N35 Induced representations and Frobenius reciprocity
In progress: these notes stop partway.
Setup: , a representation of , and a subspace invariant under . Let be elements of each coset of in (one representative per coset).
Definition N35.1. is induced from if
So
For every and every , there is a and an such that ; then maps to :
Frobenius reciprocity
Theorem N35.2 (Frobenius reciprocity). If is a representation of and is a representation of , then every -equivariant map extends uniquely to a -equivariant map . So
which is a natural map in the category-theory sense.
Proof. Let , and let be -equivariant. We want that is -equivariant with . Then is determined for all and all :, so
∎