Langlands' philosophy and Koszul duality (Q2759673)

This paper outlines a partly conjectural contribution to the local case of the Langlands philosophy in which the author describes a simultaneous extension of \textit{J. Adams}, \textit{D. Barbasch} and \textit{D. A. Vogan jun.} [The Langlands classification and irreducible characters for real reductive groups. Prog. Math. 104, Boston etc.: Birkhäuser (1992; Zbl 0756.22004)] and \textit{A. A. Beilinson}, \textit{V. Ginzburg} and \textit{W. Soergel} [J. Am. Math. Soc. 9, 473--527 (1996; Zbl 0864.17006)]. The work of Adams-Barbasch-Vogan gives a geometric classification scheme for smooth, admissible representations of real reductive groups in which Kazhdan-Lusztig data on the representation theory side matches up with intersection cohomology data in the geometry. Without going into technicalities, the author proposes that this classification of objects is part of a much stronger equivalence of categories, whose details he describes and verifies in many cases (e.g. tori). In addition, the author explains how his conjecture fits with the character duality formulae of \textit{D. A. Vogan} jun. [Duke Math. J. 49, 943--1073 (1982; Zbl 0536.22022)]. In fact, the connection of the conjecture with the work of Beilinson-Ginzburg-Soergel is that the conjectured equivalence of categories should be an example of a Koszul duality, which is a relationship taking place in the associated derived categories.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00049].