Characteristic functions of semigroups in semi-simple Lie groups

Abstract

Let G be a noncompact semi-simple Lie group with Iwasawa decomposition G = KAN. For a semigroup S subset of G with nonempty interior we find a domain of convergence of the Helgason - Laplace transform I-s(lambda, u) = integral(S)e(lambda(a(g,u))) dg, where dg is the Haar measure of G, u is an element of K, lambda is an element of a*, a is the Lie algebra of A and gu = ke(a(g,u))n is an element of KAN. The domain is given in terms of a flag manifold of G written F-Theta(s) called the flag type of S, where Theta(S) is a subset of the simple system of roots. It is proved that I-S(lambda,u) < infinity if lambda belongs to a convex cone defined from Theta(S) and u is an element of pi(-1)(D-Theta(s)(S)), where D-Theta(s)(S) subset of F-Theta(s) is a B-convex set and pi : K -> F-Theta(s) is the natural projection. We prove differentiability of 1 5 (A, u) and apply the results to construct of a Riemannian metric in D-Theta(s)(S) invariant by the group S boolean AND S(-1 )of units of S