M\'ethode des Orbites et Repr\'esentations Quantiques

Abstract

The first part of this thesis studies the notion of a "quantum representation", introduced by J.-M. Souriau in order to provide a polarization-free characterization of the Lie group representations attached to coadjoint orbits. When the group is compact, we show that Souriau's condition does indeed select the Borel-Weil representation within sections of the line bundle over an orbit. At the other extreme, for exponential solvable groups, we give counterexamples to an analogous assertion, but show that a refined condition does select the Kirillov-Bernat representation attached to an orbit. The second part develops the symplectic analogue of Mackey's normal subgroup analysis, using the notion of an induced hamiltonian G-space introduced by Kazhdan, Kostant and Sternberg.