## Invariant forms on irreducible modules of simple algebraic groups

#### Mikko Korhonen (EPFL)

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.