Irreducible Admissible Representation Research Articles

Arithmetic branching law and generic 𝐿-packets

Let G G be a classical group defined over a local field F F of characteristic zero. For any irreducible admissible representation π \pi of G ( F ) G(F) , which is of Casselman-Wallach type if F F is archimedean, we extend the study of spectral decomposition of local descents by Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field F F . In particular, if π \pi has a generic local L L -parameter, we introduce the spectral first occurrence index f s ( π ) {\mathfrak {f}}_{\mathfrak {s}}(\pi ) and the arithmetic first occurrence index f a ( π ) {\mathfrak {f}}_{{\mathfrak {a}}}(\pi ) of π \pi and prove in this paper that f s ( π ) = f a ( π ) {\mathfrak {f}}_{\mathfrak {s}}(\pi )={\mathfrak {f}}_{{\mathfrak {a}}}(\pi ) . Based on the theory of consecutive descents of enhanced L L -parameters developed by Jiang, Liu, and Zhang [Arithmetic wavefront sets and generic L L -packets, arXiv:2207.04700], we are able to show in this paper that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result (Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535], Theorem 1.7) to broader generality.

Cyclic base change of cuspidal automorphic representations over function fields

Let $G$ be a split semisimple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi.

A Cauchy Harish-Chandra integral for a dual pair over a p-adic field, the definition and a conjecture

For a real irreducible dual pair there is an integral kernel operator which maps the distribution character of an irreducible admissible representation of the group with the smaller or equal rank to an invariant eigendistribution on the group with the larger or equal rank.The purpose of this article is to transfer this construction to the p-adic case. We provide the precise definition of the integral kernel operator and formulate a conjecture.

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.

On the duality involution for p-adic general spin groups

Let GSpinm be the quasi-split general spin group over a p-adic field F for a positive integer m. For a certain class of quasi-split reductive groups, including the general spin groups, a specific involution ι, called the duality involution, is constructed in [Pra19]–generalising the MVW involution on the classical groups. In this article, for any irreducible admissible representation (π,V) of GSpinm(F), we show that π∨≃πι. Here π∨ is the contragredient of π and πι is the composite of π with ι. We also prove the analogue of this result for the split general spin groups over Fq((t)).

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.

Contragredients and a multiplicity one theorem for general spin groups

Each orthogonal group $${\text {O}}(n)$$ has a nontrivial $${\text {GL}}(1)$$ -extension, which we call $${\text {GPin}}(n)$$ . The identity component of $${\text {GPin}}(n)$$ is the more familiar $${\text {GSpin}}(n)$$ , the general Spin group. We prove that the restriction to $${\text {GPin}}(n-1)$$ of an irreducible admissible representation of $${\text {GPin}}(n)$$ over a nonarchimedean local field of characteristic zero is multiplicity free and also prove the analogous theorem for $${\text {GSpin}}(n)$$ . Our proof uses the method of Aizenbud, Gourevitch, Rallis and Schiffman, who proved the analogous theorem for $${\text {O}}(n)$$ , and of Waldspurger, who proved that for $${\text {SO}}(n)$$ . We also give an explicit description of the contragredient of an irreducible admissible representation of $${\text {GPin}}(n)$$ and $${\text {GSpin}}(n)$$ , which is needed to apply their method to our situations.

Appearance of the Kashiwara–Saito singularity in the representation theory of p-adic GL(16)

In 1993 David Vogan proposed a basis for the vector space of stable distributions on $p$-adic groups using the microlocal geometry of moduli spaces of Langlands parameters. In the case of general linear groups, distribution characters of irreducible admissible representations, taken up to equivalence, form a basis for the vector space of stable distributions. In this paper we show that these two bases, one putative, cannot be equal. Specifically, we use the Kashiwara-Saito singularity to find a non-Arthur type irreducible admissible representation of $p$-adic $\mathop{GL}_{16}$ whose ABV-packet, as defined in earlier work, contains exactly one other representation; remarkably, this other admissible representation is of Arthur type. In the course of this study we strengthen the main result concerning the Kashiwara-Saito singularity. The irreducible admissible representations in this paper illustrate a fact we found surprising: for general linear groups, while all A-packets are singletons, some ABV-packets are not.

The local period integrals and essential vectors

AbstractBy applying the formula for essential Whittaker functions established by Matringe and Miyauchi, we study five integral representations for irreducible admissible generic representations of GLn over p‐adic fields. In each case, we show that the integrals achieve local formal L‐functions defined by Langlands parameters, when the test vector is associated to the new form. We give the relation between local periods involving essential Whittaker functions and special values of formal L‐factors at for certain distinguished or unitary representations. The period integrals are also served as standard nonzero distinguished forms.

A remark on conductor, depth and principal congruence subgroups

In this paper we study a relation between conductor, depth, and the level of principal congruence subgroups for irreducible admissible representations of GLn(F) for a non-archimedean local field F of characteristic zero. As a global application, we estimate conductors of unitary irreducible automorphic cuspidal representations of GLn(AQ) in terms of principal congruence subgroups. We also give an explicit formula for the dimension of fixed vectors with respect to principal congruence subgroups for irreducible admissible representations of GL2(F) as a local application.

Characters and quantum reduction for orthosymplectic Lie superalgebras

In this study, we study principal admissible representations for the affine Lie superalgebra [Formula: see text]. Using the character formula of irreducible admissible representations of [Formula: see text], we calculate a character formula of [Formula: see text]-modules which are obtained from the quantized Drinfeld–Sokolov reduction and principal admissible representations. As a by-product, we obtain the minimal series modules of the Neveu–Schwarz algebra through the [Formula: see text]-modules arising from the principal admissible modules over [Formula: see text].

Sign signatures and characters of Weyl Groups

Let GR be the set of real points of a complex linear reductive group and Gˆλ, its classes of irreducible admissible representations with infinitesimal integral regular character λ. In this case, each cell of representations is associated to a special nilpotent orbit. This helps organize the corresponding set of irreducible Harish-Chandra modules. The goal of this paper is to bypass the need for the character table of the Weyl group associated with Gˆλ in the Springer correspondence, by describing a new and less computationally intensive parametrization of irreducible representations of simple complex Weyl groups.

Theta correspondence for p-adic dual pairs of type I

Abstract We describe explicitly the Howe correspondence for the symplectic-orthogonal and unitary dual pairs over a nonarchimedean local field of characteristic zero. More specifically, for every irreducible admissible representation of these groups, we find its first occurrence index in the theta correspondence and we describe, in terms of their Langlands parameters, the small theta lifts on all levels.

Représentations et quasi-caractères de niveau 0 ; endoscopie

Let F be a finite extension of ℚ p and let G be a connected reductive group over F. We assume that p is large relatively to G. Let G ′ be an endoscopic group of G. Following Arthur, we have, roughly speaking, a spectral transfer morphism, denoted by transfert, which, to a stable finite linear combination of irreducible admissible representations of G ′ (F), associates a finite linear combination of irreducible admissible representations of G(F). Let p 0,G be the Bernstein’s projector such that, for an irreducible admissible representation π of G(F), we have p 0,G (π)=π if π has level 0 and p 0,G (π)=0 if π has strictly positive level. Define similarly p 0,G ′ . We prove that p 0,G ′ preserves the space of stable finite linear combinations of irreducible admissible representations of G ′ (F) and that p 0,G ∘transfert=transfert∘p 0,G ′ .

Admissible restrictions of irreducible representations of reductive Lie groups: symplectic geometry and discrete decomposability

Let $G$ be a real reductive Lie group, $L$ a compact subgroup, and $\pi$ an irreducible admissible representation of $G$. In this article we prove a necessary and sufficient condition for the finiteness of the multiplicities of $L$-types occurring in $\pi$ based on symplectic techniques. This leads us to a simple proof of the criterion for discrete decomposability of the restriction of unitary representations with respect to noncompact subgroups (the author, Ann. Math. 1998), and also provides a proof of a reverse statement which was announced in [Proc.ICM 2002, Thm.D]. A number of examples are presented in connection with Kostant's convexity theorem and also with non-Riemannian locally symmetric spaces.

DUALIZING INVOLUTIONS ON THE METAPLECTIC GL(2) à la TUPAN

AbstractLet F be a non-Archimedean local field of characteristic zero. Let G = GL(2, F) and $3\widetildeG = \widetilde{GL}(2,F)$ be the metaplectic group. Let τ be the standard involution on G. A well-known theorem of Gelfand and Kazhdan says that the standard involution takes any irreducible admissible representation of G to its contragredient. In such a case, we say that τ is a dualizing involution. In this paper, we make some modifications and adapt a topological argument of Tupan to the metaplectic group $\widetildeG$ and give an elementary proof that any lift of the standard involution to $\widetildeG$ ; is also a dualizing involution.

Dualizing involutions on the metaplectic GL(2)

Let F be a non-Archimedean local field of characteristic zero. Let G=GL(2,F) and G˜=GL˜(2,F) be the metaplectic group. Let τ be the standard involution on G. A well known theorem of Gelfand and Kazhdan says that the standard involution takes any irreducible admissible representation of G to its contragredient. In such a case, we say that τ is a dualizing involution. In this paper, we show that any lift of the standard involution to G˜ is also a dualizing involution.

Distinguished representations, Shintani base change and a finite field analogue of a conjecture of Prasad

Let E/F be a quadratic extension of fields, and G a connected quasi-split reductive group over F. Let Gop be the opposition group obtained by twisting G by the duality involution considered by Prasad. Assume that the field F is finite. Let π be an irreducible generic representation of G(E). When π is a Shintani base change lift of some representation of Gop(F), we give an explicit nonzero G(F)-invariant vector in terms of the Whittaker vector of π. This shows particularly that π is G(F)-distinguished.When the field F is p-adic, the paper also proves that the duality involution takes an irreducible admissible generic representation of G(F) to its contragredient. As a special case of this result, all generic representations of G2,F4 or E8 are self-dual.

Stability of Asai local factors for GL(2)

Let [Formula: see text] be a non-archimedean local field of characteristic not equal to [Formula: see text] and let [Formula: see text] be a quadratic algebra. We prove the stability of local factors attached to irreducible admissible (complex) representations of [Formula: see text] via the Rankin–Selberg method under highly ramified twists. This includes both the Asai as well as the Rankin–Selberg local factors attached to pairs. Our method relies on expressing the gamma factor as a Mellin transform using Bessel functions.

Derivatives and exceptional poles of the local exterior square L-function for $$GL_m$$

Let $$\pi $$ be an irreducible admissible representation of $$GL_m(F)$$, where F is a non-archimedean local field of characteristic zero. In 1990’s Jacquet and Shalika established an integral representation for the exterior square L-function. We complete, following the method developed by Cogdell and Piatetski-Shapiro, the computation of the local exterior square L-function $$L(s,\pi ,\wedge ^2)$$ via the integral representation in terms of L-functions of supercuspidal representations by a purely local argument. With this result, we show the equality of the local analytic L-functions $$L(s,\pi ,\wedge ^2)$$ via the integral representation for the irreducible admissible representation $$\pi $$ for $$GL_m(F)$$ and the local arithmetic L-functions $$L(s, \wedge ^2(\phi (\pi )))$$ of its Langlands parameter $$\phi (\pi )$$ through local Langlands correspondence.