometric wave-front set may not be a singleton

We show that the geometric wave-front set of specific half-integral-depth supercuspidal representations of ramified p p -adic unitary groups is not a singleton.

Local exterior square and Asai L-functions for GL(n) in odd characteristic

Let $F$ be a non-archimedean local field of odd characteristic $p > 0$. In this paper, we consider local exterior square $L$-functions $L(s,\pi,\wedge^2)$, Bump-Friedberg $L$-functions $L(s,\pi,BF)$, and Asai $L$-functions $L(s,\pi,As)$ of an irreducible admissible representation $\pi$ of $GL_m(F)$. In particular, we establish that those $L$-functions, via the theory of integral representations, are equal to their corresponding Artin $L$-functions $L(s,\wedge^2(\phi(\pi)))$, $L(s+1/2,\phi(\pi))L(s,\wedge^2(\phi(\pi)))$, and $L(s,As(\phi(\pi)))$ of the associated Langlands parameter $\phi(\pi)$ under the local Langlands correspondence. These are achieved by proving the identity for irreducible supercuspidal representations, exploiting the local to global argument due to Henniart and Lomeli.

Supercuspidal representations in non-defining characteristics

We show that a mod-ℓ-representation of a p-adic group arising from the analogue of Yu's construction is supercuspidal if and only if it arises from a supercuspidal representation of a finite reductive group. This has been previously shown by Henniart and Vigneras under the assumption that the second adjointness holds.

A local Langlands parameterization for generic supercuspidal representations of $p$-adic $G_2$

We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / \QQ_p$ we will construct a bijection \[ \CL_g : \CA^0_g(G_2,K) \rightarrow \CG^0(G_2,K) \] from the set of generic supercuspidal representations of $G_2(K)$ to the set of irreducible continuous homomorphisms $\rho : W_K \to G_2(\CC)$ with $W_K$ the Weil group of $K$. The construction of the map is simply a matter of assembling arguments that are already in the literature, together with a previously unpublished theorem of G. Savin on exceptional theta correspondences, included as an appendix. The proof that the map is a bijection is arithmetic in nature, and specifically uses automorphy lifting theorems. These can be applied thanks to a recent result of Hundley and Liu on automorphic descent from $GL(7)$ to $G_2$.

Airy sheaves for reductive groups

We construct a class of ℓ $\ell$ -adic local systems on A 1 $\mathbb {A}^1$ that generalizes the Airy sheaves defined by N. Katz to reductive groups. These sheaves are finite field analogues of generalizations of the classical Airy equation y ′ ′ ( z ) = z y ( z ) $y^{\prime \prime }(z)=zy(z)$ . We employ the geometric Langlands correspondence to construct the sought-after local systems as eigenvalues of certain rigid Hecke eigensheaves, following the methods developed by Heinloth, Ngô, and Yun. The construction is motivated by a special case of Adler and Yu's construction of tame supercuspidal representations. The representations that we consider can be viewed as deeper analogues of simple supercuspidals. For GL n $\mathrm{GL}_n$ , we compute the Frobenius trace of the local systems in question and show that they agree with Katz's Airy sheaves. We make precise conjectures about the ramification behavior of the local systems at ∞ $\infty$ . These conjectures, in particular, imply cohomological rigidity of Airy sheaves.

Top degree Ext-groups and relatively supercuspidal spectra

Relatively supercuspidal representations are analogue of supercuspidal representations in the relative Langlands program. This work studies relatively supercuspidal representations using top degree Ext-groups via the Schneider-Stuhler duality. As examples, the relatively supercuspidal spectra for the Flicker-Rallis case and the diagonal case are determined.

Stable Vectors in Dual Vinberg Representations of F4

AbstractThis paper gives a classification of stable vectors in dual Vinberg representations coming from a graded Lie algebra of type F4 in a way that is independent of the field of definition. Relating these gradings to Moy–Prasad filtrations, we obtain the input for Reeder–Yu’s construction of epipelagic supercuspidal representations. As a corollary, this construction gives new supercuspidal representations of $F_{4}(\mathbb {Q}_{p})$ F 4 ( ℚ p ) when p is small.

Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by a unitary involution

Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by a unitary involution

Simple Supercuspidals and the Langlands Correspondence

In this paper, we survey some mathematical developments that followed from the discovery of simple supercuspidal representations of p-adic groups.