The F-valued points of the algebra ofstrongly regular functions of a Kac-Moody group

Abstract

Let G m , resp. G f , be the minimal, resp. formal, Kac-Moody group, associated to a symmetrizable generalized Cartan matrix, over a field F of characteristic 0. Let F[G m ] be the algebra of strongly regular functions on G m . We denote by G m , resp. G f , certain monoid completions of G m , resp. G f , built by using the faces of the Tits cone. We show that there is an action of G f x G f on the spectrum of F-valued points of F[G m ]. As a G f x Gf-set it can be identified with a certain quotient of the G f x G f -set G f x G f , built by using G m . We prove a Birkhoff decomposition for the F-valued points of F[G m ]. We describe the stratification of the spectrum of F-valued points of F[G m ] in G f x G f -orbits. We show that every orbit can be covered by suitably defined big cells.