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.

  • 1
    $\begingroup$ My guess is that this is too good to be true. $\endgroup$
    – user125639
    Commented Sep 1, 2018 at 1:10
  • 1
    $\begingroup$ 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. $\endgroup$ Commented Sep 1, 2018 at 21:34
  • 1
    $\begingroup$ 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.) $\endgroup$
    – user148212
    Commented Sep 2, 2018 at 10:20

1 Answer 1


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.)

  • 2
    $\begingroup$ Deligne--Lusztig characters are indeed defined to be alternating sums of true characters.They may not be alternating sums of distinct irreducible characters, but the OP didn't claim this. $\endgroup$ Commented Sep 2, 2018 at 8:23
  • 1
    $\begingroup$ @AStasinski That does seem like an odd phrasing though, seeing as any virtual character is such an alternating sum (since it is just a difference of characters). $\endgroup$ Commented Sep 3, 2018 at 9:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.