Orbital Measures on SU(2) / SO(2)

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Abstract

We let U=SU(2), K=SO(2) and denote by NU(K) the normalizer of K in U. For a an element of U\NU(K), we let μa 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 μa (p), the convolution of p copies of μa, 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\NU(K) and every integer p≥3, the Radon–Nikodym derivative of μa (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.

Original languageEnglish
Pages (from-to)493-520
Number of pages28
JournalMonatshefte fur Mathematik
Volume178
Issue number4
DOIs
Publication statusPublished - Dec 1 2015

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Keywords

  • Absolutely continuous measure
  • Analytic combinatorics
  • Bi-invariant measure
  • Dichotomy conjecture
  • Exponential sums
  • Harmonic analysis
  • Symmetric space

ASJC Scopus subject areas

  • Mathematics(all)

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