Weil representations of unitary groups over ramified extensions of finite local rings with odd nilpotency length

TL;DR

This paper decomposes the Weil representation of unitary groups over certain ramified local rings, revealing its irreducible components and multiplicity properties, especially when restricted to special unitary subgroups.

Contribution

It provides the first explicit irreducible decomposition of Weil representations for unitary groups over ramified extensions with odd nilpotency, including multiplicity results.

Findings

Weil representation is multiplicity free.

Decomposition into irreducible components is explicitly characterized.

Restriction to special unitary groups preserves irreducibility.

Abstract

We find the irreducible decomposition of the Weil representation of the unitary group $\mathrm{U}_{2n}(A)$, where $A$ is a ramified quadratic extension of a finite, commutative, local, principal ideal ring $R$ and the nilpotency degree of the maximal ideal of $A$ is odd. We show in particular that this Weil representation is multiplicity free. Restriction to the special unitary group $\mathrm{SU}_{2n}(A)$ preserves irreducibility and multiplicity freeness provided $n>1$.