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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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1435-5337
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Volume 31, Issue 4

# Characteristic functions of semigroups in semi-simple Lie groups

Luiz A. B. San Martin
/ Laercio J. Santos
Published Online: 2019-03-21 | DOI: https://doi.org/10.1515/forum-2018-0243

## Abstract

Let G be a noncompact semi-simple Lie group with Iwasawa decomposition $G=KAN$. For a semigroup $S\subset G$ with nonempty interior we find a domain of convergence of the Helgason–Laplace transform ${I}_{S}\left(\lambda ,u\right)={\int }_{S}{e}^{\lambda \left(𝖺\left(g,u\right)\right)}𝑑g$, where dg is the Haar measure of G, $u\in K$, $\lambda \in {𝔞}^{\ast }$, $𝔞$ is the Lie algebra of A and $gu=k{e}^{𝖺\left(g,u\right)}n\in KAN$. The domain is given in terms of a flag manifold of G written ${𝔽}_{\mathrm{\Theta }\left(S\right)}$ called the flag type of S, where $\mathrm{\Theta }\left(S\right)$ is a subset of the simple system of roots. It is proved that ${I}_{S}\left(\lambda ,u\right)<\mathrm{\infty }$ if λ belongs to a convex cone defined from $\mathrm{\Theta }\left(S\right)$ and $u\in {\pi }^{-1}\left({\mathcal{𝒟}}_{\mathrm{\Theta }\left(S\right)}\left(S\right)\right)$, where ${\mathcal{𝒟}}_{\mathrm{\Theta }\left(S\right)}\left(S\right)\subset {𝔽}_{\mathrm{\Theta }\left(S\right)}$ is a B-convex set and $\pi :K\to {𝔽}_{\mathrm{\Theta }\left(S\right)}$ is the natural projection. We prove differentiability of ${I}_{S}\left(\lambda ,u\right)$ and apply the results to construct of a Riemannian metric in ${\mathcal{𝒟}}_{\mathrm{\Theta }\left(S\right)}\left(S\right)$ invariant by the group $S\cap {S}^{-1}$ of units of S.

MSC 2010: 22E46; 22F30; 22E30

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Revised: 2019-02-07

Published Online: 2019-03-21

Published in Print: 2019-07-01

The first author was supported by CNPq grant no. 303755/09-1, FAPESP grant no. 2012/18780-0 and CNPq/Universal grant no 476024/2012-9.

Citation Information: Forum Mathematicum, Volume 31, Issue 4, Pages 815–842, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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