Using as

We just introduced , and now the idea is to try to substitute into our covariant ansatz (3).

It turns out that this works for some class of representations (but not all). We will explain later what is the condition on , but for now just proceed with the general for as long as we can.

We want to apply the ansatz of the form (3) for . This means that we need to find a fixed vector representing a cohomology class of the covariant complex (6).

Since is in , we have to start with understanding the structure of .