Orbital Measures on SU(2) / SO(2)

Abstract

We let $U=SU(2)$, $K=SO(2)$ and denote by $N_{U} (K)$ the normalizer of K in U. For a an element of $U\backslash N_{U} (K)$, we let $\mu _{a}$ be the normalized singular measure supported in KaK. For p a positive integer, it was proved by Gupta and Hare (Monatsh Math 159:27–59, 2010) that $\mu _{a}^{(p)},$ the convolution of p copies of $\mu _{a}$, is absolutely continuous with respect to the Haar measure of the group U as soon as $p \ge 2$. The aim of this paper is to go a step further by proving the following two results : (i) for every a in $U\backslash N_{U}(K)$ and every integer $p \ge 3$, the Radon–Nikodym derivative of $\mu _{a}^{(p)}$ with respect to the Haar measure $m_{U}$ on U, namely $\hbox {d}\mu _{a}^{(p)} / \hbox {d}m_{U}$, is in $L^{2}(U)$, and (ii) there exist a in $U\backslash N_{U}( K)$ for which $\hbox {d}\mu _{a}^{(2)}/\hbox {d}m_{U}$ is not in $L^{2}(U),$ hence a counter example to the dichotomy conjecture stated by Gupta and Hare (Bull Aust Math Soc 79:513–522, 2009). Since $L^{2}(U) \subseteq L^{1} (U)$, our result gives in particular a new proof of the main result of Gupta and Hare (Monatsh Math 159:27–59, 2010) when $p>2$.