### Abstract

In [2] and [1], the regularity of the Radon-Nikodym derivative of the convolutions of orbital measures on a compact symmetric space of rank one was studied. The aim of this paper is to extend the results obtained in [1] to the case of complex Grassmannians. More precisely, let M = U/K , where U = SU(p + q) and K = S(U(p)× U(q)), be the complex Grassmannian of a p-plane in Cp+^{q} , p ? q ? 2, a_{1}, ..., a_{r} be r points in U , and consider the convolution product ?_{a} _{1} ? ... ? ?_{ar} of the orbital measures ?_{a}1 , ..., ?_{ar} supported on Ka_{1}K, ..., Ka_{r}K . By a result of Ragozin [10], if r ? dim M, then ?_{a} _{1} ? ... ? ?_{ar} is absolutely continuous with respect to the Haar measure of U . The aim of this paper is to investigate the C^{k}?regularity of the Radon-Nikodym derivative of ?_{a} _{1} ? ... ? ?_{ar} with respect to the Haar measure of U . Mathematics Subject Classification 2010: Primary 43A77, 43A90; Secondary 53C35, 28C10.

Original language | English |
---|---|

Pages (from-to) | 695-713 |

Number of pages | 19 |

Journal | Journal of Lie Theory |

Volume | 21 |

Issue number | 3 |

Publication status | Published - Jan 1 2018 |

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### Keywords

- Convolution of orbital measures
- Grassmannians
- Radon-Nikodym derivative
- Spherical functions

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Lie Theory*,

*21*(3), 695-713.

**Convolution of orbital measures on complex grassmannians.** / Al-Hashami, Mahmoud; Anchouche, Boudjemâa.

Research output: Contribution to journal › Article

*Journal of Lie Theory*, vol. 21, no. 3, pp. 695-713.

}

TY - JOUR

T1 - Convolution of orbital measures on complex grassmannians

AU - Al-Hashami, Mahmoud

AU - Anchouche, Boudjemâa

PY - 2018/1/1

Y1 - 2018/1/1

N2 - In [2] and [1], the regularity of the Radon-Nikodym derivative of the convolutions of orbital measures on a compact symmetric space of rank one was studied. The aim of this paper is to extend the results obtained in [1] to the case of complex Grassmannians. More precisely, let M = U/K , where U = SU(p + q) and K = S(U(p)× U(q)), be the complex Grassmannian of a p-plane in Cp+q , p ? q ? 2, a1, ..., ar be r points in U , and consider the convolution product ?a 1 ? ... ? ?ar of the orbital measures ?a1 , ..., ?ar supported on Ka1K, ..., KarK . By a result of Ragozin [10], if r ? dim M, then ?a 1 ? ... ? ?ar is absolutely continuous with respect to the Haar measure of U . The aim of this paper is to investigate the Ck?regularity of the Radon-Nikodym derivative of ?a 1 ? ... ? ?ar with respect to the Haar measure of U . Mathematics Subject Classification 2010: Primary 43A77, 43A90; Secondary 53C35, 28C10.

AB - In [2] and [1], the regularity of the Radon-Nikodym derivative of the convolutions of orbital measures on a compact symmetric space of rank one was studied. The aim of this paper is to extend the results obtained in [1] to the case of complex Grassmannians. More precisely, let M = U/K , where U = SU(p + q) and K = S(U(p)× U(q)), be the complex Grassmannian of a p-plane in Cp+q , p ? q ? 2, a1, ..., ar be r points in U , and consider the convolution product ?a 1 ? ... ? ?ar of the orbital measures ?a1 , ..., ?ar supported on Ka1K, ..., KarK . By a result of Ragozin [10], if r ? dim M, then ?a 1 ? ... ? ?ar is absolutely continuous with respect to the Haar measure of U . The aim of this paper is to investigate the Ck?regularity of the Radon-Nikodym derivative of ?a 1 ? ... ? ?ar with respect to the Haar measure of U . Mathematics Subject Classification 2010: Primary 43A77, 43A90; Secondary 53C35, 28C10.

KW - Convolution of orbital measures

KW - Grassmannians

KW - Radon-Nikodym derivative

KW - Spherical functions

UR - http://www.scopus.com/inward/record.url?scp=85043977027&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85043977027&partnerID=8YFLogxK

M3 - Article

VL - 21

SP - 695

EP - 713

JO - Journal of Lie Theory

JF - Journal of Lie Theory

SN - 0949-5932

IS - 3

ER -