Is the Adjoint Action self dual over finite fields?

Question

Given a finite group $G$, and representation $\rho: G \to H(\mathbb{F}_q)$ where $H$ is some classical algebraic group ($Gl$, $Sl$, $O$, $SO$, $SP$, $GSP$, $U$, etc), is the induced Adjoint representation:

$$Ad(\rho): G \to End(\mathfrak{h}(\mathbb{F}_q))$$ $$Ad(\rho)(g)\circ x = \rho(g)x\rho(g^{-1})$$

self dual? (i.e. $Ad(\rho) \cong Ad(\rho)^*$).

Does it depend on the group? In particular, I care about $GSp_4$, but the general question is interesting as well. A follow up question (which might be easier to answer): If $Ad(\rho)$ as above is self dual (or not), are the $G$-equivariant subspaces of $\mathfrak{h}(\mathbb{F}_q)$ self dual?

Answer 1

To expand on Marty's brief comment, self-duality is not possible for any $p$ and for many if not most finite groups. (Also, your tag 'lie-groups' is not meaningful here, since a finite group of rational points requires starting with an algebraic group. It does turn out after a lot of work that semisimple complex Lie groups are essentially the same as semisimple algebraic groups over $\mathbb{C}$, but for this question you need tags 'finite-groups' and 'algebraic-groups' rather than 'finite-fields' and 'lie-groups'.)

Starting with a classical algebraic group $H$ on your list (or even a Spin group), the question makes sense over any algebraically closed field such as an algebraic closure of a finite field. In this situation, the adjoint representation of $H$ has been thoroughly explored in older work by Hogeweij and Hiss. As in characteristic 0, $H$ has a finite center equal to the kernel of the adjoint representation. But the Lie algebra sometimes has a center of positive dimension, e.g., $\mathfrak{sl}_n$ when the characteristic $p$ divides $n$. Aside from this possibility, the modular representations of an arbitrary finite group are typically not self-dual---indeed, the module obtained is often indecomposable but not irreducible. There are lots of examples for finite classical groups $G$ of Lie type, even when they are embedded in the rational points of a suitable $H$ over a finite field.

An easy example to keep in mind involves the principal series representations of the finite group of Lie type $G=\mathrm{SL}_2(\mathbb{F}_p)$: here you typically have two composition factors, with a nonsplit extension, and the dual module has the same two (self-dual!) composition factors but arranged in the opposite order.

Note that Jantzen has worked out many details for such principal series in a more general setting here.