Generically multiple transitive algebraic group actions
With every nontrivial connected algebraic group G we associate a positive integer 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 Xn admits an open orbit. We show that gtd(G) ≤2 (respectively, gtd(G) = 1) for all solvable (respectively, nilpotent) G, and we calculate 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 M. Burger.