The smoothness of convolutions of orbital measures on complex Grassmannian symmetric spaces

It is well known that if $G\mathrm{/}K$ is any irreducible symmetric space and $\mu _{a}$isacontinuousorbitalmeasuresupportedonthedoublecoset$KaK,$ then the convolution product, ${\mu }_a^k,$ is absolutely continuous for some suitably large $k\le \dim G\mathrm{/}K$. The minimal value of $k$ is known in some symmetric spaces and in the special case of groups or rank one symmetric spaces it has even been shown that ${\mu }_a^k$ belongs to the smaller space $L^2$ for some $k$. Here we prove that this $L^2$ property holds for all the compact, complex Grassmanian symmetric spaces, $$. Moreover, for the orbital measures at a dense set of points $a$, we prove that ${\mu }_a^2\in L^2$ (or ${\mu }_a^3\in L^2$ if $p=q$).