### 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 |

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### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**The absolute continuity of convolutions of orbital measures in symmetric spaces.** / Gupta, Sanjiv Kumar; Hare, Kathryn E.

Research output: Contribution to journal › Article

*Journal of Mathematical Analysis and Applications*, vol. 450, no. 1, pp. 81-111. https://doi.org/10.1016/j.jmaa.2017.01.027

}

TY - JOUR

T1 - The absolute continuity of convolutions of orbital measures in symmetric spaces

AU - Gupta, Sanjiv Kumar

AU - Hare, Kathryn E.

PY - 2017/6/1

Y1 - 2017/6/1

N2 - 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 An or D3, when rank G/K +1 is needed.

AB - 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 An or D3, when rank G/K +1 is needed.

KW - Absolute continuity

KW - Double coset

KW - Orbital measure

KW - Symmetric space

UR - http://www.scopus.com/inward/record.url?scp=85009754101&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85009754101&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2017.01.027

DO - 10.1016/j.jmaa.2017.01.027

M3 - Article

VL - 450

SP - 81

EP - 111

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -