Exceptional representations of simple algebraic groups in prime characteristic

Abstract

Let G be a simply connected simple algebraic group over an algebraically closed field K of characteristic p>0 with root system R, and let $\mathfrak{g}=\mathcal{L}(G)$ be its restricted Lie algebra. Let V be a finite dimensional $\mathfrak{g}$-module over K. For any point $v\inV$, the {\it isotropy subalgebra} of $v$ in $\mathfrak{g}$ is ${\mathfrak{g}}_v=\{x\in \mathfrak{g}/x\cdot v=0\}$. A restricted $\mathfrak{g}$-module V is called exceptional if for each $v\in V$ the isotropy subalgebra ${\mathfrak{g}}_v$ contains a non-central element (that is, ${\mathfrak{g}}_v⊈\mathfrak{z}(\mathfrak{g})$). This work is devoted to classifying irreducible exceptional $\mathfrak{g}$-modules. A necessary condition for a $\mathfrak{g}$-module to be exceptional is found and a complete classification of modules over groups of exceptional type is obtained. For modules over groups of classical type, the general problem is reduced to a short list of unclassified modules. The classification of exceptional modules is expected to have applications in modular invariant theory and in classifying modular simple Lie superalgebras.

162 pages, 11 tables, Thesis submitted to the University of Manchester for the degree of Doctor of Philosophy of the Faculty of Science, under the supervision of A. Premet