Related papers: Hodge theory, intertwining functors, and the Orbit…

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…

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…

The Hodge filtration of the module of derivations on the orbit space of a finite real reflection group acting on an $\ell$-dimensional Euclidean space was introduced and studied by K. Saito. The filtration is equivalent data to the flat…

Given a smooth algebraic variety X with an action of a connected reductive linear algebraic group G, and an equivariant D-module M, we study the G-decompositions of the associated V-, Hodge, and weight filtrations. If M is the localization…

We show that the Hodge filtration of a tempered Hodge module is generated by the lowest piece of its Hodge filtration. As a consequence, we prove the main conjecture of [SV] in the special case of tempered representations of real reductive…

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration…

We present an application of Hodge theory towards the study of irreducible unitary representations of reductive Lie groups. We describe a conjecture about such representations and discuss some progress towards its proof.

We compute the Hodge filtration on cohomology groups of complements of complex coordinate subspace arrangements. By means of this result we construct integral representations of holomorphic functions such that kernels of these…

Let $G$ be a complex reductive algebraic group with Lie algebra $\mathfrak{g}$ and let $G_{\mathbb{R}}$ be a real form of $G$ with maximal compact subgroup $K_{\mathbb{R}}$. Associated to $G_{\mathbb{R}}$ is a $K \times…

We prove results concerning the behavior of Hodge ideals under restriction to hypersurfaces or fibers of morphisms, and addition. The main tool is the description of restriction functors for mixed Hodge modules by means of the…

We describe locally the representation varieties of fundamental groups for smooth complex varieties at representations coming from the monodromy of a variation of mixed Hodge structure. Given such a manifold $X$ and such a linear…

We introduce a category of possibly irregular holonomic D-modules which can be endowed in a canonical way with an irregular Hodge filtration. Mixed Hodge modules with their Hodge filtration naturally belong to this category, as well as…

We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us…

For $i : Z \to X$ a closed immersion of smooth varieties, we study how the $V$-filtration along $Z$ and the Hodge filtration on a mixed Hodge module $M$ on $X$ interact with each other. We also give a formula for the functors $i^*$, $i^!$…

Let $G$ be a connected reductive algebraic group $G$ over an algebraically closed field $k$ of prime characteristic $p$, and $\ggg=\Lie(G)$. In this paper, we study modular representations of the reductive Lie algebra $\ggg$ with…

Given a mixed Hodge module and a meromorphic function f on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module, which extends the construction of…

We consider mixed Hodge module structures on GKZ-hypergeometric differential systems. We show that the Hodge filtration on these D-modules is given by the order filtration, up to suitable shift. As an application, we prove a conjecture on…

We study the vanishing of (co)homology along ring homomorphisms for modules that admit certain filtrations, and generalize a theorem of O. Celikbas-Takahashi. Our work produces new classes of rigid and test modules, in particular over local…

We construct Hodge filtered cohomology groups for complex manifolds that combine the topological information of generalized cohomology theories with geometric data of Hodge filtered holomorphic forms. This theory provides a natural…

For an algebraic vector bundle $E$ over a smooth algebraic variety $X$, a monodromic $D$-module on $E$ is decomposed into a direct sum of some $O$-modules on $X$. We show that the Hodge filtration of a monodromic mixed Hodge module is…