Harish-Chandra restriction (${}^{\ast }{R}_{L\subset P}^G(V)=V^{U^F}$)

Restriction first views a $G^F$-module as a $P^F$-module, then takes the subspace fixed by the unipotent radical. The resulting $L^F$-module is adjoint to Harish-Chandra induction by ordinary Frobenius reciprocity. Over characteristic zero, averaging makes invariants exact and identifies them with coinvariants.

An irreducible complex $G^F$-module is cuspidal if its Harish-Chandra restriction is zero for every proper rational parabolic subgroup. By adjunction, it occurs in no Harish-Chandra induction from a proper rational Levi subgroup. This is a finite-group notion, distinct from a cuspidal automorphic representation.