In Hodge theory, there are several categories of objects that turn out to be (neutral) Tanakian, for example, split Hodge structures, mixed Hodge structures, variations of Hodge structure, etc. As such these categories are equivalent to the category of representations of their Tanakian galois groups. Unfortunately, most of these groups seem difficult to describe explicitly. However, there is an easy description of the category of split real Hodge structures. It is the category of representations of group Deligne called S: the Weil restriction of scalars from C to R of the multiplicative group. Deligne also described real mixed Hodge structures. But here the group involved is more complicated: it is the semi-direct product of S with a pro-unipotent group scheme U. Nilpotent orbits are certain variations of Hodge structure, which can be defined in terms of linear algerbaic data. The simplest of these are the SL2 orbits introduced by Schmid. It turns out that the category of SL2 orbits is equivalent to the category of representations of a certain real reductive algebraic group over R which is a semi-direct product of SL2 and Deligne's group S. I will describe this and a related group which controls certain nilpotent orbits. This gives a group-theoretic understanding of certain operations on variations of mixed Hodge structure, such as, taking the limit mixed Hodge structure.

The content of this talk is joint work with Gregory Pearlstein.