| Résume | Abstract: An automorphism $\theta$ of order $m$ of a simple Lie algebra $\mathfrak{g}$ can alternatively be thought of as a $Z/mZ$-grading $\mathfrak{g} = \mathfrak{g}(0) + \mathfrak{g}(1) + \ldots + \mathfrak{g}(m-1)$, where $\mathfrak{g}(0)$ is the set of fixed points for the automorphism. Hence $\mathfrak{g}(0)$ acts on each of the subspaces $\mathfrak{g}(i)$. It was established by Vinberg in characteristic zero that the corresponding representations for $\mathfrak{g}(0)$ have many of the well-known properties of the adjoint representation of $\mathfrak{g}$. In particular, the ring of invariants is isomorphic to a polynomial ring, closed orbits are given by semisimple elements of $\mathfrak{g}(i)$, and the ``little Weyl group'' is a complex reflection group. In this talk I will give an overview of the above results and explain how they can be extended to the case where the ground field is of characteristic $p$. Subsequently I will try to outline how these results can be applied to study the representation theory of reductive groups over finite extensions of the $p$-adic numbers. |
