Borel-de Siebenthal Positive Root Systems

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.