Hodge theory, intertwining functors, and the Orbit Method for real reductive groups

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.