Let $\mathbb{F}_q$ be a finite field with $q=p^r$ and $p$ prime. Let $G$ be a connected reductive group over $\mathbb{F}_q$. Is there a difference between the theory of unipotent cuspidal representation of $G(\mathbb{F}_q)$ when $p$ is large and when $p$ is small?

For example, B. Srinivasan constructed the unipotent cuspidal representation $\theta_{10}$ of $Sp_4(\mathbb{F}_q)$ when $p>3$. What happened when $p=2$? Actually, when $p=2$, the irreducible representation of $Sp_4(\mathbb{F}_q)$ was classified by Enomoto. Is there a $\theta_{10}$ representation in this case? Enomoto does not use $\theta_{10}$ notation, but I suspect his $\theta_5$ is the unipotent cuspidal representation. Can someone confirm?

In Carter's book, page 460, there is a table for unipotent cuspidal representations of $G_2(\mathbb{F}_q)$, i.e., $G_2[1], G_2[-1], G_2[\theta], G_2[\theta^2]$. Is there a requirement for $p$ in Carter's notation? According my understanding, when $p>3$, the irreducible representations of $G_2[\mathbb{F}_q]$ are classified by Chang-Ree; when when $p=2,3$, this was done by Enomoto. Could some one provide a comparison of Carter's notations with Change-Ree's notations (should be $X_{17}, X_{18}, X_{19}, \overline{X}_{19}$) and Enomoto's notations?

It looks like the general theory of representations of $G(\mathbb{F}_q)$ can be very different since the group structure are different. On the other hand, representations of $G(\mathbb{F}_q)$ should be classified via ${}^L(G)$ (I believe this was proved by Lusztig), which is a complex group. Thus there is no difference on the geometric side for $p$ small or large.

This question is not of research level. If it is not appropriate here, I will delete it.

  • 1
    $\begingroup$ I think in Lusztig's classification the dual group is defined over F_q, not the complex field. $\endgroup$
    – user148212
    Apr 21, 2020 at 18:41


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