Abstract
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
{*}
be the involution that A inherits from
{\mathcal{O}}
.
We consider various unitary groups
{\mathcal{U}_{m}(A)}
of rank m over A, depending on the nature of
{*}
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.
Exponential sums and total Weil representations of finite symplectic and unitary groups