Generalized Whittaker vectors for holomorphic and quaternionic representations

Abstract

The conjugacy class of parabolic subgroups with Heisenberg unipotent radical in a simple Lie groups over ³ not of type $C\_{n}$ contains an element defined over  for each quaternionic real form. In this paper we study the Whittaker models for quaternionic discrete series of these real forms and prove results analogous and by analogous methods to the case of simple Lie groups over  that are the automorphism groups of tube type Hermitian symmetric domain and (so-called Bessel models) for holomorphic representations. In particular we calculate the decomposition of the space of Whittaker vectors under the action of the stabilizer of the corresponding character in a Levi factor of the Heisenberg parabolic subgroup.