# Patrick Brosnan

### Degenerations of Hodge structure and real algebraic groups

**Abstract**: In his papers on mixed Hodge theory, Deligne observed that the category of split real mixed Hodge structures is equivalent to the category of a representations of the Weil restriction of the multiplicative group of the complex numbers to the reals. The category of mixed Hodge structures has a slightly more complicated description, also due to Deligne: it is the category of finite dimensional representations of an affine group scheme which is a semi-direct product of the group S and a pro-unipotent group U whose associated Lie algebra is free on one generator in each bidegree (p,q) with p and q negative integers.

This talk is about joint work with Greg Pearlstein proving an analogue of Deligne's result for certain degenerating families of mixed Hodge structures: the nilpotent orbits with limit split over the reals. We

find a semi-direct product similar to the group for real mixed Hodge structures but with S replace by a certain real reductive group of rank 3 and derived group SL_2. Our original motivation for this was to understand another (unpublished) result of Deligne which is very useful for studying degenerations.