Double Coset Decompositions of Reductive Lie Groups Arising from Two Involutions

Abstract

(G)0 ⊂ H ⊂ G and (G )0 ⊂ L ⊂ G . Here G = {g ∈ G | ρ(g) = g} for an automorphism ρ of G and F0 denote the connected component of F containing the identity e for a Lie group F . In [8], we gave standard representatives of double coset decompositions H\G/L for some typical triples (G,H,L) by only using “algebraic” method. As the most elementary example, we studied the case (G,H,L) = (GL(n, IF), GL(p, IF)×GL(n−p, IF), GL(r, IF)× GL(n − r, IF)) for an arbitrary field IF. We also studied in [8] other examples related to quadratic forms on vector spaces over the fields IR,C and IH. In this paper, we assume that G is a reductive Lie group and we will describe the structure of the double coset decomposition H\G/L for an arbitrary (G,H,L). First we study the compact case (Section 3) and next the noncompact case (Section 4). In Section 3, we assume that the semisimple part Gs of G is compact. (Gs is the analytic subgroup of G for gs = [g, g].) When G = Gs and