Irreducible representations of unipotent subgroups of symplectic and unitary groups defined over rings

Abstract

Abstract. Let A be a ring with 1 ¤ 0, not necessarily finite, endowed with an involution , that is, an anti-automorphism of order Ä 2. Let H n .A/ be the additive group of all n n hermitian matrices over A relative to . Let U n .A/ be the subgroup of GL n .A/ of all upper triangular matrices with ones along the main diagonal. Let P D H n .A/ Ì U n .A/, where U n .A/ acts on H n .A/ by -congruence transformations. We may view P as a unipotent subgroup of either a symplectic group Sp 2n .A/, if D 1 A (in which case A is commutative), or a unitary group U 2n .A/ if ¤ 1 A . In this paper we construct and classify a family of irreducible representations of P over a field F that is essentially arbitrary. In particular, when A is finite and F D C, we obtain irreducible representations of P of the highest possible degree.