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Characteristic functions of semigroups in semi-simple Lie groups

  • Luiz A. B. San Martin EMAIL logo and Laercio J. Santos
Published/Copyright: March 21, 2019
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 4

Abstract

Let G be a noncompact semi-simple Lie group with Iwasawa decomposition G=K⁢A⁢N. For a semigroup S⊂G 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, u∈K, λ∈𝔞∗, 𝔞 is the Lie algebra of A and g⁢u=k⁢e𝖺⁢(g,u)⁢n∈K⁢A⁢N. 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 u∈π-1⁢(𝒟Θ⁢(S)⁢(S)), where 𝒟Θ⁢(S)⁢(S)⊂𝔽Θ⁢(S) is a B-convex set and π:K→𝔽Θ⁢(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 S∩S-1 of units of S.

Keywords: Semi-simple Lie groups; Helgason–Laplace transform; semigroups; flag manifolds, parametric statistical model
MSC 2010: 22E46; 22F30; 22E30

Communicated by Karl-Hermann Neeb

Funding statement: 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.

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Received: 2018-10-08
Revised: 2019-02-07
Published Online: 2019-03-21
Published in Print: 2019-07-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Characteristic functions of semigroups in semi-simple Lie groups
  3. Damping estimates for oscillatory integral operators with real-analytic phases and its applications
  4. Formality properties of finitely generated groups and Lie algebras
  5. Toric actions in cosymplectic geometry
  6. On classification of typical representations for GL3(F)
  7. Integrable generators of Lie algebras of vector fields on ℂn
  8. Expanding phenomena over matrix rings
  9. On sup-norm bounds part II: GL(2) Eisenstein series
  10. Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces
  11. Partial and full boundary regularity for non-autonomous functionals with Φ-growth conditions
  12. Radial averaging operator acting on Bergman and Lebesgue spaces
  13. The 𝑝-adic variation of the Gross–Kohnen–Zagier theorem
https://doi.org/10.1515/forum-2018-0243
Keywords for this article
Semi-simple Lie groups; Helgason–Laplace transform; semigroups; flag manifolds, parametric statistical model

Articles in the same Issue

  1. Frontmatter
  2. Characteristic functions of semigroups in semi-simple Lie groups
  3. Damping estimates for oscillatory integral operators with real-analytic phases and its applications
  4. Formality properties of finitely generated groups and Lie algebras
  5. Toric actions in cosymplectic geometry
  6. On classification of typical representations for GL3(F)
  7. Integrable generators of Lie algebras of vector fields on ℂn
  8. Expanding phenomena over matrix rings
  9. On sup-norm bounds part II: GL(2) Eisenstein series
  10. Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces
  11. Partial and full boundary regularity for non-autonomous functionals with Φ-growth conditions
  12. Radial averaging operator acting on Bergman and Lebesgue spaces
  13. The 𝑝-adic variation of the Gross–Kohnen–Zagier theorem
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