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

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 $n>1$.