Commutative Nilpotent Closed Algebras and Weil Representations

Let 𝔽 be the field GF(q 2) of q 2 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 End𝔽 V 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 End𝔽 V 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.