Convolution of $χ$-orbital Measures on Complex Grassmannians

Abstract

Let $p$ and $q$ be integers such that $p\geq q \geq 1$ and let\\ $SU(p+q)/ S\left(U(p)\times U(q) \right) $ be the corresponding complex Grassmannian. The aim of this paper is to extend the main result in \cite{anchouche1}, \cite{Alhashami} to the case of convolution of $χ$-orbital measures where $χ$ is a character of $S\left(U(p)\times U(q) \right) $. More precisely, we give sufficient conditions for the $C^ν$-smoothness of the Radon Nikodym derivative $f_{ a_{1},...,a_{r}, χ}=d\left(μ_{a_1, χ}\ast...\astμ_{a_r, χ}\right) /dμ_{SU(p+q)}$ of the convolution $μ_{a_1, χ}\ast...\astμ_{a_r, χ}$ of some orbital measures $μ_{a_j, χ}$ (see the definition below) with respect to the Haar measure $μ_{\mathsf{SU(p+q)}}$ of $SU(p+q)$.