Regular irreducible characters of a hyperspecial compact group and Weil representations over finite fields

TL;DR

This paper introduces a method for constructing irreducible unitary representations of hyperspecial compact subgroups of reductive p-adic groups, utilizing Clifford theory and Weil representations over finite fields, under certain Schur multiplier assumptions.

Contribution

It provides a novel approach to representation construction for these groups, especially under the assumption of trivial Schur multipliers, with evidence from classical groups.

Findings

Method successfully constructs irreducible representations for specific groups.

Triviality of Schur multipliers is verified for classical groups.

Approach relies on Clifford theory and Weil representations over finite fields.

Abstract

A method to construct irreducible unitary representations of a hyperspecial compact subgroup of a reductive group over p-adic field with odd p is presented. Our method is based upon Cliffods 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 regular cases in general. We will give several examples of classical groups where the Schur multipliers are actually trivial.