### Abstract

Let G/K be a noncompact symmetric space, G_{c}/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 a_{c} =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 G_{c}, then v^{r}
_{a} is absolutely continuous with respect to the left Haar measure on G, where a_{c} (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)