Borel-de Siebenthal Positive Root Systems

TL;DR

This paper classifies all Borel-de Siebenthal positive root systems for certain Lie groups and applies this classification to determine the number of unitary equivalence classes of specific discrete series representations.

Contribution

It explicitly determines all Borel-de Siebenthal positive root systems for the given Lie algebra setting, assuming their existence.

Findings

All Borel-de Siebenthal positive root systems are classified.

Number of unitary equivalence classes of certain discrete series representations is determined.

Application to non-Hermitian symmetric spaces is provided.

Abstract

Let $G$ be a connected simple Lie group with finite centre, $K$ be a maximal compact subgroup of $G,$ and rank$(G)=$ rank$(K).$ Let $\frak{g}_0=$Lie$(G), \frak{k}_0=$Lie$(K) \subset \frak{g}_0, \frak{t}_0$ be a maximal abelian subalgebra of $\frak{k}_0, \frak{g}=\frak{g}_0^\mathbb{C}, \frak{k}=\frak{k}_0^\mathbb{C},$ and $\frak{h}=\frak{t}_0^\mathbb{C}.$ The existence of a Borel-de Siebenthal positive root system of $\Delta(\frak{g}, \frak{h})$ is proved by Borel and de Siebenthal. In this article, we have determined all Borel-de Siebenthal positive root systems of $\Delta(\frak{g}, \frak{h}),$ assuming the existence. As an application, we have determined the number of unitary equivalence classes of all Borel-de Siebenthal discrete series representations of $G$ (if $G/K$ is not Hermitian symmetric) with a fixed infinitesimal character.