Exterior square gamma factors for cuspidal representations of $$\mathrm {GL}_n$$ GL n : simple supercuspidal representations

We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations.

     

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.

     

Irreducible unitary representations of SU(2, 2)

We give an explicit classification of the irreducible unitary representations of the simple Lie group SU(2, 2).     

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     

Average of Dirichlet coefficients of cuspidal representations related to GL(2)

The Ramanujan Journal - Let $$\pi $$ be a cuspidal representation on $${{\,\mathrm{GL}\,}}(2,\mathbb {A}_{\mathbb {Q}}).$$ We give nontrivial lower and upper bounds for average of absolute values...     

The characters of supercuspidal representations as weighted orbital integrals

Weighted orbital integrals are the terms which occur on the geometric side of the trace formula. We shall investigate these distributions on ap-adic group. We shall evaluate the weighted orbital integral of a supercuspidal matrix coefficient as a multiple of the corresponding character. Supported in part by NSERC Operating Grant A3483.     

An integral transform in L2-cohomology for the ladder representations of U(p, q)

Starting from the realization of the Fock space as L²-cohomology of $\text{C}$^{p + q}, $\text{H}$^0,p($\text{C}$^{p + q}) = ⊕_m?$\text{Z}$$\text{H}$_m^0,p($\text{C}$^{p + q}), an integral transform is constructed which is a direct-image mapping from $\text{H}$_m^{0,p($\text{C}$p + q) into the space of holomorphic sections of some vector bundle E_m over M ≈ U(p, q)/(U(q) × U(p)), m ? 0}. The transform intertwines the natural actions of U(p, q) and is injective if m ? 0, so it provides a geometric realization of the ladder representations of U(p, q). The sections in the image of the transform satisfy certain linear differential equations, which are explicitly described. For example, Maxwell's equations are of this form if p = q = 2 and m = 2. Thus, this transform is analogous to the Penrose correspondence.     

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.

     

Fourier-Jacobi type spherical functions for principal series representations of Sp(2, R)

In this paper, we study the Fourier-Jacobi type spherical functions on Sp(2, R) for irreducible principal series representations. We give the multiplicity theorem and an explicit formula for this function.     

On the ramsey numbers N(3,3,…3;2)

The main results of this paper are N(3,3,3,3;2) > 50 andf(k+1)≥3 f(k)+f(k?2), where $\text{f(k) = N}\text{3,3,…;2)}\text{k}\text{times}\text{?1 for k ≥ 3}$.