Generically multiple transitive algebraic group actions

Abstract

With every nontrivial connected algebraic group $G$ we associate a positive integer $\mathrm{gtd}(G)$ called the generic transitivity degree of $G$ and equal to the maximal $n$ such that there is a nontrivial action of $G$ on an irreducible algebraic variety $X$ for which the diagonal action of $G$ on $X^n$ admits an open orbit. We show that $\mathrm{gtd}(G)⩽2$ (respectively, $\mathrm{gtd}(G)=1$) for all solvable (respectively, nilpotent) $G$, and we calculate $\mathrm{gtd}(G)$ for all reductive $G$. We prove that if $G$ is nonabelian reductive, then the above maximal $n$ is attained for $X=G/P$ where $P$ is a proper maximal parabolic subgroup of $G$ (but not only for such homogeneous spaces of $G$). For every reductive $G$ and its proper maximal parabolic subgroup $P$, we find the maximal $r$ such that the diagonal action of $G$ (respectively, a Levi subgroup $L$ of ~$P$) on $(G/P{)}^r$ admits an open $G$-orbit (respectively, $L$-orbit). As an application, we obtain upper bounds for the multiplicities of trivial components in some tensor product decompositions. As another application, we classify all the pairs $(G,P)$ such that the action of $G$ on $(G/P{)}^3$ admits an open orbit, answering a question of {\sc M. Burger}.

Revised version, 30 pages

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