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

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