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

Sanjiv Kumar Gupta, Kathryn Hare

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 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.

Original languageEnglish
Pages (from-to)841-875
Number of pages35
JournalCanadian Journal of Mathematics
Volume68
Issue number4
DOIs
Publication statusPublished - Aug 1 2016

Fingerprint

Absolute Continuity
Convolution
Lie Algebra
Absolutely Continuous
Orbit
Simple Lie Algebra
Interior
Lebesgue Measure
Fold
Roots
Integer
Invariant

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.

In: Canadian Journal of Mathematics, Vol. 68, No. 4, 01.08.2016, p. 841-875.

Research output: Contribution to journalArticle

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