000228625 001__ 228625 000228625 005__ 20181203024721.0 000228625 0247_ $$2doi$$a10.1016/j.jalgebra.2017.02.031 000228625 022__ $$a0021-8693 000228625 02470 $$2ISI$$a000400219200017 000228625 037__ $$aARTICLE 000228625 245__ $$aInvariant forms on irreducible modules of simple algebraic groups 000228625 260__ $$aSan Diego$$bElsevier$$c2017 000228625 269__ $$a2017 000228625 300__ $$a38 000228625 336__ $$aJournal Articles 000228625 520__ $$aLet 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. 000228625 6531_ $$aAlgebraic groups 000228625 6531_ $$aChevalley groups 000228625 6531_ $$aClassical groups 000228625 6531_ $$aQuadratic forms 000228625 6531_ $$aRepresentation theory of algebraic groups 000228625 700__ $$0248357$$aKorhonen, Mikko$$g243747 000228625 773__ $$j480$$q385-422$$tJournal Of Algebra 000228625 909C0 $$0252563$$pGR-TES$$xU12576 000228625 909CO $$ooai:infoscience.tind.io:228625$$pSB$$particle 000228625 917Z8 $$x133751 000228625 937__ $$aEPFL-ARTICLE-228625 000228625 973__ $$aEPFL$$rREVIEWED$$sPUBLISHED 000228625 980__ $$aARTICLE