2009/01/28 by Bent Ørsted, Joseph A. Wolf, Orsted, Bent +1
Mathematics · #22E30 #22E46 (Primary) #32L10 #32M10 (Secondary) #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.0901.4505
openalex publication_date 2009/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Δ+ 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. \medskip 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.