Hodge theory, intertwining functors, and the Orbit Method for real reductive groups
Dougal Davis, Lucas Mason-Brown
TL;DR
By unifying the Orbit Method with Schmid–Vilonen Hodge theory for real reductive groups, the paper shows that the Hodge filtration on unipotent representations coincides with Orbit Method quantization, enabling unitarity proofs and explicit $K$-type descriptions. A central innovation is a Hodge-theoretic upgrade of Beilinson–Bernstein intertwining functors, realized via pro-objects and an affine Hecke algebra action on filtered mixed Hodge modules, connecting localization, monodromy, and Z-algebra actions. The authors prove a Cohen–Macaulay property for the Hodge associated graded of irreducibles with very weakly unipotent annihilators, which yields precise quantization of $K$-types and, in particular, a vector-bundle description on nilpotent $K$-orbits for small-boundary unipotent representations. They also establish cohomology vanishing on partial flag varieties and derive a framework that confirms long-standing unitarity conjectures for unipotent representations, including special unipotent cases for complex groups and corresponding real forms. Overall, the work provides a robust, geometry-infused approach to the unitary dual problem, with broad implications for representation theory and geometric Langlands-type perspectives.
Abstract
We study the Hodge filtrations of Schmid and Vilonen on unipotent representations of real reductive groups. We show that for various well-defined classes of unipotent representations (including, for example, the oscillator representations of metaplectic groups, the minimal representations of all simple groups, and all unipotent representations of complex groups) the Hodge filtration coincides with the quantization filtration predicted by the Orbit Method. We deduce a number of longstanding conjectures about such representations, including a proof that they are unitary and a description of their $K$-types in terms of co-adjoint orbits. The proofs rely heavily on certain good homological properties of the Hodge filtrations on weakly unipotent representations, which are established using a Hodge-theoretic upgrade of the Beilinson-Bernstein theory of intertwining functors for $\mathcal{D}$-modules on the flag variety. The latter consists of an action of the affine Hecke algebra on a category of filtered monodromic $\mathcal{D}$-modules, which we use to compare Hodge filtrations coming from different localizations of the same representation. As an application of the same methods, we also prove a new cohomology vanishing theorem for mixed Hodge modules on partial flag varieties.