∗ products and representations of nilpotent groups

Didier Arnal

Vol. 114 (1984), No. 2, 285–308

DOI: 10.2140/pjm.1984.114.285

Abstract

On each orbit W of the coadjoint representation of a nilpotent, connected and simply connected Lie group G, there exist ∗ products which are relative quantizations for the Lie algebra g of G. Choosing one of these ∗ products, we first define a ∗-exponential for each X in g. These ∗-exponentials are formal power series and, with the ∗ product, they form a group. Thanks to that, we are able to define a representation of G in a “ ∗ polarization” and to intertwine it with the unitary irreducible one associated to W. Finally, we study the uniqueness of our construction.

Mathematical Subject Classification 2000

Primary: 22E25

Secondary: 58F06, 81D07

Milestones

Received: 8 July 1982

Revised: 1 January 1983

Published: 1 October 1984