# 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

Alternatively have a look at the program.

## Cohomology, Nilpotent Elements, and Module Invariants

Let (${g}$,$[p]$) be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field $k$ of characteristic $p > 0$. This talk is concerned with the role of the nullcone $\mathcal{N}_p({g}) := \{x \in {g} \ ; \ x^{[p]} = 0\}$ within the representation theory of $({g},[p])$. I will begin by delineating the historical development of the theory of cohomological support varieties and rank varieties (closed conical subsets of $\mathcal{N}_p({g})$), as expounded by Friedlander-Parshall and Jantzen.

## Modular invariant theory of finite groups and Galois ring extensions

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.

## Representations and cohomology of general linear groups and symmetric groups

This is a report on joint work with Haralampos Geranios. The connection between representations of general linear groups and symmetric groups goes back to Schur's Thesis in 1901, where it was used to describe the representations of general linear groups over fields of characteristic $0$. Much can still be gained from this connection in positive characteristic, especially as there are now many methods available on the general linear side not available to Schur, such as high weight theory, cohomology theory and the algebraic geometry of quotient spaces.

## On a relative version of Serre's notion of G-complete reducibility

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.

## Homogeneous vector bundles over abelian varieties via representation theory

Let $A$ be an abelian variety over a field. The homogeneous (or translation-invariant) vector bundles over $A$ form an abelian category; the Fourier-Mukai transform yields an equivalence of this category with that of coherent sheaves with finite support on the dual abelian variety. The talk will present an alternative approach to the category of homogeneous vector bundles, based on an equivalence with the category of finite-dimensional representations of a commutative affine group scheme (the "affine fundamental group" of $A$).

## First cohomology of Frobenius kernels and truncated invariants

$k$ 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 $k[{g}]$.

## Tangle categories and endomorphism algebras of certain infinite dimensional $U_q(\mathrm{sl}_2)$ modules

We demonstrate an equivalence of categories between a certain categories of infinite dimensional representations of quantum $\mathrm{sl}_2$ and two parametrised versions of the Temperley-Lieb category of type $B$. This may be used to discuss semi-simplicity. This is joint work with K. Iohara and R. Zhang.

## On the cohomology of locally symmetric spaces

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.

## Étale and affine representations of Lie algebras and algebraic groups

A rational representation $p : G \to \mathrm{Aff}(V )$ of an algebraic group $G$ for the affine group over a complex vector space $V$ is called an étale affine representation of $G$ if there exists a point $v \in V$ such that the orbit $\rho(G)v$ is Zariski-open in $V$ and the isotropy group $G_v$ at $v$ is a finite group.

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