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Let us consider a split adjoint simple group $G$ over $\overline{\mathbb{F}_q}$. Then we have a Frobenius map $F$, and we can consider a finite group of Lie type, $G^F$. (We can assume that the characteristic is very good if you want.)

Unipotent characters of $G^F$ are important materials, but I believe that they are mysterious. So my interest is almost unipotent characters, i.e., $$ \rho_\chi:= \frac{1}{|W|}\sum_{w\in W}\chi(w)R_w(1), $$ where $W$ is the Weyl group of $G$, $\chi$ an irreducible character of $W$ and $R_w$ the Deligne-Lusztig character over the $w$-twisted torus.

My question is the following. Are there any conditions on $\chi$ that make $\rho_\chi$ an actual unipotent character of $G^F$?

If $G=\mathrm{GL}_n$, it is well-known that every $\rho_\chi$ is actual unipotent character. However, this is not true in general.

I hope this is not a silly question, and I sincerely appreciate any of your comments in advance!

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    $\begingroup$ I guess "fake degree" is a keyword you may want to know in this context (but probably you already do know it!). $\endgroup$ Commented Sep 15 at 13:35
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    $\begingroup$ Lusztig has partitioned the set of irreducible characters of $W$ into families. Your almost character will be irreducible exactly when $\chi$ lies in a family by itself. As @SamHopkins says, you basically need to use the Hecke algebra to work this out. Your character must at least be special, so $a_{\chi} = b_{\chi}$, and I believe the characters you are looking for are exactly those special characters whose generic denominator $n_{\chi}$ is $1$. $\endgroup$ Commented Sep 15 at 16:10
  • $\begingroup$ @SamHopkins I sincere appreciate your kind reply! I will study fake degree further more! $\endgroup$ Commented Sep 17 at 6:25
  • $\begingroup$ @JayTaylor Thank you for your reply very much! Your comment is exactly what I want. I will study this more. $\endgroup$ Commented Sep 17 at 6:26

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