Closed orbits of Borel subgroups

V. L. Popov

Sbornik. Mathematics, Tome 63 (1989) no. 2, pp. 375-392

Résumé

The author considers an algebraic action of a connected reductive algebraic group $G$ defined over an algebraically closed field $k$ on an affine irreducible algebraic variety $X$, and studies the question of when the action of a Borel subgroup $B$ of $G$ on $X$ is stable, i.e., the $B$-orbit of any point belonging to some nonempty open subset of $X$ is closed in $X$. A criterion for stability is obtained: Suppose that $\operatorname{char}k=0$. In order that the action of $B$ on $X$ be stable it is necessary, and, if $G$ is semisimple and the group of divisor classes $\mathrm{Cl}X$ is periodic, also sufficient that $X$ contain a point with a finite $G$-stabilizer. For an action $G:V$ defined by a linear representation $G\to GL(V)$ the cases when $B:V$ is not stable and either $G$ is simple or $G$ is semisimple and the action $G:V$ is irreducible are listed. A general criterion for an orbit of a connected solvable group acting on an affine variety to be closed is also obtained, and it is used to obtain a simple sufficient condition for an orbit of such a group, acting linearly, to be closed. Bibliography: 30 titles.