Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 31 (2021), No. 2, 335--349Copyright Heldermann Verlag 2021 The Smoothness of Convolutions of Singular Orbital Measures on Complex Grassmannians Sanjiv Kumar Gupta Dept. of Mathematics, Sultan Qaboos University, Sultanate of Oman gupta@squ.edu.om Kathryn E. Hare Dept. of Pure Mathematics, University of Waterloo, Canada kehare@uwaterloo.ca [Abstract-pdf] It is well known that if $G/K$ is any irreducible symmetric space and $\mu_{a}$ is a continuous orbital measure supported on the double coset $KaK$, then the convolution product, $\mu _{a}^{k},$ is absolutely continuous for some suitably large number $k\leq \dim G/K$. The minimal value of $k$ is known in some symmetric spaces and in the special case of compact groups or rank one compact 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 Grassmannian symmetric spaces, $SU(p+q)/S(U(p)\times U(q))$. 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 $p=q$). Keywords: Orbital measure, spherical functions, complex Grassmannian symmetric space, absolute continuity. MSC: 43A90, 43A85; 58C35, 33C50. [ Fulltext-pdf  (155  KB)] for subscribers only.