Bilinear and quadratic forms on rational modules of split reductive groups. (Q2798991)

Let \(G\) be a reductive and split algebraic group over a field \(k\). For a representation \(V\) of \(G\), let \(V^*\) denote the dual module, \(\text{Bil\,}V:=V^*\otimes V^*\) denote the bilinear forms on \(V\), and \(\text{Quad\,}V:=S^2(V^*)\) denote the quadratic forms on \(V\), where \(S^2(-)\) denotes the second symmetric power. When the characteristic is not 2, one has the decomposition \(\text{Bil\,}V\simeq\Lambda^2V^*\oplus\text{Quad\,}V\), where \(\Lambda^2(-)\) denotes the second exterior power. The main focus of this paper is on understanding the \(G\)-invariant bilinear or quadratic forms on certain fundamental \(G\)-modules in positive characteristic: irreducible modules, standard induced modules (from a Borel), Weyl modules, and tilting modules. Each of these families of modules is parameterized by the set of dominant weights. For a dominant weight \(\lambda\), denote the corresponding modules (respectively) by \(L(\lambda)\), \(H^0(\lambda)\), \(V(\lambda)\), \(T(\lambda)\). A related focal point of the paper is determining when a representation \(V\) is symplectic (i.e., \(\Lambda^2(V^*)^G\neq 0\)) or orthogonal (i.e., \((\text{Quad\,}V)^G\neq 0\)).NEWLINENEWLINE For an irreducible module, the question of whether \((\text{Bil\,}L(\lambda))^G\neq 0\) is known independent of characteristic, and using the structure of \(V(\lambda)\) (with \(L(\lambda)\) as its head), one has an isomorphism \((\text{Bil\,}V(\lambda))^G\simeq(\text{Bil\,}L(\lambda))^G\). The case of induced and tilting modules is more complex, and the authors give new and interesting results. For induced modules (i.e., an \(H^0(\lambda)\)) they relate this to cohomology over the associated Borel subgroup of \(G\), and, for tilting modules (i.e., a \(T(\lambda)\)), a formula is obtained involving the multiplicities of standard induced modules in a good filtration of a tilting module.NEWLINENEWLINE For quadratic forms, the case of Weyl modules has been known for characteristics other than 2. Here, in characteristic 2, the authors give a complete determination of when a Weyl module \(V(\lambda)\) is orthogonal and precisely identify \((\text{Quad\,}V(\lambda))^G\) in that case. They also give some sufficient conditions for an irreducible module \(L(\lambda)\) to be orthogonal; with the determination of necessary conditions (over any characteristic) remaining an open question.NEWLINENEWLINE The methods used in the paper are partially cohomological in nature. An outcome of this approach is that the authors are able to make computations of first cohomology groups \(\text{H}^1(G,\Lambda^2(V))\) when \(V\) is an irreducible, induced, or tilting module as above. Throughout the paper, the authors illustrate the ideas with examples and provide applications of their results.