Characteristic functions of semigroups in semi-simple Lie groups

Author(s)

San Martin, Luiz A. B. ^[1] ; Santos, Laercio J. ^[2]

Affiliation

^[1] Univ Estadual Campinas, Inst Math, Campinas, SP - Brazil

^[2] Univ Fed Juiz de Fora, Dept Math, Juiz De Fora, MG - Brazil

• Document type: Journal article
• Source: FORUM MATHEMATICUM; v. 31, n. 4, p. 815-842, JUL 2019.
• Web of Science Citations: 0

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. (AU)

• FAPESP's process: 12/18780-0 - Geometry of control systems, dynamical and stochastics systems
• Grantee: Marco Antônio Teixeira
• Support Opportunities: Research Projects - Thematic Grants