“Smoothness of convolution products of orbital measures on rank one compact symmetric spaces”
A result of Gupta and Hare established a dichotomy for the convolution product of orbital measures on compact Lie groups. They showed that the convolution product of these measures were either singular or in L2. A natural extension of this question is to symmetric spaces, where it was found that in SU(2)/SO(2), this dichotomy failed to hold.
In this talk, I will discuss the situation for all rank one compact symmetric spaces, where we have found characterizations of exactly when these convolution products are absolutely continuous or in L2. We give a condition in terms of the dimensions of the corresponding double cosets that the orbital measures are supported on.