THE PLANCHEREL FORMULA FOR COUNTABLE GROUPS

International audienceWe discuss a Plancherel formula for countable groups, which provides a canonical decomposition of the regular representation of such a group Γ into a direct integral of factor representations. Our main result gives a precise description of this decomposition in terms of the Plancherel formula of the FC-center of Γ (that is, the normal sugbroup of Γ consisting of elements with a finite conjugacy class); this description involves the action of an appropriate totally disconnected compact group of automorphisms of the FC-center. As an application, we determine the Plancherel formula for linear groups. In an appendix, we use the Plancherel formula to provide a unified proof for Thoma's and Kaniuth's theorems which respectively characterize countable groups which are of type I and those whose regular representation is of type II