### 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 |
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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.