COMMUTATIVE SUBALGEBRAS AND SPHERICITY IN ${\mathbb Z}_2$-GRADED LIE ALGEBRAS

Abstract

Let $\mathfrak{g}=\mathfrak{g}_0\oplus\mathfrak{g}_1$ be a $\mathbb Z_2$-graded simple Lie algebra. Fix a Borel subalgebra $\mathfrak b_0\subset\mathfrak{g}_0$. Let $\mathfrak a\subset\mathfrak{g}_1$ be a $\mathfrak b_0$-stable subalgebra. Then $\mathfrak a$ is automatically commutative. It is known that if $\overline{G{\cdot}\mathfrak a}$ is a spherical $G$-variety, then $G_0{\cdot}\mathfrak a$ is a spherical $G_0$-variety. We describe all $\mathbb Z_2$-gradings having the property that $\overline{G{\cdot}\mathfrak a}$ is a spherical $G$-variety for any $\mathfrak a$.