Jimmy He - April 22, 2016
Over the summer I worked with Professor Hare on two related projects. Both involved examining the convolutions of so-called orbital measures on Lie groups and symmetric spaces and determining their smoothness properties. These problems are of great interest because they shed light onto the topological properties of products of conjugacy classes in the Lie group and sums of adjoint orbits in the Lie algebra.
For the first project, I looked at when convolutions of orbital measures on exceptional symmetric spaces were absolutely continuous with respect to the Haar measure. Due to the inaccessible nature of these spaces, I had to develop new techniques that would allow us to understand the exceptional Lie groups better. In particular, I found a method of transferring results from certain simpler exceptional symmetric spaces to more complicated ones using a Freudenthal "magic square" construction.
This allowed us to lift results for the simpler spaces, obtained using combinatorial techniques, to larger spaces where these techniques now failed. Using these tools, we were able to characterize exactly when convolution powers of orbital measures were absolutely continuous with respect to the Haar measure in all exceptional symmetric spaces.
The second project I worked on involved looking at when convolution products of orbital measures on rank one symmetric spaces were square-integrable, and in particular when these convolution powers were absolutely continuous with respect to the Haar measure but not squareintegrable. This was inspired by a paper which examined the special case for the space SU(2)/SO(2). When I started working on the problem not much was known for any symmetric space other than SU(2)/SO(2).
I obtained asymptotic estimates for the spherical functions on each space, which were derived using classical estimates for the asymptotics of hypergeometric functions. Using these estimates, we were ultimately able to characterize exactly when convolution powers of orbital measures were square-integrable. We also determined which rank one symmetric spaces allowed for orbital measures whose convolution power was absolutely continuous but not square-integrable.