Orbit method and nondegenerate series

Abstract

1. If G is a reductive Lie group, then its Plancherel formula ([1], [2], [8]) involves a of representations for each conjugacy class of Cartan subgroups. These series are realized [8] by the action of G on square integrable cohomology of partially holomorphic vector bundles over certain G-orbits on complex flag manifolds. That is similar to their realization by the Kostant-Kirillov orbit method using semisimple orbits. The differences occur when G has noncommutative Cartan subgroups, and also for representations with singular infinitesimal character, i.e. when the semisimple orbit is not regular. Recently Wakimoto [6] used possibly-nonsemisimple orbits to realize the principal series, which is the for a maximally noncompact Cartan subgroup H, when G is a connected semisimple group and H is commutative (e.g. when G is linear). Here we use our method [8] to extend Wakimoto's procedure and realize all but a few members of every nondegenerate of unitary representation classes for a reductive group. In the case of regular infinitesimal character there is no essential change from [8]. But in the case of singular infinitesimal character we rely on results of Ozeki and Wakimoto ([4], [6]), using nonsemisimple orbits in an interesting way. To avoid repetition we assume some acquaintance with [8].