Irreducible Supercuspidal Representations Research Articles

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.

On the types for supercuspidal representations of inner forms of GLN

Let F be a non-Archimedean local field, A be a central simple F-algebra, and G be the multiplicative group of A. It is known that for every irreducible supercuspidal representation π, there exists a [G,π]G-type (J,λ), called a (maximal) simple type. We will show that [G,π]G-types defined over some maximal compact subgroup are unique up to G-conjugations under some unramifiedness assumption on a simple stratum.

Restricting supercuspidal representations via a restriction of data

Let $F$ be a non-archimedean local field of residual characteristic $p$. Let $\mathbb{G}$ be a reductive group defined over $F$ which splits over a tamely ramified extension and set $G=\mathbb{G}(F)$. We assume that $p$ does not divide the order of the Weyl group of $\mathbb{G}$. Given a closed connected $F$-subgroup $\mathbb{H}$ that contains the derived subgroup of $\mathbb{G}$, we study the restriction to $H$ of an irreducible supercuspidal representation $\pi=\pi_G(\Psi)$ of $G$, where $\Psi$ is a $G$-datum as per the J.K. Yu Construction. We provide a full description of $\pi|_H$ into irreducible components, with multiplicity, via a restriction of data which constructs $H$-data from $\Psi$. Analogously, we define a restriction of Kim-Yu types to study the restriction of irreducible representations of $G$ which are not supercuspidal.

A reducibility problem for even unitary groups: The depth zero case

We study a problem concerning parabolic induction in certain p-adic unitary groups. More precisely, for E/F a quadratic extension of p-adic fields the associated unitary group G=U(n,n) contains a parabolic subgroup P with Levi component L isomorphic to GLn(E). Let π be an irreducible supercuspidal representation of L of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation ιPGπ is reducible.

Types for tame $p$-adic groups

Let $k$ be a non-archimedean local field with residual characteristic $p$. Let $G$ be a connected reductive group over $k$ that splits over a tamely ramified field extension of $k$. Suppose $p$ does not divide the order of the Weyl group of $G$. Then we show that every smooth irreducible complex representation of $G(k)$ contains an 𝔰-type of the form constructed by Kim--Yu and that every irreducible supercuspidal representation arises from Yu's construction. This improves an earlier result of Kim, which held only in characteristic zero and with a very large and ineffective bound on $p$. By contrast, our bound on $p$ is explicit and tight, and our result holds in positive characteristic as well. Moreover, our approach is more explicit in extracting an input for Yu's construction from a given representation.

Some results on reducibility of parabolic induction for classical groups

Given a (complex, smooth) irreducible representation $\pi$ of the general linear group over a non-archimedean local field and an irreducible supercuspidal representation $\sigma$ of a classical group, we show that the (normalized) parabolic induction $\pi\rtimes\sigma$ is reducible if there exists $\rho$ in the supercuspidal support of $\pi$ such that $\rho\rtimes\sigma$ is reducible. In special cases we also give irreducibility criteria for $\pi\rtimes\sigma$ when the above condition is not satisfied.

An induction theorem for groups acting on trees

If G G is a group acting on a locally finite tree X X , and S \mathscr {S} is a G G -equivariant sheaf of vector spaces on X X , then its compactly-supported cohomology is a representation of G G . Under a finiteness hypothesis, we prove that if H c 0 ( X , S ) H_c^0(X, \mathscr {S}) is an irreducible representation of G G , then H c 0 ( X , S ) H_c^0(X, \mathscr {S}) arises by induction from a vertex or edge stabilizing subgroup. If G \boldsymbol {\mathrm {G}} is a reductive group over a nonarchimedean local field F F , then Schneider and Stuhler realize every irreducible supercuspidal representation of G = G ( F ) G = \boldsymbol {\mathrm {G}}(F) in the degree-zero cohomology of a G G -equivariant sheaf on its reduced Bruhat-Tits building X X . When the derived subgroup of G \boldsymbol {\mathrm {G}} has relative rank one, X X is a tree. An immediate consequence is that every such irreducible supercuspidal representation arises by induction from a compact-mod-center open subgroup.

A local converse theorem for $${\mathrm {Sp}}_{2r}$$ Sp 2 r

In this paper, we prove the local converse theorem for $${\mathrm {Sp}}_{2r}(F)$$ over a p-adic field F. More precisely, given two irreducible supercuspidal representations of $${\mathrm {Sp}}_{2r}(F)$$ with the same central character such that they are generic with the same additive character and they have the same gamma factors when twisted with generic irreducible representations of $${\mathrm {GL}}_n(F)$$ for all $$1\le n\le r$$ , then these two representations must be isomorphic. Our proof is based on the local analysis of the local integrals which define local gamma factors. A key ingredient of the proof is certain partial Bessel function property developed by Cogdell–Shahidi–Tsai recently. The same method can give the local converse theorem for $${\mathrm {U}}(r,r)$$ .

On Residues of Intertwining Operators in Cases with Prehomogeneous Nilradical

AbstractLet P = M N be a Levi decomposition of a maximal parabolic subgroup of a connected reductive group G over a p-adic field F. Assume that there exists w0∊ G(F) that normalizes M and conjugates P to an opposite parabolic subgroup. When N has a Zariski dense Int M-orbit, F. Shahidi and X. Yu described a certain distribution D on M(F), such that, for irreducible unitary supercuspidal representations π of M(F) withis irreducible if and only if D( f )≠ 0 for some pseudocoefficient f of π. Since this irreducibility is conjecturally related to π arising via transfer from certain twisted endoscopic groups of M, it is of interest to realize D as endoscopic transfer from a simpler distribution on a twisted endoscopic group H of M. This has been done in many situations where N is abelian. Here we handle the standard examples in cases where N is nonabelian but admit a Zariski dense Int M-orbit.

L-Functoriality for Local Theta Correspondence of Supercuspidal Representations with Unipotent Reduction

AbstractThe preservation principle of local theta correspondences of reductive dual pairs over ap-adic field predicts the existence of a sequence of irreducible supercuspidal representations of classical groups. Adams and Harris-Kudla-Sweet have a conjecture about the Langlands parameters for the sequence of supercuspidal representations. In this paper we prove modified versions of their conjectures for the case of supercuspidal representations with unipotent reduction.

On the generic local Langlands correspondence for 𝐺𝑆𝑝𝑖𝑛 groups

In the case of split G S p i n GSpin groups, we prove an equality of L L -functions between automorphic local L L -functions defined by the Langlands-Shahidi method and local Artin L L -functions. Our method of proof is based on previous results of the first author which allow us to reduce the problem to supercuspidal representations of Levi subgroups of G S p i n GSpin , by constructing Langlands parameters for general generic irreducible admissible representations of G S p i n GSpin from the one for generic irreducible supercuspidal representations of its Levi subgroups.

Supercuspidal representations and preservation principle of theta correspondence

Abstract The preservation principle of the local theta correspondence predicts the existence of a chain of irreducible supercuspidal representations of p-adic classical groups. In this paper, we give an explicit characterization of the chain starting from an irreducible supercuspidal representations of a unitary group of one variable or an orthogonal group of two variables. In particular, we define the Lusztig-like correspondence of generic cuspidal data for p-adic groups and establish its relation with local theta correspondence of supercuspidal representations for p-adic dual pairs.

On the degree of certain localL-functions

Let be an irreducible supercuspidal representation of GLn.F/, where F is a p-adic field. By a result of Bushnell and Kutzko, the group of unramified self-twists of has cardinality n=e, where e is the oF-period of the principal oF-order in Mn.F/ attached to . This is the degree of the local Rankin‐ Selberg L-function L.s; _ /. In this paper, we compute the degree of the Asai, symmetric square, and exterior square L-functions associated to . As an application, assuming p is odd, we compute the conductor of the Asai lift of a supercuspidal representation, where we also make use of the conductor formula for pairs of supercuspidal representations due to Bushnell, Henniart, and Kutzko (1998).

Towards the Jacquet conjecture on the Local Converse Problem for $p$-adic $\mathrm {GL}_n$

The Local Converse Problem is to determine how the family of the local gamma factors \gamma(s,\pi\times\tau,\psi) characterizes the isomorphism class of an irreducible admissible generic representation \pi of \mathrm {GL}_n(F) , with F a non-archimedean local field, where \tau runs through all irreducible supercuspidal representations of \mathrm {GL}_r(F) and r runs through positive integers. The Jacquet conjecture asserts that it is enough to take r=1,2,\ldots,\left[\frac{n}{2}\right] . Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture, we formulate a general approach to prove the Jacquet conjecture. With this approach, the Jacquet conjecture is proved under an assumption which is then verified in several cases, including the case of level zero representations.

On the set of maximal nilpotent supports of supercuspidal representations

Let G be a quasisplit reductive group over a p-adic field k, T a maximal unramified anisotropic torus ofG.k/, and a character ofT.k/ satisfying certain conditions. Assume the residue characteristicp ofk is large enough. It was shown by DeBacker and Reeder that the irreducible supercuspidal representation of G.k/ associated to .T.k/;/ is generic if and only if B.T;k/ is a special vertex of B.G;k/. We compute the set of maximal nilpotent support Nwh;max. / when B.T;k/ is not a special point in B.G;k/.

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.

Théorie de Lubin–Tate non abélienne ℓ-entière

For two distinct primes p and l, we investigate the Z_l-cohomology of the Lubin-Tate towers of a p-adic field. We prove that it realizes some version of Langlands and Jacquet-Langlands correspondences for flat families of irreducible supercuspidal representations parametrized by a Z_l-algebra R, in a way compatible with extension of scalars. When R is a field of characteristic l, this gives a cohomological realization of the Langlands-Vigneras correspondence for supercuspidals, and a new proof of its existence. When R runs over complete local algebras, this provides bijections between deformations of matching mod-l representations. Roughly speaking, we can decompose "the supercuspidal part" of the l-integral cohomology as a direct sum, indexed by irreducible supercuspidals \pi mod l, of tensor products of universal deformations of \pi and of its two mates. Besides, we also get a virtual realization of both the semi-simple Langlands-Vigneras correspondence and the l-modular Langlands-Jacquet transfer for all representations, by using the cohomology complex and working in a suitable Grothendieck group.

The tempered spectrum of quasi-split classical groups III: The odd orthogonal groups

Abstract. We study the tempered spectrum of special orthogonal groups of odd degree over p-adic fields of characteristic zero. Residues of intertwining operators for representations parabolically induced from arbitrary maximal parabolic subgroups are determined in terms of non-vanishing of weighted sums of twisted orbital integrals. This is connected with the theory of twisted endoscopy, and poles of the Langlands L-functions, in particular, with the Rankin product L-function between irreducible (generic) supercuspidal representations of the classical group SO2m+1(F) and those of GLk(F). These poles are described for, any m and k, explicitly in terms of twisted endoscopy. Moreover, we show these poles describe precisely when a supercuspidal representation of GLk(F) is the local component of the transfer from a special odd orthogonal group.

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.

On Types for Unramified p-Adic Unitary Groups

AbstractLet F be a non-archimedean local field of residue characteristic neither 2 nor 3 equipped with a galois involution with fixed field F0, and let G be a symplectic group over F or an unramified unitary group over F0. Following the methods of Bushnell–Kutzko for GL(N, F), we define an analogue of a simple type attached to a certain skew simple stratum, and realize a type in G. In particular, we obtain an irreducible supercuspidal representation of G like GL(N, F).