On $\theta$-stable Borel subalgebras of large type for real reductive groups

Abstract

Vogan-Zuckerman's standard representation $X$ for a real reductive group $G(R)$ is constructed from a $\theta$-stable parabolic subalgebra $\mathfrak{q}$ of the complexified Lie algebra $\mathfrak{g}$ of $G(R)$. Adams and Vogan showed that the set of $\mathfrak{g}$-principal $K$-orbits in the associated variety $\mathrm{Ass}(X)$ of $X$ is in one-to-one correspondence with the set $\mathcal{B}_{\mathfrak{g}^-}^L/K$ of $K$-conjugacy classes of $\theta$-stable Borel subalgebras of large type having representatives in the opposite parabolic subalgebra $\mathfrak{q}^-$ of $\mathfrak{q}$. In this paper, we give a description of $\mathcal{B}_{\mathfrak{q}}^L/K$ and show that $\mathcal{B}_{\mathfrak{q}}^L/K\ne\emptyset$ under certain condition on the positive system of imaginary roots contained in $\mathfrak{q}$. Furthermore, we construct a finite group which acts on $\mathcal{B}_{\mathfrak{q}}^L/K$ transitively.