MULTITRANSITIVITY OF CALOGERO-MOSER SPACES

Abstract

Let G be the group of unimodular automorphisms of a free associative ℂ-algebra on two generators. A theorem of G. Wilson and the first author [BW] asserts that the natural action of G on the Calogero-Moser spaces C _n is transitive for all n ϵ ℕ. We extend this result in two ways: first, we prove that the action of G on C _n is doubly transitive, meaning that G acts transitively on the configuration space of ordered pairs of distinct points in C _n ; second, we prove that the diagonal action of G on ${C}_{n_1}\times {C}_{n_2}\times \cdots \times {C}_{n_m}$ is transitive provided n ₁, n ₂, …, n _m are pairwise distinct numbers.