Mixed Hodge modules and real groups

Dougal Davis, Kari Vilonen

Published 2022-02-17Version 1

Let $G$ be a complex reductive group, $\theta \colon G \to G$ an involution, and $K = G^\theta$. In arXiv:1206.5547, W. Schmid and the second named author proposed a program to study unitary representations of the corresponding real form $G_\mathbb{R}$ 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 are polarizations of the underlying Hodge modules. Our third main result states that, for regular Beilinson-Bernstein data, the minimal $K$-type of an irreducible Harish-Chandra module lies in the lowest piece of the Hodge filtration of the corresponding Hodge module. An immediate consequence of our results is a Hodge-theoretic proof of the signature multiplicity formula of arXiv:1212.2192, which was the inspiration for this work.