## Tohoku Mathematical Journal 2000 March SECOND SERIES VOL. 52, NO. 1

 Tohoku Math. J. 52 (2000), 127-152

Title ON $\theta$-STABLE BOREL SUBALGEBRAS OF LARGE TYPE FOR REAL REDUCTIVE GROUPS

Author Takuya Ohta

(Received September 17, 1998, revised May 13, 1999)
Abstract. Vogan-Zuckerman's standard representation $X$ for a real reductive group $G(\boldsymbol{R})$ is constructed from a $\theta$-stable parabolic subalgebra $\mathfrak{q}$ of the complexified Lie algebra $\mathfrak{g}$ of $G(\boldsymbol{R})$. 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.

1991 Mathematics Subject Classification. Primary 17B45; Secondary 22E45, 22E46, 22E47.