Clifford theory of Weil representations of unitary groups

TL;DR

This paper provides a detailed Clifford theory analysis of Weil representations of unitary groups over certain rings, describing their irreducible components relative to various congruence subgroups.

Contribution

It offers a comprehensive Clifford theory framework for understanding Weil representations of unitary groups over rings with involution, including both abelian and nonabelian congruence subgroups.

Findings

Clifford theory description of all irreducible components

Analysis includes abelian and nonabelian congruence subgroups

Applicable to unitary groups over rings with involution

Abstract

Let ${\mathcal O}$ be an involutive discrete valuation ring with residue field of characteristic not 2. Let $A$ be a quotient of ${\mathcal O}$ by a nonzero power of its maximal ideal and let $*$ be the involution that $A$ inherits from ${\mathcal O}$. We consider various unitary groups ${\mathcal U}_m(A)$ of rank $m$ over $A$, depending on the nature of $*$ and the equivalence type of the underlying hermitian or skew hermitian form. Each group ${\mathcal U}_m(A)$ gives rise to a Weil representation. In this paper, we give a Clifford theory description of all irreducible components of the Weil representation of ${\mathcal U}_m(A)$ with respect to all of its abelian congruence subgroups and a third of its nonabelian congruence subgroups.