Geometry of the Borel-de Siebenthal Discrete Series

Bent Ørsted¹ ; Joseph A. Wolf²

¹Department of Mathematics, Aarhus University, 8000 Aarhus C, Denmark
²Department of Mathematics, University of California, Berkeley, CA 94720--3840, U.S.A.

Journal of Lie Theory, Tome 20 (2010) no. 1, pp. 175-212

Résumé

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.

Détail

Classification : 22E46, 22E30, 32L10, 32M10 <!--
Mots-clés : Discrete series, cohomology, compact subvarieties, relative invariants