### 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 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Gupta, S. K., Hare, K. E., & Seyfaddini, S. (2009). L2-singular dichotomy for orbital measures of classical simple Lie algebras.

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