Convolution of Orbital Measures on Complex Grassmannians

摘要

In [2] and [1], the regularity of the Radon-Nikodym derivative of the convolutions of orbital measures on a compact symmetric space of rank one was studied. The aim of this paper is to extend the results obtained in [1] to the case of complex Grassmannians. More precisely, let M = U/K, where U = SU(p + q) and K = S(U(p) x (U(q)), be the complex Grassmannian of a p-plane in C~(p+q), p ≥ q ≥ 2, a_1,...,a_r be r points in U, and consider the convolution product v_(a1) * ...* v_(ar). of the orbital measures v_(a1),..., v_(ar) supported on K_(a1) K,..., K_(ar)K. By a result of Ragozin [10], if r ≥ dim M, then v_(a1) * ... * v_(ar) is absolutely continuous with respect to the Haar measure of U. The aim of this paper is to investigate the C~k - regularity of the Radon-Nikodym derivative of v_(a1) * ... * v_(ar) with respect to the Haar measure of U.