On θ-stable Borel subalgebras of large type for real reductive groups

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概要

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• Vogan-Zuckerman's standard representation $X$ for a real reductive group $G(\textbf{\textit R})$ is constructed from a θ-stable parabolic subalgebra $\mathfrak{q}$ of the complexified Lie algebra $\mathfrak{g}$ of $G(\textbf{\textit{R}})$. Adams and Vogan showed that the set of $\mathfrak{g}$-principal $K$-orbits in the associated variety $Ass(X)$ of $X$ is in one-to-one correspondence with the set $\mathcal{B} {\mathfrak{g}^-}^L/K$ of $K$-conjugacy classes of θ-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.

• 東北大学の論文

著者

• Ohta Takuya

  Department Of Mathematics Tokyo Denki University

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