Equivariant analytic localization of group representations (Q2750967)

The study of infinite dimensional representations of a real connected semisimple Lie group \(G_0\) with finite center involves many technical difficulties that do not appear in the study of the finite dimensional irreducible representations of \(G_0\) (Borel-Weil theorem). One of the aspects which are important in this study is the classical problem of geometric realizations of these representations by \(G_0\) modules. Some results have been obtained by J. Wolf and W. Schmid by constructing the maximal globalizations of Vogan-Zuckerman standard modules. This approach is related, by duality, to the geometric construction method developed by H. Hecht and J. Taylor, called the analytic localization. The author presents a refinement of Hecht and Taylor's method by considering the equivariant analytic localization. This method improves the geometric realizations presented before and is easier to relate to algebraic localization results. The memoir has nine chapters with the titles: 1. Preliminaries, 2. The category \({\mathcal T}\), 3. The equivalence categories, 4. The category \(D^b_{G_0} ({\mathcal U}_0 ({\mathcal G}))\), 7. Localization, 8. Our main equivalence categories, 9. Equivalence for any regular weight \(\lambda\).