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. 



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



a question on Deligne-Lusztig characters
No, your parenthetic comment at the end indicates some confusion about the nature of Deligne-Lusztig virtual (= generallized) characters: these are defined to be $\mathbb{Z}$-linear combinations of actual characters, not necessarily "alternating sums" (meaning coefficients are $\pm 1$). Already in 1974 Chang-Ree (before the publication of the 1976 Deligne-Lusztig paper) followed the pattern of Ian Macdonald's conjectural program in their study of the characters of the finite groups $G\_2(q)$ [*here*](https://mathscinet.ams.org/mathscinet-getitem?mr=0364419).
Their four ordinary irreducible characters denoted $X\_{17}, X\_{18}, X\_{19}, \overline{X}\_{19}$ turn out to be cuspidal and don't combine in a simple way. Later Lusztig himself worked out similar but more complicated cuspidal characters. (Roger Carter's 1985 treatise [*here*](https://mathscinet.ams.org/mathscinet-getitem?mr=794307) includes many details about the Deligne-Lusztig construction.)


