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Forum Mathematicum

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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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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


Let G be a noncompact semi-simple Lie group with Iwasawa decomposition G=KAN. For a semigroup SG with nonempty interior we find a domain of convergence of the Helgason–Laplace transform IS(λ,u)=Seλ(𝖺(g,u))𝑑g, where dg is the Haar measure of G, uK, λ𝔞, 𝔞 is the Lie algebra of A and gu=ke𝖺(g,u)nKAN. The domain is given in terms of a flag manifold of G written 𝔽Θ(S) called the flag type of S, where Θ(S) is a subset of the simple system of roots. It is proved that IS(λ,u)< if λ belongs to a convex cone defined from Θ(S) and uπ-1(𝒟Θ(S)(S)), where 𝒟Θ(S)(S)𝔽Θ(S) is a B-convex set and π:K𝔽Θ(S) is the natural projection. We prove differentiability of IS(λ,u) and apply the results to construct of a Riemannian metric in 𝒟Θ(S)(S) invariant by the group SS-1 of units of S.

Keywords: Semi-simple Lie groups; Helgason–Laplace transform; semigroups; flag manifolds, parametric statistical model

MSC 2010: 22E46; 22F30; 22E30


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About the article

Received: 2018-10-08

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, DOI: https://doi.org/10.1515/forum-2018-0243.

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