Bass-Serre theory for groupoids and Calogero-Moser spaces

Abstract: Let C_n be the n-th Calogero-Moser space: the space of conjugacy classes of pairs of matrices (X,Y) in Mat_n(C) satisfying condition rk( XY-YX+Id_n)=1. The group G of symplectic automorphisms of C[x,y] act transitively on C_n. Let G_n denotes the stabilizer of a point in C_n under G. In this talk, I will explain how to compute G_n using the Bass-Serre theory of groupoids. We associate to each action groupoid C_n x G an " orbifold" (graph of groups) consisting of orbits of certain subgroups of G. The group G_n can be identified with the fundamental group of this orbifold. Our computation are motivated by the fact that G_n are precisely the automorphism groups of (non-isomorphic) domains Morita equivalent to the first Weyl algebra. The problem of description of these automorphism groups was originally posed by T.Stafford. (This joint work with Y.Berest and A.Eshmatov.)