In my new preprint Bilinear and quadratic forms on rational modules of split reductive groups, written with Daniel K. Nakano, we extend the beautiful and classical theory of orthogonal and symplectic representations of split reductive groups in characteristic zero to prime characteristic. The two main new technical challenges are that orthogonal representations in characteristic 2 are connected with quadratic forms rather than bilinear forms, and that there are four reasonable generalizations of the characteristic zero notion of irreducible representation (irreducible representations and standard, Weyl, and tilting modules).

My favorite result in our paper is for Weyl modules over fields of characteristic 2. If the complex irrep with the same highest weight is self-dual, then the Weyl module is orthogonal in characteristic 2, except for a unique (up to some sense of uniqueness) exception, the natural representation of the symplectic group.