On massive higher spin supermultiplets in d=4

In this work we discuss the cubic interactions for massless spin 3/2 gravitino with massive higher spin supermultiplets using three superblocks (2,3/2), (5/2,2) and (3,5/2) as the first non-trivial examples. We use gauge invariant formalism for the massive higher spin fields and, as it common in such cases, we face an ambiguity related with the possible field redefinitions due to the presence of Stueckelberg fields. From one hand, we show how this ambiguity can be used as one more way to classify possible cubic vertices. We also note that all these field redefinitions do not change the part of the Lagrangian which appears in the unitary gauge (where all Stueckelberg fields are set to zero) so we still have some important independent results. From the other hand, we show how using the so-called unfolded formalism one can fix these ambiguities and obtain consistent deformations for all massive field gauge invariant curvatures which is the most important step in the Fradkin-Vasiliev formalism. Unfortunately, this works for the massive fields only so the way to construct deformations for the massless field curvatures is still has to be found.