Orbital Measures on SU(2) / SO(2)

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\N_U(K), we let μ_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 μ_a ^(p), the convolution of p copies of μ_a, is absolutely continuous with respect to the Haar measure of the group U as soon as p≥2. The aim of this paper is to go a step further by proving the following two results : (i) for every a in U\N_U(K) and every integer p≥3, the Radon–Nikodym derivative of μ_a ^(p) with respect to the Haar measure (Formula Presented.), and (ii) there exist a in (Formula Presented.), hence a counter example to the dichotomy conjecture stated by Gupta and Hare (Bull Aust Math Soc 79:513–522, 2009). Since (Formula Presented.), 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.