### 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\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.

Original language | English |
---|---|

Pages (from-to) | 493-520 |

Number of pages | 28 |

Journal | Monatshefte fur Mathematik |

Volume | 178 |

Issue number | 4 |

DOIs | |

Publication status | Published - 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)

### Cite this

*Monatshefte fur Mathematik*,

*178*(4), 493-520. https://doi.org/10.1007/s00605-015-0812-x

**Orbital Measures on SU(2) / SO(2).** / Anchouche, Boudjemâa; Gupta, Sanjiv Kumar; Plagne, Alain.

Research output: Contribution to journal › Article

*Monatshefte fur Mathematik*, vol. 178, no. 4, pp. 493-520. https://doi.org/10.1007/s00605-015-0812-x

}

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

UR - http://www.scopus.com/inward/citedby.url?scp=84947486873&partnerID=8YFLogxK

U2 - 10.1007/s00605-015-0812-x

DO - 10.1007/s00605-015-0812-x

M3 - Article

AN - SCOPUS:84947486873

VL - 178

SP - 493

EP - 520

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

IS - 4

ER -