Representations of algebraic groups and principal bundles

Abstract

In this talk we discuss the relations between representations of algebraic groups and principal bundles on algebraic varieties, especially in characteristic $p$. We quickly review the notions of stable and semistable vector bundles and principal $G$-bundles, where $G$ is any semisimple group. We define the notion of a low height representation in characteristic $p$ and outline a proof of the theorem that a bundle induced from a semistable bundle by a low height representation is again semistable. We include applications of this result to the following questions in characteristic $p$: 1) Existence of the moduli spaces of semistable $G$-bundles on curves. 2) Rationality of the canonical parabolic for nonsemistable principal bundles on curves. 3) Luna's etale slice theorem. We outline an application of a recent result of Hashimoto to study the singularities of the moduli spaces in (1) above, as well as when these spaces specialize correctly from characteristic 0 to characteristic $p$. We also discuss the results of Laszlo-Beauville-Sorger and Kumar-Narasimhan on the Picard group of these spaces. This is combined with the work of Hara and Srinivas-Mehta to show that these moduli spaces are $F$-split for $p$ very large. We conclude by listing some open problems, in particular the problem of refining the bounds on the primes involved.