### Abstract

Let g be a compact simple Lie algebra of dimension d. It is a classical result that the convolution of any d non-trivial, G-invariant, orbital measures is absolutely continuous with respect to Lebesgue measure on g, and the sum of any d non-trivial orbits has non-empty interior. The number d was later reduced to the rank of the Lie algebra (or rank +1 in the case of type A_{n}). More recently, the minimal integer k = k(X) such that the k-fold convolution of the orbital measure supported on the orbit generated by X is an absolutely continuous measure was calculated for each X ∈ g. In this paper g is any of the classical, compact, simple Lie algebras. We characterize the tuples (X_{1}, . . . , X_{L}), with X_{i} ∈ g, which have the property that the convolution of the L-orbital measures supported on the orbits generated by the X_{i} is absolutely continuous, and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of g and the structure of the annihilating roots of the X_{i}. Such a characterization was previously known only for type A_{n}.

Original language | English |
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Pages (from-to) | 841-875 |

Number of pages | 35 |

Journal | Canadian Journal of Mathematics |

Volume | 68 |

Issue number | 4 |

DOIs | |

Publication status | Published - Aug 1 2016 |

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### Keywords

- Absolutely continuous measure
- Compact Lie algebra
- Orbital measure

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Characterizing the absolute continuity of the convolution of orbital measures in a classical Lie algebra.** / Gupta, Sanjiv Kumar; Hare, Kathryn.

Research output: Contribution to journal › Article

*Canadian Journal of Mathematics*, vol. 68, no. 4, pp. 841-875. https://doi.org/10.4153/CJM-2015-018-2

}

TY - JOUR

T1 - Characterizing the absolute continuity of the convolution of orbital measures in a classical Lie algebra

AU - Gupta, Sanjiv Kumar

AU - Hare, Kathryn

PY - 2016/8/1

Y1 - 2016/8/1

N2 - Let g be a compact simple Lie algebra of dimension d. It is a classical result that the convolution of any d non-trivial, G-invariant, orbital measures is absolutely continuous with respect to Lebesgue measure on g, and the sum of any d non-trivial orbits has non-empty interior. The number d was later reduced to the rank of the Lie algebra (or rank +1 in the case of type An). More recently, the minimal integer k = k(X) such that the k-fold convolution of the orbital measure supported on the orbit generated by X is an absolutely continuous measure was calculated for each X ∈ g. In this paper g is any of the classical, compact, simple Lie algebras. We characterize the tuples (X1, . . . , XL), with Xi ∈ g, which have the property that the convolution of the L-orbital measures supported on the orbits generated by the Xi is absolutely continuous, and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of g and the structure of the annihilating roots of the Xi. Such a characterization was previously known only for type An.

AB - Let g be a compact simple Lie algebra of dimension d. It is a classical result that the convolution of any d non-trivial, G-invariant, orbital measures is absolutely continuous with respect to Lebesgue measure on g, and the sum of any d non-trivial orbits has non-empty interior. The number d was later reduced to the rank of the Lie algebra (or rank +1 in the case of type An). More recently, the minimal integer k = k(X) such that the k-fold convolution of the orbital measure supported on the orbit generated by X is an absolutely continuous measure was calculated for each X ∈ g. In this paper g is any of the classical, compact, simple Lie algebras. We characterize the tuples (X1, . . . , XL), with Xi ∈ g, which have the property that the convolution of the L-orbital measures supported on the orbits generated by the Xi is absolutely continuous, and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of g and the structure of the annihilating roots of the Xi. Such a characterization was previously known only for type An.

KW - Absolutely continuous measure

KW - Compact Lie algebra

KW - Orbital measure

UR - http://www.scopus.com/inward/record.url?scp=84978873784&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84978873784&partnerID=8YFLogxK

U2 - 10.4153/CJM-2015-018-2

DO - 10.4153/CJM-2015-018-2

M3 - Article

VL - 68

SP - 841

EP - 875

JO - Canadian Journal of Mathematics

JF - Canadian Journal of Mathematics

SN - 0008-414X

IS - 4

ER -