Clifford theory of Weil representations of unitary groups

Let $\mathcal{O}$ be an involutive discrete valuation ring with residue field of characteristic not 2. Let $A$ be a quotient of $\mathcal{O}$ by a nonzero power of its maximal ideal and let $\ast$ be the involution that $A$ inherits from $\mathcal{O}$. We consider various unitary groups ${\mathcal U}_m(A)$ofrank$m$over$A$,dependingonthenatureof$*$ and the equivalence type of the underlying hermitian or skew hermitian form. Each group ${\mathcal{U}}_m(A)$ gives rise to a Weil representation. In this paper, we give a Clifford theory description of all irreducible components of the Weil representation of ${\mathcal{U}}_m(A)$ with respect to all of its abelian congruence subgroups and a third of its nonabelian congruence subgroups.