Overgroups of Irreducible Linear Groups, I

In work spread over several decades, Dynkin ([4, 3]), Seitz ([10, 11]), and Testerman ([16]) classified the maximal closed connected subgroups of simple algebraic groups. Their analyses for the classical group cases were based primarily on a striking result: If G is a simple algebraic group and φ : G SL V is a tensor indecomposable irreducible rational representation, then with specified exceptions the image of G is maximal among closed connected subgroups of one of the classical groups SL V , Sp V , or SO V . From a slightly different perspective, the question they answered was: Given an irreducible, closed, connected subgroup G of SL V for some vector space V , find all possibilities for closed, connected overgroups Y of G in SL V . This question of irreducible overgroups, or the restriction of irreducible modules to subgroups, appears in other contexts as well. In this paper we present some results in the absence of the connectedness requirement for the subgroup: The eventual goal is to classify all possible triples G Y V with G Aut Y both closed irreducible subgroups of SL V , Y SL V SO V , or Sp V , and Y a simple group of classical type. In this paper and [5], we give complete results for the case when G is not connected but has simple identity component X , and the TY -high weight and TX -high weights of V are restricted. Specifically, the papers are concerned with the proof of Theorem 1 below. 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).