### 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 |
---|---|

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 |

### Fingerprint

### 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

**L2-singular dichotomy for orbital measures of classical compact Lie groups.** / Gupta, Sanjiv Kumar; Hare, Kathryn E.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 222, no. 5, pp. 1521-1573. https://doi.org/10.1016/j.aim.2009.06.008

}

TY - JOUR

T1 - L2-singular dichotomy for orbital measures of classical compact Lie groups

AU - Gupta, Sanjiv Kumar

AU - Hare, Kathryn E.

PY - 2009/12/1

Y1 - 2009/12/1

N2 - 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 L2-singular dichotomy: Either μh k ∈ L2 (G) or μh k is singular to the Haar measure on G. The minimum exponent k for which μh k ∈ L2 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 μk ∈ L1 (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.

AB - 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 L2-singular dichotomy: Either μh k ∈ L2 (G) or μh k is singular to the Haar measure on G. The minimum exponent k for which μh k ∈ L2 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 μk ∈ L1 (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.

KW - Compact Lie group

KW - Conjugacy class

KW - Orbital measure

KW - Singular measure

KW - Tangent space

UR - http://www.scopus.com/inward/record.url?scp=69849108766&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69849108766&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2009.06.008

DO - 10.1016/j.aim.2009.06.008

M3 - Article

VL - 222

SP - 1521

EP - 1573

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 5

ER -