Unbounding Ext

We produce examples in the cohomology of algebraic groups which answer two questions of Parshall and Scott. Specifically, if $G=SL_2$, then we show: (a) \dim \Ext_G^2(L,L) can be arbitrarily large for a simple module $L$; and (b) the sequence $\max_{L-\text{irred}}\dim H^k(G,L)$ grows exponentially fast with $k$.Comment: 14 pages; version to appear in J. Al

On the minimal modules for exceptional Lie algebras: Jordan blocks and stabilisers

Let G be a simple simple-connected exceptional algebraic group of type G_2, F_4, E_6 or E_7 over an algebraically closed field k of characteristic p>0 with \g=Lie(G). For each nilpotent orbit G.e of \g, we list the Jordan blocks of the action of e on the minimal induced module V_min of \g. We also establish when the centralisers G_v of vectors v\in V_min and stabilisers \Stab_G of 1-spaces \subset V_min are smooth; that is, when \dim G_v=\dim\g_v or \dim \Stab_G=\dim\Stab_\g.Comment: This contains corrections and should be used instead of the published versio

Classification of the maximal subalgebras of exceptional Lie algebras over fields of good characteristic

Let $G$ be an exceptional simple algebraic group over an algebraically closed field $k$ and suppose that the characteristic $p$ of $k$ is a good prime for $G$. In this paper we classify the maximal Lie subalgebras $\mathfrak{m}$ of the Lie algebra $\mathfrak{g}={\rm Lie}(G)$. Specifically, we show that one of the following holds: $\mathfrak{m}={\rm Lie}(M)$ for some maximal connected subgroup $M$ of $G$, or $\mathfrak{m}$ is a maximal Witt subalgebra of $\mathfrak{g}$, or $\mathfrak{m}$ is a maximal \it{\mbox{exotic semidirect product}}. The conjugacy classes of maximal connected subgroups of G are known thanks to the work of Seitz, Testerman and Liebeck--Seitz. All maximal Witt subalgebras of $\mathfrak{g}$ are $G$-conjugate and they occur when $G$ is not of type ${\rm E}_6$ and $p-1$ coincides with the Coxeter number of $G$. We show that there are two conjugacy classes of maximal exotic semidirect products in $\mathfrak{g}$, one in characteristic $5$ and one in characteristic $7$, and both occur when $G$ is a group of type ${\rm E}_7$.Comment: This version is accepted for publication in Journal of the American Mathematical Society; 40 page