a question on Deligne-Lusztig characters

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.

• My guess is that this is too good to be true. – user125639 Sep 1 '18 at 1:10
• This is also false for $\theta_{10}$, Srinivasen's Sp(4,q) cuspidal unipotent representation. See, for example, Aubert's Complex Modular Representations of the Group Sp(4,q). It appears in two Deligne-Lusztig characters, both of which are composed of more than two irreducible representations, and in both there's a mixture of plus and minus signs. I'm pretty sure this is a minimal example with respect to rank. – Dror Speiser Sep 1 '18 at 21:34
• Your assertion "It is known that if T^F is anisotropic, then ±R_T,θ is cuspidal." is incorrect in general. (Already not true for SL_2.) – user148212 Sep 2 '18 at 10:20

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. 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 includes many details about the Deligne-Lusztig construction.)