Mixed Hodge modules and real groups

Abstract

Let G be a complex reductive group, θ:G→G an involution, and K=Gθ. In [29], W. Schmid and the second named author proposed a program to study unitary representations of the corresponding real form GR using K-equivariant twisted mixed Hodge modules on the flag variety of G and their polarizations. In this paper, we make the first significant steps towards implementing this program. Our first main result gives an explicit combinatorial formula for the Hodge numbers appearing in the composition series of a standard module in terms of the Lusztig-Vogan polynomials. Our second main result is a polarized version of the Jantzen conjecture, stating that the Jantzen forms on the composition factors a..