Harish-Chandra induction ($R_{L\subset P}^G$)

For a rational parabolic subgroup $P=L⋉U$, use inflation of a group representation for a complex $L^F$-module across $P^F\to L^F$ and induce it to $G^F$. This is Harish-Chandra induction. Its adjoint, Harish-Chandra restriction, takes $U^F$-invariants. These are ordinary finite-group group representation functors and should not be confused with general Deligne-Lusztig induction from a nonrational Borel.

For rational parabolic subgroups $P=LU$ and $Q=MV$, define $\mathcal{S}(M,L)=\{x:M\cap {}^{x}L\text{ contains a maximal algebraic torus of }G\}$. Then ${}^{\ast }{R}_M^G{R}_L^G$ is the sum over $M^F\backslash \mathcal{S}(M,L{)}^F/L^F$ of $R_{M\cap {}^{x}L}^M{}^{\ast }{R}_{M\cap {}^{x}L}^{{}^{x}L}\mathrm{ad}x$, with the intersection parabolics understood. It follows from ordinary finite-group Mackey restriction formula, taking $V^F$-invariants, and the compatible-Levi intersection lemma for parabolics.

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.