Commutative nilpotent closed algebras and weil representations

Abstract

Let F be the field GF(q2) of q2 elements, q odd, and let V be an F-vector space endowed with a nonsingular Hermitian form φ. Let σ be the adjoint involutory antiautomorphism of EndFV associated to the form, and let U(φ) be the corresponding unitary group. We ask whether the restrictions of the Weil representation of U(φ) to certain subgroups are multiplicity-free. These subgroups consist of the members of U(φ) in subalgebras of the form FI + N, where N is a σ-stable commutative nilpotent subalgebra of EndFV with the further property that N contains its annihilator. We give a necessary condition for multiplicity-freeness that depends on the dimensions of N and that annihilator. Moreover, the case that N is conjugate to its regular representation is completely settled. Several other classes of subalgebra are discussed in detail. © Taylor & Francis Group, LLC.