= Solution
Write $\chi=\varepsilon R_T^G(\theta)$, where $\varepsilon\in\{1,-1\}$ and $\chi$ is the character of an <irreducible representation>. An irreducible representation is a <cuspidal representation of a finite reductive group> exactly when its <Harish-Chandra restriction> to every proper rational <Levi subgroup> of a rational <parabolic subgroup> is zero.
The needed identity is the <Harish-Chandra restriction of a Deligne-Lusztig character>. For a rational <parabolic subgroup> $P=M\ltimes V$, the torus case of the induction-restriction formula gives
$$
{}^*R_{M\subset P}^{G}R_T^G(\theta)=\sum_{x\in M^F\backslash\{g\in G^F:{}^gT\subset M\}/T^F}R_{{}^xT}^{M}({}^x\theta).
$$
Here ${}^x\theta(xtx^{-1})=\theta(t)$. The geometric meaning of the indexing condition is that only a rational conjugate of the inducing <maximal algebraic torus> lying in the Levi can survive taking $V^F$-invariants. This is the established torus case of <Deligne-Lusztig induction>, valid without a restriction on $q$; it is not an appeal to an unrestricted induction-restriction conjecture. See https://webusers.imj-prg.fr/~jean.michel/papiers/lectures_beijing_2015.pdf[Dudas and Michel's lectures, Theorem 14.4].
Suppose first that $T$ is an <elliptic maximal torus>, meaning it lies in no proper rational <parabolic subgroup>. For any proper rational $P=M\ltimes V$, the indexing set in the displayed sum is empty. Indeed ${}^xT\subset M$ with $x\in G^F$ would imply $T\subset{}^{x^{-1}}P$, itself a proper rational <parabolic subgroup>. Hence ${}^*R_{M\subset P}^{G}R_T^G(\theta)=0$, and multiplication by $\varepsilon$ gives ${}^*R_{M\subset P}^{G}\chi=0$. This proves cuspidality.
Conversely suppose $T\subset P$ for a proper rational <parabolic subgroup>. There is a unique <Levi subgroup> of $P$ containing this particular <maximal algebraic torus>: relative to $T$, it is generated by $T$ and those root subgroups for which both the root and its negative occur in $P$. Call it $M$. Uniqueness and $F(T)=T$, $F(P)=P$ imply $F(M)=M$, so this is a proper rational Levi. Transitivity of <Deligne-Lusztig induction> through this rational parabolic says
$$
R_T^G(\theta)=R_{M\subset P}^{G}\bigl(R_T^M(\theta)\bigr).
$$
Set $\psi=R_T^M(\theta)$, allowing it to be a <virtual character>. Adjunction of <Harish-Chandra induction> and <Harish-Chandra restriction>, extended linearly to <virtual characters>, gives
$$
\left\langle{}^*R_{M\subset P}^{G}\chi,\psi\right\rangle_{M^F}=\left\langle\chi,R_{M\subset P}^{G}\psi\right\rangle_{G^F}=\left\langle\chi,R_T^G(\theta)\right\rangle_{G^F}=\varepsilon\langle\chi,\chi\rangle_{G^F}=\varepsilon\ne0.
$$
Thus this <Harish-Chandra restriction> cannot vanish, so $\chi$ is not cuspidal. No positivity or irreducibility of $\psi$ is needed, and the sign in $\chi$ causes no cancellation problem. We have proved
$$
\boxed{\varepsilon R_T^G(\theta)\text{ irreducible is cuspidal}\quad\Longleftrightarrow\quad T\text{ is elliptic}.}
$$
The term <elliptic maximal torus> is relative to the ambient group: split central factors are allowed. The actual condition is absence of a proper rational <parabolic subgroup> containing $T$.
Back to article page
