The smoothness of convolutions of singular orbital measures on complex grassmannians

It is well known that if G/K is any irreducible symmetric space and μa is a continuous orbital measure supported on the double coset KaK, then the convolution product, μk a, is absolutely continuous for some suitably large k ≤ dimG/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 μk a belongs to the smaller space L2 for some k . Here we prove that this L2 property holds for all the compact, complex Grassmannian symmetric spaces, SU(p + q)/S(U(p) × U(q)) . Moreover, for the orbital measures at a dense set of points a, we prove that μ2 a ϵ L2 (or μ3 a ϵ L2 if p = q ).