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

Abstract

Let $p$ and $q$ be integers such that $p\ge q\ge 1$ 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 ${\mu }_{\mathsf{SU}(\mathsf{p}+\mathsf{q})}$ of $SU(p+q)$.