TY - JOUR
T1 - Orbital Measures on SU(2) / SO(2)
AU - Anchouche, Boudjemâa
AU - Gupta, Sanjiv Kumar
AU - Plagne, Alain
PY - 2015/12/1
Y1 - 2015/12/1
N2 - 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.
AB - 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.
KW - Absolutely continuous measure
KW - Analytic combinatorics
KW - Bi-invariant measure
KW - Dichotomy conjecture
KW - Exponential sums
KW - Harmonic analysis
KW - Symmetric space
UR - http://www.scopus.com/inward/record.url?scp=84947486873&partnerID=8YFLogxK
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U2 - 10.1007/s00605-015-0812-x
DO - 10.1007/s00605-015-0812-x
M3 - Article
AN - SCOPUS:84947486873
SN - 0026-9255
VL - 178
SP - 493
EP - 520
JO - Monatshefte fur Mathematik
JF - Monatshefte fur Mathematik
IS - 4
ER -