# cuspidal unipotent representation in small characteristic

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.

• I think in Lusztig's classification the dual group is defined over F_q, not the complex field. Apr 21 '20 at 18:41