Why there is nothing in of

Let us study the representations of SUGRA field configurations. Bosonic SUGRA fields are various tensor fields. Let us for a moment forget about the equations of motion, and even about the gauge transformations. Under the algebra of rotations of the , the SUGRA tensor fields form some representation . We observe:
  • A tensor field transforms in some representation of the little algebra and the corresponding component of is the so-called “induced representation”:

  • The Frobenius reciprocity is a relation between the intertwining operators:


Let us pause and understand this relation a bit better.