### 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 |
---|---|

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 |

### Fingerprint

### Keywords

- convolution
- double coset
- orbital measures
- symmetric spaces

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Convolution of orbital measures in symmetric spaces.** / Anchouche, Boudjemǎa; Gupta, Sanjiv Kumar.

Research output: Contribution to journal › Article

*Bulletin of the Australian Mathematical Society*, vol. 83, no. 3, pp. 470-485. https://doi.org/10.1017/S0004972710002017

}

TY - JOUR

T1 - Convolution of orbital measures in symmetric spaces

AU - Anchouche, Boudjemǎa

AU - Gupta, Sanjiv Kumar

PY - 2011/6

Y1 - 2011/6

N2 - 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.

AB - 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.

KW - convolution

KW - double coset

KW - orbital measures

KW - symmetric spaces

UR - http://www.scopus.com/inward/record.url?scp=79956151490&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79956151490&partnerID=8YFLogxK

U2 - 10.1017/S0004972710002017

DO - 10.1017/S0004972710002017

M3 - Article

VL - 83

SP - 470

EP - 485

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

SN - 0004-9727

IS - 3

ER -