Hesselink normal forms of unipotent elements in some representations of classical groups in characteristic two

Abstract

Let G be a simple linear algebraic group over an algebraically closed field K of characteristic two. Any non-trivial self-dual irreducible K[G]-module W admits a non-degenerate G-invariant alternating bilinear form, thus giving a representation f:G→Sp(W). In the case where G=SLn(K) and W has highest weight ϖ1+ϖn−1, and in the case where G=Sp2n(K) and W has highest weight ϖ2, we determine for every unipotent element u∈G the conjugacy class of f(u) in Sp(W). As a part of this result, we describe the conjugacy classes of unipotent elements of Sp(V1)⊗Sp(V2) in Sp(V1⊗V2).