Invariant forms on irreducible modules of simple algebraic groups

Mikko Korhonen (EPFL)

Frank Adams 1,

Let \(G\) be a simple linear algebraic group over an algebraically closed field \(K\) of characteristic \(p\geq 0\) and let \(V\) be an irreducible rational \(G\)-module with highest weight \(\lambda\). When \(V\) 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 \(p\neq 2\), the answer is well known and easily described in terms of \(\lambda\). In the case where \(p=2\), we know that if \(V\) 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. I will present results which settle the problem for some families of \(\lambda\). For example, an answer can be given in the case where \(G\) is of classical type and \(\lambda\) is a fundamental highest weight.

Import this event to your Outlook calendar
▲ Up to the top