Deformation program for principal series representations
The author begins with a parametrization of the principal series orbits of a semisimple Lie group \(G\) in terms of a cotangent bundle of a nilpotent Lie group and of a coadjoint orbit of a compact Lie group. Then he combines Berezin symbolic calculus on compact orbits and some generalization of Weyl correspondence to a cotangent bundle to obtain a symbolic calculus on the principal series orbits which is adapted to the description of the principal series representations. He shows that the \(*\)-product on principal series orbits associated to this symbolic calculus can be interpreted in terms of a Berezin product and on S. Gutt's star product on the cotangent bundle of a nilpotent Lie group. Finally, the author applies his construction to define the adapted Fourier transform of \(G\) and he gives in the case where \(G\) has a unique class of Cartan subalgebras an elegant retranscription of the Plancherel formula.
scientific article; zbMATH DE number 843629
scientific article