Regular irreducible characters of a hyperspecial compact group

TL;DR

This paper provides a new parametrization of irreducible unitary representations of hyperspecial compact groups over non-archimedean local fields, using Clifford theory and Weil representations, under certain triviality assumptions.

Contribution

It introduces a novel parametrization method for these representations based on character groups of finite abelian groups, extending previous theories.

Findings

Parametrization aligns with regular adjoint orbits.

Method applies to general linear groups with strong evidence.

Relies on triviality of certain Schur multipliers.

Abstract

A parametrization of irreducible unitary representations associated with the regular adjoint orbits of a hyperspecial compact subgroup of a reductive group over a non-dyadic non-archimedean local filed is presented. The parametrization is given by means of (a subset of) the character group of certain finite abelian groups arising from the reductive group. Our method is based upon Cliffod's theory and Weil representations over finite fields. It works under an assumption of the triviality of certain Schur multipliers defined for an algebraic group over a finite field. The assumption of the triviality has good evidences in the case of general linear groups and highly probable in general.