MULTITRANSITIVITY OF CALOGERO-MOSER SPACES

Let G be the group of unimodular automorphisms of a free associative ?-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 ? ?. 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.