The absolute continuity of convolutions of orbital measures in symmetric spaces

Sanjiv Kumar Gupta, Kathryn E. Hare

Research output: Contribution to journalArticle

Abstract

We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces G/K of Cartan type III, where G is a non-compact, connected Lie group and K is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, KzK, in G. The characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements z. A consequence of our work is to show that the convolution product of any rank G/K, continuous, K-bi-invariant measures is absolutely continuous in any of these symmetric spaces, other than those whose restricted root system is type An or D3, when rank G/K +1 is needed.

Original languageEnglish
Pages (from-to)81-111
Number of pages31
JournalJournal of Mathematical Analysis and Applications
Volume450
Issue number1
DOIs
Publication statusPublished - Jun 1 2017

Keywords

  • Absolute continuity
  • Double coset
  • Orbital measure
  • Symmetric space

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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