Convolution of $\chi$-orbital measures on complex Grassmannians

Let {\scriptsize $SU(p+q)/S(U(p)\times U(q))$} be the Grassmannian of complex $p$-dimensional subspaces of $\mathbb{C}^{p+q}$, where $p$ and $q$ are integers such that $p\geq q \geq 2$. The aim of this paper is to extend the main result in~\cite{anchouche2},~\cite{Alhashami} to the case of convolution of $\chi$-orbital measures where $\chi$ is a character of $S(U(p)\times U(q))$. More precisely, we give sufficient conditions for the $\C^{\nu}$-smoothness of the Radon Nikodym derivative $$f_{ a_{1},...,a_{r}, \chi}=d(\mu_{a_1, \chi}*...*\mu_{a_r, \chi}) /d\mu_{{SU(p+q)}}$$ of the convolution $\mu_{a_1, \chi}*...*\mu_{a_r, \chi}$ with respect to the Haar measure $\mu_{SU(p+q)}$ of $SU(p+q)$, where $\mu_{a_j, \chi}$ are some orbital measures defined below.