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$(K)\mathrm{.}$ Let ${\mathfrak{g}}_0=$Lie$(G),{\mathfrak{k}}_0=$Lie$(K)\subset {\mathfrak{g}}_0,{\mathfrak{t}}_0$ be a maximal abelian subalgebra of ${\mathfrak{k}}_0,\mathfrak{g}={\mathfrak{g}}_0^{\mathbb{C}},\mathfrak{k}={\mathfrak{k}}_0^{\mathbb{C}},$ and $\mathfrak{h}={\mathfrak{t}}_0^{\mathbb{C}}\mathrm{.}$ The existence of a Borel-de Siebenthal positive root system of $\mathrm{\Delta }(\mathfrak{g},\mathfrak{h})$ is proved by Borel and de Siebenthal. In this article, we have determined all Borel-de Siebenthal positive root systems of $\mathrm{\Delta }(\mathfrak{g},\mathfrak{h}),$ 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 $G\mathrm{/}K$ is not Hermitian symmetric) with a fixed infinitesimal character.