On the ansatz (3) the BRST operator acts as follows:

    (6)

The cohomology of this complex will be denoted . This depends on the representation .

A representation is present in the spectrum of massless SUGRA on if and only if is nontrivial. This is determined in terms of the structure of the representation.

I will now give an explicit construction of for some class of representations of , a subclass of the so-called Kac modules.

Kac modules are particular cases of the induced representation which I will now review.