 Harish-Chandra restriction of a Deligne-Lusztig character

For a rational Levi subgroup $M$ of a rational parabolic subgroup and a rational maximal torus $T$, the torus case of the induction-restriction formula isIn particular it vanishes if there is no rational conjugate of $T$ in $M$. If $T\subset M$, transitivity gives $R_T^G=R_M^G{R}_T^M$. Together with adjunction these identities show that an irreducible character up to sign arising from $T$ is cuspidal exactly when $T$ is elliptic.