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Write $\chi =\epsilon R_T^G(\theta )$, where $\epsilon \in \{1,-1\}$ 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 $P=M⋉V$, the torus case of the induction-restriction formula givesHere ${}^x\theta (xt{x}^{-1})=\theta (t)$. 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 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 $P=M⋉V$, the indexing set in the displayed sum is empty. Indeed ${}^{x}T\subset M$ with $x\in G^F$ would imply $T\subset {}^{x^{-1}}P$, itself a proper rational parabolic subgroup. Hence ${}^{\ast }{R}_{M\subset P}^G{R}_T^G(\theta )=0$, and multiplication by $\epsilon$ gives ${}^{\ast }{R}_{M\subset P}^G\chi =0$. 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 saysSet $\psi =R_T^M(\theta )$, allowing it to be a virtual character. Adjunction of Harish-Chandra induction and Harish-Chandra restriction, extended linearly to virtual characters, givesThus 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 provedThe 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 $T$.