Unitary representations of real groups and localization theory for Hodge modules

TL;DR

This work proves a unitarity criterion for irreducible Harish-Chandra modules with real infinitesimal character by reading unitarity off the canonical Hodge filtration provided by Saito’s mixed Hodge modules and Beilinson–Bernstein localization. The authors develop a deformation and wall-crossing theory for mixed Hodge modules, refine localization with the Hodge filtration, and compute the Hodge filtration on tempered Hodge modules, enabling a geometrical, Hodge-theoretic route to unitary dual questions. Key outcomes include a twisted Kodaira vanishing theorem for mixed Hodge modules, fully faithful Hodge globalization in the regular dominant case, and an explicit unitarity criterion involving the Cartan involution acting on Gr^F_p(V). The results yield strong vanishing and global-generation statements for Hodge filtrations on flag varieties and establish a robust, functorial bridge between Hodge theory and representation theory with potential broad impact on understanding unitary representations via geometric methods. Overall, the paper provides a conceptually geometric framework that connects Hodge theory, localization, and unitarity in real reductive groups, with concrete vanishing and generation results supporting the unitarity criterion and its applications to tempered and discrete series representations.

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.

Theorems & Definitions (153)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6: =Theorem \ref{['thm:hodge generation']}
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9: Theorems \ref{['thm:hodge and signature K']} and \ref{["thm:hodge and signature K'"]}
• Theorem 1.10