### Abstract

It is known that all continuous orbital measures, μ,on a compact, connected, classical simple Lie group G or its Lie algebra satisfy a dichotomy: either μ^{k} εL^{2} or μ^{k} is purely singular to Haar measure. In this note we prove that the same dichotomy holds for the dual situation, continuous orbital measures on the complex group G ^{C}. We also determine the sharp exponent k such that any k-fold convolution product of continuous G-bi-invariant measures on G^{C} is absolute continuous with respect to Haar measure.

Original language | English |
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Pages (from-to) | 409-419 |

Number of pages | 11 |

Journal | Bolletino dell Unione Matematica Italiana |

Volume | 3 |

Issue number | 3 |

Publication status | Published - 2010 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Bolletino dell Unione Matematica Italiana*,

*3*(3), 409-419.

**L2- Singular dichotomy for orbital measures on complex groups.** / Gupta, S. K.; Hare, K. E.

Research output: Contribution to journal › Article

*Bolletino dell Unione Matematica Italiana*, vol. 3, no. 3, pp. 409-419.

}

TY - JOUR

T1 - L2- Singular dichotomy for orbital measures on complex groups

AU - Gupta, S. K.

AU - Hare, K. E.

PY - 2010

Y1 - 2010

N2 - It is known that all continuous orbital measures, μ,on a compact, connected, classical simple Lie group G or its Lie algebra satisfy a dichotomy: either μk εL2 or μk is purely singular to Haar measure. In this note we prove that the same dichotomy holds for the dual situation, continuous orbital measures on the complex group G C. We also determine the sharp exponent k such that any k-fold convolution product of continuous G-bi-invariant measures on GC is absolute continuous with respect to Haar measure.

AB - It is known that all continuous orbital measures, μ,on a compact, connected, classical simple Lie group G or its Lie algebra satisfy a dichotomy: either μk εL2 or μk is purely singular to Haar measure. In this note we prove that the same dichotomy holds for the dual situation, continuous orbital measures on the complex group G C. We also determine the sharp exponent k such that any k-fold convolution product of continuous G-bi-invariant measures on GC is absolute continuous with respect to Haar measure.

UR - http://www.scopus.com/inward/record.url?scp=79956099585&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79956099585&partnerID=8YFLogxK

M3 - Article

VL - 3

SP - 409

EP - 419

JO - Bolletino dell Unione Matematica Italiana

JF - Bolletino dell Unione Matematica Italiana

SN - 1972-6724

IS - 3

ER -