Wladimir Popov: Generically multiple transitive algebraic group actions

Abstract: With every nontrivial connected lalgebraic group G we
associate a positive integer gtd(G), called the generic transitivity
degree of G, that is 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 the product of n copies of X admits an
open orbit. We find gtd(G) for all reductive G and estimate gtd(G) for
nonreductive 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. For every reductive G and its proper maximal
parabolic subgroup P, we find the maximal r such that the diagonal
action of G on the product of r copies of G/P admits an open
G-orbit. As the applications, we obtain upper bounds for the
multiplicities of trivial components in some tensor product
decompositions and we classify all the pairs (G,P) such that the
action of G on the product of 3 copies of G/P admits an open orbit,
answering a question of M. Burger.