Evaluation adjunction for unnormalized induction (${\mathrm{Hom}}_G(W,{\mathrm{Ind}}_B^G\xi )\cong {\mathrm{Hom}}_B(W,\xi )$)

Evaluation of an intertwiner at the identity gives a $B$-equivariant functional. Conversely a functional $\ell$ gives $A(v)(g)=\ell (\pi (g)v)$, which has the required covariance and is fixed on the right by any compact-open subgroup fixing $v$. These inverse constructions prove the displayed adjunction. If $\xi$ is trivial on the unipotent radical, it is equivalent to a character-valued functional on the Jacquet module.