Invariant forms on irreducible modules of simple algebraic groups

Let G be a simple linear algebraic group over an algebraically dosed field K of characteristic p >= 0 and let V be an irreducible rational G-module with highest weight A. When is self-dual, a basic question to ask is whether V has a non-degenerate G-invariant alternating bilinear form or a non degenerate G-invariant quadratic form. If p not equal 2, the answer is well known and easily described in terms of A. In the case where p = 2, we know that if is self-dual, it always has a non-degenerate G-invariant alternating bilinear form. However, determining when V has a non-degenerate G-invariant quadratic form is a classical problem that still remains open. We solve the problem in the case where G is of classical type and A is a fundamental highest weight omega(i), and in the case where G is of type A(i) and lambda = omega(r) + omega(s) for 1 <= r < s <= l. We also give a solution in some specific cases when G is of exceptional type. As an application of our results, we refine Seitz's 1987 description of maximal subgroups of simple algebraic groups of classical type. One consequence of this is the following result. If X < Y < SL(V) are simple algebraic groups and V down arrow X is irreducible, then one of the following holds: (1) V down arrow Y is not self-dual; (2) both or neither of the modules V down arrow Y and V down arrow X have a non-degenerate invariant quadratic form; (3) p = 2, X = SO(V), and Y = Sp(V). (C) 2017 Elsevier Inc. All rights reserved.