Convolution of orbital measures on complex grassmannians

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)× U(q)), be the complex Grassmannian of a p-plane in Cp+^q , p ? q ? 2, a₁, ..., a_r be r points in U , and consider the convolution product ?_a1 ? ... ? ?_ar of the orbital measures ?_a1 , ..., ?_ar supported on Ka₁K, ..., Ka_rK . By a result of Ragozin [10], if r ? dim M, then ?_a1 ? ... ? ?_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 ?_a1 ? ... ? ?_ar with respect to the Haar measure of U . Mathematics Subject Classification 2010: Primary 43A77, 43A90; Secondary 53C35, 28C10.