Representation theory seminar 2021, Semester 2
Time: Thursdays 2:15pm-4:15pm.
Location: Peter Hall 213 and Zoom
Sep 23 Dougal Davis (Edinburgh) Hodge modules, Lusztig-Vogan polynomials and unitary representations of real groups.
Abstract: I will discuss joint work in progress with Kari Vilonen on K-equivariant mixed Hodge modules on the flag variety for a symmetric pair (g, K). In close analogy with Kazhdan-Lusztig theory, the K-group of mixed Hodge modules has two bases consisting of standard and irreducible objects; our first main theorem computes the change of basis matrix by adding an extra Hodge parameter to the Lusztig-Vogan polynomials. Our second main theorem is a “polarised” version of the Jantzen conjecture; following ideas of Schmid and Vilonen, it allows the signature multiplicity polynomial of Adams-van Leeuwen-Trapa-Vogan for the corresponding real group representations to be read off from the Hodge polynomial (modulo a small claim). This recovers a key formula in the ALTV algorithm for the unitary dual.