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Geometry of the Borel - de Siebenthal Discrete Series

By Bent Ørsted and Joseph A. Wolf
No. 01, February 2009

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 is a distinguished positive root system $\Delta^+$ for which there is a unique noncompact simple root $\nu$, the Borel - de Siebenthal system. There is a lot of fascinating geometry associated to the corresponding Borel - de 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 Borel - de 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 $\nu$ has coefficient $1$ in the maximal root $\mu$, so we assume that the group $G_0$ is not of hermitian type, in other words that $\nu$ has coefficient 2 in $\mu$.

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 $\mu$ is orthogonal to the compact simple roots and the inducing representation is 1-dimensional.

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