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