  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.
