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 language | English |
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Pages (from-to) | 470-485 |
Number of pages | 16 |
Journal | Bulletin of the Australian Mathematical Society |
Volume | 83 |
Issue number | 3 |
DOIs | |
Publication status | Published - Jun 2011 |
Keywords
- convolution
- double coset
- orbital measures
- symmetric spaces
ASJC Scopus subject areas
- Mathematics(all)