Deformation program for principal series representations. (English) Zbl 0843.22020
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.
Reviewer: J.Ludwig (Metz)
MSC:
| 22E46 | Semisimple Lie groups and their representations |
| 43A30 | Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. |
| 58H15 | Deformations of general structures on manifolds |
Keywords:
parametrization; principal series orbits; semisimple Lie group; cotangent bundle; nilpotent Lie group; coadjoint orbit; compact Lie group; Berezin symbolic calculus; Weyl correspondence; principal series representations; Gutt’s star product; Fourier transform; Plancherel formulaReferences:
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