Unitary representations of real groups and localization theory for Hodge modules

TL;DR

This paper proves that the unitarity of certain representations of real reductive Lie groups can be determined from the Hodge filtration, using new techniques in Hodge theory and localization.

Contribution

It introduces a Hodge-theoretic approach to analyze unitarity, including a wall crossing theory, a refinement of localization, and explicit calculations of Hodge filtrations.

Findings

Hodge filtration varies semi-continuously with jumps controlled by extension functors.

Hodge filtration satisfies cohomology vanishing and global generation properties.

Derived new vanishing results for coherent sheaves on flag varieties.

Abstract

We prove a conjecture of Schmid and the second named author that the unitarity of a representation of a real reductive Lie group with real infinitesimal character can be read off from a canonical filtration, the Hodge filtration. Our proof rests on three main ingredients. The first is a wall crossing theory for mixed Hodge modules: the key result is that, in certain natural families, the Hodge filtration varies semi-continuously with jumps controlled by extension functors. The second ingredient is a Hodge-theoretic refinement of Beilinson-Bernstein localization: we show that the Hodge filtration of a mixed Hodge module on the flag variety satisfies the usual cohomology vanishing and global generation properties enjoyed by the underlying $\mathcal{D}$-module. The third ingredient is an explicit calculation of the Hodge filtration on a tempered Hodge module. As byproducts of our work, we obtain a version of Saito's Kodaira vanishing for twisted mixed Hodge modules, a calculation of the Hodge filtration on a certain object in category $\mathcal{O}$, and a host of new vanishing results for coherent sheaves on flag varieties.