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 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)