The smoothness of orbital measures on noncompact symmetric spaces

TL;DR

This paper investigates the smoothness properties of orbital measures on noncompact symmetric spaces, showing that their convolutions become square-integrable and absolutely continuous, with specific results depending on the space's rank and type.

Contribution

It establishes decay-based criteria for the smoothness of convoluted orbital measures on symmetric spaces, including sharp results for special cases.

Findings

Convolution of r(G/K) orbital measures has density in L^2(G).

Orbital measures supported on regular elements become L^2 after convolution.

Three orbital measures are needed for certain spaces to achieve L^2 density.

Abstract

Let $G/K$ be an irreducible symmetric space where $G$ is a non-compact, connected Lie group and $K$ is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $% L^{2}(G)$ and hence is an absolutely continuous measure with respect to Haar measure. The number $r$ is approximately the rank of $G/K$. For the special case of the orbital measures, $\nu_{a_{i}}$, supported on the double cosets $Ka_{i}K$ where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu_{a_{1}}\ast \nu_{a_{2}}\in L^{2},$ except for the symmetric space of Cartan type $AI$ when the convolution of three orbital measures is needed (even though $\nu_{a_{1}}\ast \nu_{a_{2}}$ is absolutely continuous).