Theorem 15.3.1 (Frobenius reciprocity). Let be a representation of and a representation of . Then every -equivariant map extends uniquely to a -equivariant map . That is,
Proof. Choose coset representatives and write .
Uniqueness. Given , if is -equivariant with , then for any : where the last acts via the -action on . This determines on each summand, hence on all of .
Existence and -equivariance. Define . To verify -equivariance: let act on by sending to where for . For : using -equivariance of in the third equality and in the fourth. So is -equivariant.
Conversely, given , its restriction to is -equivariant. These two maps are inverse to each other. ∎
Corollary 15.3.1 (Frobenius reciprocity (character form)). For any representation of and of ,
Note. Thus, if is an irreducible representation of and an irreducible representation of , the number of times appears in equals the number of times appears in .