
Journal of Lie Theory 20 (2010), No. 1, 175212 Copyright Heldermann Verlag 2010 Geometry of the Borelde Siebenthal Discrete Series Bent Ørsted Department of Mathematics, Aarhus University, 8000 Aarhus C, Denmark orsted@imf.au.dk Joseph A. Wolf Department of Mathematics, University of California, Berkeley, CA 947203840, U.S.A. jawolf@math.berkeley.edu Let G_{0} be a connected, simply connected real simple Lie group. Suppose that G_{0} has a compact Cartan subgroup T_{0}, so it has discrete series representations. Relative to T_{0} there are several distinguished positive root systems &Delta^{+} for which there is a unique noncompact simple root ν, the "Borelde Siebenthal system". There is a lot of fascinating geometry associated to the corresponding "Borelde Siebenthal discrete series" representations of G_{0}. In this paper we explore some of those geometric aspects and we work out the K_{0}spectra of the Borelde Siebenthal discrete series representations. This has already been carried out in detail for the case where the associated symmetric space G_{0}/K_{0} is of hermitian type, i.e. where ν has coefficient 1 in the maximal root μ, so we assume that the group G_{0} is not of hermitian type, in other words that ν has coefficient 2 in μ. Several authors have studied the case where G_{0}/K_{0} 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 1dimensional. Keywords: Discrete series, cohomology, compact subvarieties, relative invariants. MSC: 22E46; 22E30, 32L10, 32M10 