Let $k$ be a finite field and $\bar k$ be its algebraic closure, and $F$ be the Frobenius map. Let $G$ be a reductive group over $\bar k$, $T$ be an $F$-invariant maximal torus of $G$, and $\theta$ be a character of $T^F$. Then there is a Deligne-Lusztig character $R_{T,\theta}$ of $G^F$. It is known that if $T^F$ is anisotropic, then $\pm R_{T,\theta}$ is cuspidal.

Question: Given an irreducible cuspidal representation $\rho$ of $G^F$, is it true that one can find an anisotropic $T^F$, a character $\theta$ of $T^F$, and another real representation (not virtual representation in Grothendieck group, but a true representation) $\rho'$ such that $\rho+\rho'=\pm R_{T,\theta}$? (In general, $R_{T,\theta}$ is an alternating sum of true representations, right?)

Thanks in advance. If this question is too simple to be here, I will remove it.

Let $k$ be a finite field and $\bar k$ be its algebraic closure, and $F$ be the Frobenius map. Let $G$ be a reductive group over $\bar k$, $T$ be an $F$-invariant maximal torus of $G$, and $\theta$ be a character of $T^F$. Then there is a Deligne-Lusztig character $R_{T,\theta}$ of $G^F$. It is known that if $T^F$ is anisotropic, then $\pm R_{T,\theta}$ is cuspidal.

Given an irreducible cuspidal representation $\rho$ of $G^F$, is it true that one can find an anisotropic $T^F$, a character $\theta$ of $T^F$, and another real representation (not virtual representation in Grothendieck group, but a true representation) $\rho'$ such that $\rho+\rho'=\pm R_{T,\theta}$? (In general, $R_{T,\theta}$ is an alternating sum of true representations, right?)

Thanks in advance. If this question is too simple to be here, I will remove it.