The absolute continuity of convolutions of orbital measures in symmetric spaces. (English) Zbl 1362.43007
Let \(G/K\) be the Riemannian symmetric space, where \(G\) is a non-compact connected Lie group and \(K\) is a compact connected subgroup fixed by a Cartan involution on \(G\). The orbital measures are the measures on \(G\) of the form \(\nu_z= m_K* \delta_z *m_K\), where \(m_K\) is the Haar measure on \(K\), \(z\in G\). These are uniform measures supported on the double cosets \(KzK\) in \(G\). They are purely singular, probability measures. The authors characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces of Cartan type III. The characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements \(z\).
Reviewer: Sergei S. Platonov (Petrozavodsk)
MSC:
| 43A85 | Harmonic analysis on homogeneous spaces |
| 22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |
| 43A05 | Measures on groups and semigroups, etc. |
| 53C35 | Differential geometry of symmetric spaces |
| 43A10 | Measure algebras on groups, semigroups, etc. |
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