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Characteristic functions of semigroups in semi-simple Lie groups by Luiz A. B. San Martin; Laercio J. Santos is a Mathematics article available to read on EtoBox.

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## Abstract Let __G__ be a noncompact semi-simple Lie group with Iwasawa decomposition G = K ⁢ A ⁢ N {G=KAN} . For a semigroup S ⊂ G {S\subset G} with nonempty interior we find a domain of convergence of the Helgason–Laplace transform I S ⁢ ( λ , u ) = ∫ S e λ ⁢ ( a ⁢ ( g , u ) ) ⁢ d g {I\_{S}(\lambda,u)=\int\_{S}e^{\lambda(\mathsf{a}(g,u))}\,dg} , where __dg__ is the Haar measure of __G__, u ∈ K {u\in K} , λ ∈ a ∗ {\lambda\in\mathfrak{a}^{\ast}} , a {\mathfrak{a}} is the Lie algebra of __A__ and g ⁢ u = k ⁢ e a ⁢ ( g , u ) ⁢ n ∈ K ⁢ A ⁢ N {gu=ke^{\mathsf{a}(g,u)}n\in KAN} . The domain is given in terms of a flag manifold of __G__ written F Θ ⁢ ( S ) {\mathbb{F}\_{\Theta(S)}} called the flag type of __S__, where Θ ⁢ ( S ) {\Theta(S)} is a subset of the simple system of roots. It is proved that I S ⁢ ( λ , u ) < ∞ {I\_{S}(\lambda,u)<\infty} if λ belongs to a convex cone defined from Θ ⁢ ( S ) {\Theta(S)} and u ∈ π - 1 ⁢ ( D Θ ⁢ ( S ) ⁢ ( S ) ) {u\in\pi^{-1}(\mathcal{D}\_{\Theta(S)}(S))} , where D Θ ⁢ ( S ) ⁢ ( S ) ⊂ F Θ ⁢ ( S ) {\mathcal{D}\_{\Theta(S)}(S)\subset\mathbb{F}\_{\Theta(S)}} is a __B__-convex set and π : K → F Θ ⁢ ( S ) {\pi:K\rightarrow\mathbb{F}\_{\Theta(S)}} is the na

Who reads Characteristic functions of semigroups in semi-simple Lie groups?

Author

Luiz A. B. San Martin; Laercio J. Santos

Publisher

Walter de Gruyter GmbH & Co. KG; Walter de Gruyter GmbH (ISSN 0933-7741)

Published

2019

Language

EN

Field

Mathematics (Physical Sciences)