Characters Of General Linear Groups Research Articles

Endo-parameters for p-adic classical groups

For a classical group over a non-archimedean local field of odd residual characteristic p, we prove that two cuspidal types, defined over an algebraically closed field {mathbf {C}} of characteristic different from p, intertwine if and only if they are conjugate. This completes work of the first and third authors who showed that every irreducible cuspidal {mathbf {C}}-representation of a classical group is compactly induced from a cuspidal type. We generalize Bushnell and Henniart’s notion of endo-equivalence to semisimple characters of general linear groups and to self-dual semisimple characters of classical groups, and introduce (self-dual) endo-parameters. We prove that these parametrize intertwining classes of (self-dual) semisimple characters and conjecture that they are in bijection with wild Langlands parameters, compatibly with the local Langlands correspondence.

The Zelevinsky classification of unramified representations of the metaplectic group

In this paper a Zelevinsky type classification of genuine unramified irreducible representations of the metaplectic group over a p-adic field with p≠2 is obtained. The classification consists of three steps. Firstly, it is proved that every genuine irreducible unramified representation is a fully parabolically induced representation from unramified characters of general linear groups and a genuine irreducible negative unramified representation of a smaller metaplectic group. Genuine irreducible negative unramified representations are described in terms of parabolic induction from unramified characters of general linear groups and a genuine irreducible strongly negative unramified representation of a smaller metaplectic group. Finally, genuine irreducible strongly negative unramified representations are classified in terms of Jordan blocks. The main technical tool is the theory of Jacquet modules.

Tensor products of unipotent characters of general linear groups over finite fields

Given unipotent characters U 1, . . . , U k of GL n \( \left( {{{\mathbb{F}}_q}} \right) \), we prove that \( \left\langle {{U_1} \otimes \cdots \cdots \otimes {U_k},1} \right\rangle \) is a polynomial in q with non-negative integer coefficients. We study the degree of this polynomial and give a necessary and sufficient condition in terms of the representation theory of symmetric groups and root systems for this polynomial to be non-zero.

A generating function of strict Gelfand patterns and some formulas on characters of general linear groups

A generating function of strict Gelfand patterns and some formulas on characters of general linear groups