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Journal of Lie Theory 20 (2010), No. 1, 175--212
Copyright Heldermann Verlag 2010

Geometry of the Borel-de Siebenthal Discrete Series

Bent ěrsted
Department of Mathematics, Aarhus University, 8000 Aarhus C, Denmark

Joseph A. Wolf
Department of Mathematics, University of California, Berkeley, CA 94720--3840, U.S.A.

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 &Delta+ 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.

Keywords: Discrete series, cohomology, compact subvarieties, relative invariants.

MSC: 22E46; 22E30, 32L10, 32M10