### Abstract

We prove that for any classical, compact, simple, connected Lie group G, the G-invariant orbital measures supported on non-trivial conjugacy classes satisfy a surprising L^{2}-singular dichotomy: Either μ_{h}
^{k} ∈ L^{2} (G) or μ_{h}
^{k} is singular to the Haar measure on G. The minimum exponent k for which μ_{h}
^{k} ∈ L^{2} is specified; it depends on Lie properties of the element h ∈ G. As a corollary, we complete the solution to a classical problem - to determine the minimum exponent k such that μ^{k} ∈ L^{1} (G) for all central, continuous measures μ on G. Our approach to the singularity problem is geometric and involves studying the size of tangent spaces to the products of the conjugacy classes.

Original language | English |
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Pages (from-to) | 1521-1573 |

Number of pages | 53 |

Journal | Advances in Mathematics |

Volume | 222 |

Issue number | 5 |

DOIs | |

Publication status | Published - Dec 1 2009 |

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### Keywords

- Compact Lie group
- Conjugacy class
- Orbital measure
- Singular measure
- Tangent space

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*222*(5), 1521-1573. https://doi.org/10.1016/j.aim.2009.06.008