Unitary groups over local rings

TL;DR

This paper explores the structural properties of unitary groups over local rings, including non-commutative cases, and applies these findings to compute group orders and analyze Weil representation constituents.

Contribution

It develops new structural insights into unitary groups over local rings, extending to non-commutative cases, and applies these to order computation and representation analysis.

Findings

Derived formulas for orders of finite unitary groups

Analyzed degrees of irreducible Weil representation constituents

Extended structural properties to non-commutative local rings

Abstract

Structural properties of unitary groups over local, not necessarily commutative, rings are developed, with applications to the computation of the orders of these groups (when finite) and to the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified extension of finite local rings.