# Regular irreducible represntations of classical groups over finite quotient rings

## TL;DR

This paper provides a new parametrization of irreducible representations of classical groups over finite quotient rings, using Weil representations and character groups of centralizers, with explicit cases for several classical groups.

## Contribution

It introduces a novel parametrization method for regular irreducible representations over finite quotient rings of classical groups, based on Weil representations and centralizer character groups.

## Key findings

- Parametrization of irreducible representations via centralizer character groups.
- Explicit descriptions for general linear, symplectic, orthogonal, and special linear groups.
- Application of Weil representations over finite fields to representation theory.

## Abstract

A parametrization of irreducible representations associated with a regular adjoint orbit of a classical group over finite quotient rings of the ring of integer of a non-dyadic non-archimedean local field is presented. The parametrization is given by means of (a subset of) the character group of the centralizer of a representative of the regular adjoint orbit. Our method is based upon Weil representations over finite fields. More explicit parametrization in terms of tamely ramified extensions of the base field is given for the general linear group, the special linear group, the symplectic group and the orthogonal group.