The Plancherel formula for complex algebraic unimodular groups
Let \(G\) be a complex algebraic unimodular group, \(\hat G\) its unitary dual. A measure \(\mu\), the Plancherel measure, is given explicitly on \(\hat G,\) so that \(\forall \phi \in C_c^{\infty}(G)\phi (e)=\int_{\hat G}\operatorname{tr}\pi (\phi)\,d\mu (\pi)\). To accomplish this, in the first part of the paper a sufficiently large subset of \(\hat G\) is constructed. Here the main tools are the ''orbit method'', a reduction to the reductive case using Mackey's theory and finally a comparison of the formulas with the known results of Harish Chandra. In the second part the methods developed by M. Duflo for general algebraic groups are adapted to the present case to obtain the Plancherel measure.
scientific article; zbMATH DE number 3889810
scientific article