The Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring

TL;DR

This paper thoroughly analyzes the Weil representation of a unitary group over a ramified quadratic extension of a finite local ring, detailing its irreducible constituents, multiplicity, and character degrees.

Contribution

It provides a complete classification of the irreducible constituents of the Weil representation for these groups, including their multiplicity and explicit description.

Findings

Weil representation is multiplicity free with monomial irreducible constituents.

Number of irreducible constituents is explicitly determined.

Character degrees are computed when the base ring is a field.

Abstract

We find all irreducible constituents of the Weil representation of a unitary group $U_m(A)$ of rank $m$ associated to a ramified quadratic extension $A$ of a finite, commutative, local and principal ring $R$ of odd characteristic. We show that this Weil representation is multiplicity free with monomial irreducible constituents. We also find the number of these constituents and describe them in terms of Clifford theory with respect to a congruence subgroup. We find all character degrees in the special case when $R$ is a field.