The local Rankin-Selberg convolution for GL(n): divisibility of the conductor

Let F be a non-Archimedean local field and an integer. Let be irreducible supercuspidal representations of GL with . One knows that there exists an irreducible supercuspidal representation of GL, with , such that the local constants (in the sense of Jacquet, Piatetskii-Shapiro and Shalika) are distinct. In this paper, we show that, when is an unramified twist of , one may here takem dividingn and , for a prime divisor ofn depending on and the order of : in particular, , where is the least prime divisor of . This follows from a result giving control of certain divisibility properties of the conductor of a pair of supercuspidal representations. Received: 11 November 2000 / Accepted: 15 January 2001 / Published online: 23 July 2001     

The supercuspidal representations of <Emphasis Type="Italic">p</Emphasis>-adic classical groups

Let G be a unitary, symplectic or special orthogonal group over a locally compact non-archimedean local field of odd residual characteristic. We construct many new supercuspidal representations of G, and Bushnell–Kutzko types for these representations. Moreover, we prove that every irreducible supercuspidal representation of G arises from our constructions.     

Construction of tame supercuspidal representations

We give a quite general construction of irreducible supercuspidal representations and supercuspidal types (in the sense of Bushnell and Kutzko) of -adic groups. In the tame case, the construction should include all known constructions, and it is expected that this gives all supercuspidal representations. We also give a conjectural Hecke algebra isomorphism, which can be used to analyze arbitrary irreducible admissible representations, following the ideas of Howe and Moy.

     

Reducibility of some induced representations of -adic unitary groups

In this paper we study reducibility of those representations of quasi-split unitary -adic groups which are parabolically induced from supercuspidal representations of general linear groups. For a supercuspidal representation associated via Howe's construction to an admissible character, we show that in many cases a criterion of Goldberg for reducibility of the induced representation reduces to a simple condition on the admissible character.

     

On Local Descent to Metaplectic Groups and Local Theta Correspondence

Let F be a p-adic field of characteristic 0.We study a twisted local descent construction for the metaplectic groups Sp_(2 n)(F),and also its relation to the corresponding local descent construction for odd special orthogonal groups via local theta correspondence.In consequence,we show that this descent construction gives irreducible supercuspidal genuine representations of Sp_(2n)(-F) parametrized by a simple local L-parameter φ_τ corresponding to an irreducible supercuspidal representation τ of GL_(2n)(F) of symplectic type,and the genericity of the representations constructed can be indicated by a local epsilon factor condition.In particular,this local descent construction recovers the local Shimura correspondence for supercuspidal representations.     

Symplectic supercuspidal representations and related problems

In this paper, we summarize the basic structures and properties of irreducible symplectic supercuspidal representations of GLn(F) over a p-adic local field F with characteristic zero, and explore possible topics for further investigation.     

Gamma factors of certain supercuspidal representations

A -factor defined by the doubling method is calculated for certain supercuspidal representations. The result is a sum over a finite group of Lie type, which may be called a non-abelian Gauss sum. It is the sum of the product of a cuspidal character and an Igusa zeta integral. In some cases, we show that our result agrees with what is expected by the local Langlands conjecture. Received: 11 July 1999 / Accepted: 29 December 1999 / Published online: 28 June 2000     

Tame Supercuspidal Representations of GL(n) Distinguished by a Unitary Group

This paper analyzes the space Hom_H(, 1), where is an irreducible, tame supercuspidal representation of GL(n) over a p-adic field and H is a unitary group in n variables contained in GL(n). It is shown that this space of linear forms has dimension at most one. The representations which admit nonzero H-invariant linear forms are characterized in several ways, for example, as the irreducible, tame supercuspidal representations which are quadratic base change lifts.Research supported in part by NSA grant #MDA904-99-1-0065.Research supported in part by NSERC     

A correspondence for the supercuspidal representations of $\overline {SL_2(F)}$, F p-adic

Following D. Manderscheid, we describe the supercuspidal representations of the n-fold metaplectic cover [`(SL<sub>2</sub>(F))]\overline {SL_2(F)}, where F is a p-adic field with (p, 2n) = 1. We prove a "Frobenius formula" for the character of a supercuspidal representation of [`(SL₂(F))]\overline {SL_2(F)}. Using this formula, we obtain a character relation between corresponding supercuspidal representations of [`(SL₂(F))]\overline {SL_2(F)} and of SL₂(F)> in the case n = 2.     

Lifting of generic depth zero representations of classical groups

We describe how generic depth zero supercuspidal representations of classical groups lift to a general linear group. The main tool is a computation of reducibility points of certain parabolically induced representations.     

On the cusp forms of congruence subgroups of an almost simple Lie group

In this paper we address the issue of existence of newforms among the cusp forms for almost simple Lie groups using the approach of the second author combined with local information on supercuspidal representations for p-adic groups known by the first author. We pay special attention to the case of \(SL_M({\mathbb {R}})\) where we prove various existence results for principal congruence subgroups.     

On cuspidal representations of general linear groups over discrete valuation rings

We define a new notion of cuspidality for representations of GL _n over a finite quotient o _k of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups G _λ of torsion o-modules. When n is a prime, we show that this notion of cuspidality is equivalent to strong cuspidality, which arises in the construction of supercuspidal representations of GL _n (F). We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations of GL _n (o _k ) for k ≥ 2 for all n is equivalent to the construction of the representations of all the groups G _λ . A functional equation for zeta functions for representations of GL _n (o _k ) is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for GL₄(o₂) are constructed. Not all these representations are strongly cuspidal.     

Distance-transitive representations of the sporadic groups

A permutation representation of a finite group is multiplicity-free if all the irreducible constituents in the permutation character are distinct. There are three main reasons why these representations are interesting: it has been checked that all finite simple groups have such permutation representations, these are often of geometric interest, and the actions on vertices of distance-transitive graphs are multiplicity-free.

In this paper we classify the primitive multiplicity-free representations of the sporadic simple groups and their automorphism groups. We determine all the distance-transitive graphs arising from these representations. Moreover, we obtain intersection matrices for most of these actions, which are of further interest and should be useful in future investigations of the sporadic simple groups.     

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 ?? 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.     

Test vectors for trilinear forms when at least one representation is not supercuspidal

Given three irreducible, admissible, infinite dimensional complex representations of GL₂(F), with F a local non-Archimedean field, the space of trilinear functionals invariant by the group has dimension at most one. When it is one we provide an explicit vector on which the functional does not vanish assuming that not all three representations are supercuspidal.     

Self-contragredient supercuspidal representations of

Let be a non-archimedean local field of residual characteristic . Then has tamely ramified self-contragredient supercuspidal representations if and only if or is even. When such representations exist, they do so in abundance.

     

Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique

Let F be a finite extension of ℚ _p . For each integer n≥1, we construct a bijection from the set ?_F ⁰ (n) of isomorphism classes of irreducible degree n representations of the (absolute) Weil group of F, onto the set ? _F ⁰ (n) of isomorphism classes of smooth irreducible supercuspidal representations of GL _n (F). Those bijections preserve epsilon factors for pairs and hence we obtain a proof of the Langlands conjectures for GL _n over F, which is more direct than Harris and Taylor’s. Our approach is global, and analogous to the derivation of local class field theory from global class field theory. We start with a result of Kottwitz and Clozel on the good reduction of some Shimura varieties and we use a trick of Harris, who constructs non-Galois automorphic induction in certain cases. Oblatum 1-III-1999 & 21-VII-1999 / Published online: 29 November 1999