### Abstract

We prove that the G-invariant orbital measures supported on adjoint orbits in the Lie algebra of a classical, compact, connected, simple Lie group satisfy a smoothness dichotomy: Either μ^{k} is singular to Lebesgue measure or μ^{k} ∈ L^{2}. The minimum k for which μ^{k} ∈ L^{2} is specified and is also the minimum k such that the k-fold sum of the orbit has positive measure.

Original language | English |
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Pages (from-to) | 91-124 |

Number of pages | 34 |

Journal | Mathematische Zeitschrift |

Volume | 262 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 |

### Fingerprint

### Keywords

- Adjoint orbit
- Classical Lie algebra
- Compact Lie group
- Conjugacy class
- Orbital measure

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematische Zeitschrift*,

*262*(1), 91-124. https://doi.org/10.1007/s00209-008-0364-z

**L2-singular dichotomy for orbital measures of classical simple Lie algebras.** / Gupta, Sanjiv Kumar; Hare, Kathryn E.; Seyfaddini, Sobhan.

Research output: Contribution to journal › Article

*Mathematische Zeitschrift*, vol. 262, no. 1, pp. 91-124. https://doi.org/10.1007/s00209-008-0364-z

}

TY - JOUR

T1 - L2-singular dichotomy for orbital measures of classical simple Lie algebras

AU - Gupta, Sanjiv Kumar

AU - Hare, Kathryn E.

AU - Seyfaddini, Sobhan

PY - 2009

Y1 - 2009

N2 - We prove that the G-invariant orbital measures supported on adjoint orbits in the Lie algebra of a classical, compact, connected, simple Lie group satisfy a smoothness dichotomy: Either μk is singular to Lebesgue measure or μk ∈ L2. The minimum k for which μk ∈ L2 is specified and is also the minimum k such that the k-fold sum of the orbit has positive measure.

AB - We prove that the G-invariant orbital measures supported on adjoint orbits in the Lie algebra of a classical, compact, connected, simple Lie group satisfy a smoothness dichotomy: Either μk is singular to Lebesgue measure or μk ∈ L2. The minimum k for which μk ∈ L2 is specified and is also the minimum k such that the k-fold sum of the orbit has positive measure.

KW - Adjoint orbit

KW - Classical Lie algebra

KW - Compact Lie group

KW - Conjugacy class

KW - Orbital measure

UR - http://www.scopus.com/inward/record.url?scp=69849110551&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69849110551&partnerID=8YFLogxK

U2 - 10.1007/s00209-008-0364-z

DO - 10.1007/s00209-008-0364-z

M3 - Article

VL - 262

SP - 91

EP - 124

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 1

ER -