INTRODUCTION

In, Dynkin, Seitz and Testerman classified the maximal closed connected subgroups of the simple algebraic groups over an algebraically closed field K of characteristic p ≥ 0. The hard part of their analyses, for subgroups of the groups of classical type, concerns an irreducible, closed, connected subgroup G of SL(V) for some K-vector space V. They determine explicitly all possibilities for closed connected overgroups Y of G in I(V) (where I(V) = SL(V), SO(V), or Sp(V) depending upon the form on V preserved by G); the results appear in tables giving the high weights of the modules V|G and V|Y.

The question of inclusion relations among irreducible subgroups of SL(V), in addition to having implications for the subgroup structures of classical groups, is of interest in its own right. In this paper we present some recent results concerning this question when we allow subgroups that are not connected (the full proofs may be found in). Specifically, in Sections 2 and 3 we discuss the structure and some of the methods of the proof of Theorem 1. Let G be a non-connected algebraic group with simple identity component X. Let V be an irreducible KG-module with restricted X-high. weight(s).