Geodesic
Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups
Canadian journal of mathematics, Tome 68 (2016) no. 2, pp. 395-421

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DOI

The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the questions of whether a given complex representation is symplectic or orthogonal have been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from 2 hold by using similar proofs. This paper considers analogues of these results for simple, induced, and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.
DOI : 10.4153/CJM-2015-042-5
Mots-clés : 20G05, 11E39, 11E88, 15A63, 20G15, orthogonal representations, symmetric tensors, alternating forms, characteristic 2, splitreductive groups 
Garibaldi, Skip; Nakano, Daniel K. Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups. Canadian journal of mathematics, Tome 68 (2016) no. 2, pp. 395-421. doi: 10.4153/CJM-2015-042-5
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     title = {Bilinear and {Quadratic} {Forms} on {Rational} {Modules} of {Split} {Reductive} {Groups}},
     journal = {Canadian journal of mathematics},
     pages = {395--421},
     year = {2016},
     volume = {68},
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