Convolution of $\chi$-orbital Measures on Complex Grassmannians

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 $\chi$-orbital measures where $\chi$ is a character of $S\left(U(p)\times U(q) \right)$. More precisely, we give sufficient conditions for the $C^{\nu}$-smoothness of the Radon Nikodym derivative $f_{ a_{1},...,a_{r}, \chi}=d\left(\mu_{a_1, \chi}\ast...\ast\mu_{a_r, \chi}\right) /d\mu_{{SU(p+q)}}$ of the convolution $\mu_{a_1, \chi}\ast...\ast\mu_{a_r, \chi}$ of some orbital measures $\mu_{a_j, \chi}$ (see the definition below) with respect to the Haar measure $\mu_{\mathsf{SU(p+q)}}$ of $SU(p+q)$.