The Schmid-Vilonen Conjecture for Verma Modules of Highest Antidominant Weight and Discrete Series Representations

In his study of the unitary dual of a real semisimple Lie group $G_{\mathbb{R}}$, Vogan and his co-workers introduced a hermitian form $(\cdot,\cdot)_{\mathfrak{u}_{\mathbb{R}}}$ on a Harish-Chandra module $I$ that is invariant under the action of the maximal compact of the complexification $G$ of $G_{\mathbb{R}}$. Understanding the signature of this form is related to the problem of determining if the Harish-Chandra module $I$ is unitary or not. In their paper \cite{VS}, Vilonen and Schmid use the Beilinson-Bernstein localization theorem along with a twisted version of Saito's theory of mixed Hodge modules to extend the definition of this invariant form to a set of irreducible modules that are not necessarily in the Harish-Chandra category anymore. This set consists of modules that are ``geometrically constructible", meaning they are induced from locally closed subvarieties of the flag variety using the Grothendieck functors for $\mathcal{D}$-modules. The corresponding form turns out to be the integral of the polarization.\parFrom Saito's theory, these modules are endowed with natural Hodge filtrations. Schmid and Vilonen posted a conjecture that aims to study the signature of this form on the graded pieces of the Hodge filtration. In this thesis we explain all the constructions of Schmid and Vilonen with particular emphasis in the case when the modules are induced from closed subvarities of the flag variety. We also provide geometric proofs of the conjecture in the cases when $I$ is a Verma module of antidominant highest weight or a discrete series representation.