### Abstract

We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces G/K of Cartan type III, where G is a non-compact, connected Lie group and K is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, KzK, in G. The characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements z. A consequence of our work is to show that the convolution product of any rank G/K, continuous, K-bi-invariant measures is absolutely continuous in any of these symmetric spaces, other than those whose restricted root system is type A_{n} or D_{3}, when rank G/K +1 is needed.

Original language | English |
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Pages (from-to) | 81-111 |

Number of pages | 31 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 450 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jun 1 2017 |

### Keywords

- Absolute continuity
- Double coset
- Orbital measure
- Symmetric space

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics