= Harish-Chandra restriction of a Deligne-Lusztig character
{c}
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 is
$$
{}^*R_M^G R_T^G(\theta)=\sum_{x\in M^F\backslash\{g\in G^F:{}^gT\subset M\}/T^F}R_{{}^xT}^{M}({}^x\theta).
$$
In 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.
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