# Characters of tori in finite reductive group

Let $$G$$ be a connected split reductive group over a finite field $$k$$. Suppose $$G$$ has connected centre. Let $$T$$ be a maximal split torus with Weyl group $$W$$. Note that $$W$$ acts on the finite group $$T(k)$$; thus, it acts on characters of $$T(k)$$.

Let $$\theta: T(k) \rightarrow \mathbb{C}^\times$$ be a $$W$$-invariant character.

Question: Does $$\theta$$ extend to a character of $$G(k)$$?

• Isn't the order $2$ character of the split torus of $SL_2(\mathbb F_p)$ a counterexample? Jun 9, 2022 at 23:08
• But the centre of SL_2 is disconnected (in characteristic not equal to 2). Jun 10, 2022 at 0:25
• What about $\textbf{PGL}_2$, then? Jun 10, 2022 at 1:57
• I think the order 2 character of maximal torus of PGL_2 actually extends to all of PGL_2(k). (Note abelianization of PGL_2(k) is nontrivial) Jun 10, 2022 at 6:28
• @Dr.Evil, indeed, for $p \ne 2$ (otherwise there is no non-trivial character!) the desired extension is $g \mapsto \operatorname{sgn}_k(\det(g))$. Jun 12, 2022 at 13:06

Edit. The answer below is incorrect. The correct computation is the determinant of the Cartan matrix, which happens to equal $$1$$ for $$E_8$$.
I believe the answer is "no" for $$G$$ equal to the split form of $$E_8$$ over a finite field $$k$$ of odd characteristic. The set of rational points is a simple finite group. The center of $$E_8$$ is trivial. For a maximal split torus $$T$$, I believe the action of the Weyl group $$W$$ on $$T(k)/(T(k))^2$$ is trivial, since $$W$$ is generated by simple reflections arising from copies of $$\textbf{SL}_2$$ in the group (the "root groups"), and the action of the Weyl group is trivial for $$\textbf{SL}_2$$. So this would mean that every nontrivial character of $$T(k)/(T(k))^2$$ gives a counterexample.
• Thanks. I'm not following the argument that $W$ acts trivially $T(k)/(T(k))^2$. I would appreciate more details. Jun 13, 2022 at 0:04