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

Sanjiv Kumar Gupta, Kathryn E. Hare

Research output: Contribution to journalArticle

12 Citations (Scopus)

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 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.

Original languageEnglish
Pages (from-to)1521-1573
Number of pages53
JournalAdvances in Mathematics
Volume222
Issue number5
DOIs
Publication statusPublished - 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)

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