Journal of Lie Theory

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 are several distinguished positive root systems ⁺ 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.