ar X iv : 0 90 1 . 45 05 v 1 [ m at h . R T ] 2 8 Ja n 20 09 Geometry of the Borel – de Siebenthal Discrete Series

Let G0 be a connected, simply connected real simple Lie group. Suppose that G0 has a compact Cartan subgroup T0, so it has discrete series representations. Relative to T0 there is a distinguished positive root system ∆ for which there is a unique noncompact simple root ν, the “Borel – de Siebenthal system”. There is a lot of fascinating geometry associated to the corresponding “Borel – de Siebenthal discrete series” representations of G0. In this paper we explore some of those geometric aspects and we work out the K0–spectra of the Borel – de Siebenthal discrete series representations. This has already been carried out in detail for the case where the associated symmetric space G0/K0 is of hermitian type, i.e. where ν has coefficient 1 in the maximal root μ, so we assume that the group G0 is not of hermitian type, in other words that ν has coefficient 2 in μ. Several authors have studied the case where G0/K0 is a quaternionic symmetric space and the inducing holomorphic vector bundle is a line bundle. That is the case where μ is orthogonal to the compact simple roots and the inducing representation is 1–dimensional.