Orthogonal, symplectic and unitary representations of finite groups

Let $K$ be a field, $G$ a finite group, and $\rho: G \to \mathbf{GL}(V)$ a linear representation on the finite dimensional $K$-space $V$. The principal problems considered are:

I. Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms $h: V \times V \rightarrow K$ which are $G$-invariant.

II. If $h$ is such a form, enumerate the equivalence classes of representations of $G$ into the corresponding group (orthogonal, symplectic or unitary group).

III. Determine conditions on $G$ or $K$ under which two orthogonal, symplectic or unitary representations of $G$ are equivalent if and only if they are equivalent as linear representations and their underlying forms are ``isotypically'' equivalent.

This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) $KG$-module components of their spaces are equivalent.

We assume throughout that the characteristic of $K$ does not divide $2\vert G\vert$.

Solutions to I and II are given when $K$ is a finite or local field, or when $K$ is a global field and the representation is ``split''. The results for III are strongest when the degrees of the absolutely irreducible representations of $G$ are odd - for example if $G$ has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index - and, in the case that $K$ is a local or global field, when the representations are split.