Convolution of orbital measures in symmetric spaces

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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 ac r is absolutely continuous with respect to the Haar measure on Gc, then vr a 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

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Keywords

  • convolution
  • double coset
  • orbital measures
  • symmetric spaces

ASJC Scopus subject areas

  • Mathematics(all)

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