##
**Continuous measures on compact Lie groups.**
*(English)*
Zbl 0969.43001

An analog of Wiener’s classical characterization of continuous measures on the circle group is obtained for compact semisimple Lie groups (and symmetric spaces of compact type):
\[
\lim_{n\to\infty}|A_n|^{-1}\sum_{\lambda\in A_n}d_\lambda^{-1}\|\widehat\mu(\pi_\lambda)\|^2_2= \sum_{g\in G}|\mu(\{g\})|^2.
\]
Here \(A_n\) is an “admissible” sequence of subsets of the set of dominant weights parametrizing the irreducible unitary representations of \(G\) and \(\mu\) is a measure on \(G\). For a compact set \(C\) of continuous measures on \(G\) the existence of a central singular measure \(\nu\) is shown, such that \(\mu*\nu\) is absolutely continuous for all \(\mu\in C\). This multiplier theorem is obtained by first constructing a singular measure using Riesz products on the maximal torus of \(G\). Furthermore it is proved that any finite linear combination of characters on \(G\) \(f\) can be represented as \(f=\mu*\nu\), where \(\mu\) and \(\nu\) are singular central measures.

Reviewer: M.Blümlinger (Wien)

### MSC:

43A05 | Measures on groups and semigroups, etc. |

43A80 | Analysis on other specific Lie groups |

22E30 | Analysis on real and complex Lie groups |

43A85 | Harmonic analysis on homogeneous spaces |

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