Can transform in a Kac module?The answer is no. Suppose that it was a Kac module. For example, consider the following state in the Kac module which would have corresponded to :
Here is an element of the universal enveloping . The corresponding vertex is constant in AdS, because all the AdS (greek) indices are contracted. At the same time it transforms as a 5-tensor under the rotations of . There are no such deformations in SYM. Therefore it must be true that this element of the Kac module gives a BRST-exact vertex. But we have not checked it explicitly.
We suspect that the representation of is a factormodule of the Kac module, but probably not an irreducible representation.