Convolution of orbital measures in symmetric spaces

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3 Citations (Scopus)

Abstract

Let G/K be a noncompact symmetric space, Gc/K its compact dual, g = t ⊕ p the Cartan decomposition of the Lie algebra g of G, a a maximal abelian subspace of a, H be an element of a, a=exp (H) , and ac =exp (iH). In this paper, we prove that if for some positive integer r, v acr is absolutely continuous with respect to the Haar measure on Gc, then vra is absolutely continuous with respect to the left Haar measure on G, where ac (respectively a) is the K-bi-invariant orbital measure supported on the double coset KacK (respectively KaK). We also generalize a result of Gupta and Hare ['Singular dichotomy for orbital measures on complex groups', Boll.Unione Mat.Ital.(9)III(2010), 409-419] to general noncompact symmetric spaces and transfer many of their results from compact symmetric spaces to their dual noncompact symmetric spaces.

Original languageEnglish
Pages (from-to)470-485
Number of pages16
JournalBulletin of the Australian Mathematical Society
Volume83
Issue number3
DOIs
Publication statusPublished - Jun 2011
Externally publishedYes

Keywords

  • convolution
  • double coset
  • orbital measures
  • symmetric spaces

ASJC Scopus subject areas

  • Mathematics(all)

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