A geometric proof of the \(L^{2}\)-singular dichotomy for orbital measures on Lie algebras and groups



Let G be a compact, connected simple Lie group and \(\mathfrak {g}\) its Lie algebra. It is known that if \(\mu \) is any G-invariant measure supported on an adjoint orbit in \(\mathfrak {g}\), then for each integer k, the k-fold convolution product of \(\mu \) with itself is either singular or in \( L^{2}\). This was originally proven by computations that depended on the Lie type of \(\mathfrak {g}\), as well as properties of the measure. In this note, we observe that the validity of this dichotomy is a direct consequence of the Duistermaat–Heckman theorem from symplectic geometry and that, in fact, any convolution product of (even distinct) orbital measures is either singular or in \(L^{2+\varepsilon }\) for some \(\varepsilon >0\). An abstract transference result is given to show that the \(L^{2}\)-singular dichotomy holds for certain of the G-invariant measures supported on conjugacy classes in G.


Orbital measure Duistermaat–Heckman theorem 

Mathematics Subject Classification

Primary 43A80 Secondary 22E30 53D05 



We are grateful to A. Wright for providing us with remarks from M. Vergne pointing out the connection with the Duistermaat–Heckman theorem and thank S. Gupta for helpful conversations.


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© Unione Matematica Italiana 2018

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of MathematicsStanford UniversityStanfordUSA

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