  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 two 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. 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, for example, homogeneous vector bundles on flag
varieties.

Representation Theory;Algebraic Geometry