A bilinear form on a representation is -invariant if for all and . Recall from Chapter 9 that a linear map determines the bilinear form , that every bilinear form arises in this way from exactly one , and that is nondegenerate if and only if is an isomorphism.
Lemma 16.4.1 (Invariant forms and equivariant maps). Let carry the dual representation, . The bilinear form determined by is -invariant if and only if is -equivariant. So the -invariant bilinear forms on correspond to the elements of .
Proof. For and , we have and . So is equivariant if and only if for all ; substituting , this says for all .∎
Proposition 16.4.2 (Invariant forms on an irreducible representation). Let be an irreducible representation of .
if and only if is real-valued.
If is not real-valued, the only -invariant bilinear form on is .
If is real-valued, has a nonzero -invariant bilinear form , unique up to a scalar factor. Every nonzero -invariant bilinear form on is nondegenerate, and is either symmetric () or skew-symmetric ().
Proof. (i) We have (Proposition 13.4.3), and two representations are isomorphic if and only if they have the same character (Corollary 13.5.6). So if and only if .
(ii) and (iii) The dual is irreducible, since . By Lemma 16.4.1, invariant forms correspond to elements of . By Schur’s lemma this space is if . If is an isomorphism of representations, then every satisfies , so is 1-dimensional. A nonzero is an isomorphism by Schur’s lemma, so every nonzero invariant form is nondegenerate. (Directly: is a subrepresentation, since , and it is not all of when , so it is .)
Finally, write with Both (symmetric) and (skew-symmetric) are -invariant bilinear forms, so and for some with . If and were both nonzero, then would be symmetric and skew-symmetric, so and , a contradiction. Hence or .∎
Proposition 16.4.3 (Real if and only if symmetric). An irreducible complex representation of is real if and only if carries a nondegenerate -invariant symmetric bilinear form .
Proof.() Let . By averaging, has a -invariant inner product . Extend it -bilinearly: This is a symmetric, -invariant, -bilinear form on . It is nondegenerate: if for all , then , so for all , and .
() Choose a -invariant Hermitian inner product on (Lemma 13.2.1), conjugate-linear in the first variable. The map from to is antilinear and injective, hence bijective, and it is -equivariant: . The form gives a -equivariant linear isomorphism ,. Composing with the inverse of gives a -equivariant antilinear bijection , characterized by The map is -linear and -equivariant, so by Schur’s lemma for some . For , using the symmetry of in the third equality and (as is bijective) at the end. So is real and positive. The map is still antilinear and -equivariant, and because is real. By Proposition 16.3.1, is real. ∎