Abstracts for Conference on: Algebraic Groups, Lie Algebras and their Representations on the occasion of Jens Carsten Jantzen's 70th birthday, November 22 - 24, 2018

This is about a joint project with my colleague C.F. Woodcock (Kent). We investigate trace surjective commutative $G$-algebras defined by non-linear actions of a finite group $G$. These arise in the analysis of invariants of "dehomogenized" graded group actions and related localizations. Every finite unipotent group has a faithful (non-linear) representation on a polynomial ring with invariant ring being again a polynomial ring.

We first review some basic results related to Serre's notion of $G$-complete reducibility for a reductive algebraic group $G$. We then discuss a relative variant of this concept where we let $K$ be a reductive subgroup of $G$, and consider subgroups of $G$ which normalise the identity component $K^o$ of $K$. We show that such a subgroup is relatively $G$-completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^o$ is completely reducible in the sense of Serre.

$k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a connected reductive group over $k$ with Lie algebra ${g}$. Consider the rings $k[G]$ and $k[{g}]$ of regular functions on $G$ and ${g}$ as $G$-modules via the conjugation action. They have been studied extensively, for example in Kostant’s 1963 paper. I will discuss the result that, under some mild assumptions, the first restricted cohomology of these modules is zero. After this I will discuss the problem of describing the invariants in a certain finite dimensional quotient of $k[{g}]$.

I will discuss joint work in progress with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne on potential automorphy over CM fields. I will give an overview of this work and mention some applications. I will then focus on how to prove one of our key inputs, namely local-global compatibility at $l=p$ for certain Galois representations attached to torsion classes in the cohomology of locally symmetric spaces. This relies crucially on some ingredients from representation theory in positive characteristic.