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 phi. Let sigma be the adjoint involutory antiautomorphism of End(F) V associated to the form, and let U(phi) be the corresponding unitary group. We ask whether the restrictions of the Weil representation of U(phi) to certain subgroups are multiplicity-free. These subgroups consist of the members of U(phi) in subalgebras of the form F I + N, where N is a sigma-stable commutative nilpotent subalgebra of End(F) 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.