Clifford theory of Weil representations of unitary groups

Abstract

Abstract Let 𝒪 {\mathcal{O}} be an involutive discrete valuation ring with residue field of characteristic not 2. Let A be a quotient of 𝒪 {\mathcal{O}} by a nonzero power of its maximal ideal, and let * {*} be the involution that A inherits from 𝒪 {\mathcal{O}} . We consider various unitary groups 𝒰 m ⁢ ( A ) {\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 𝒰 m ⁢ ( A ) {\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 𝒰 m ⁢ ( A ) {\mathcal{U}_{m}(A)} with respect to all of its abelian congruence subgroups and a third of its nonabelian congruence subgroups.