Let $S$ be a simple group of Lie type defining over $\mathbb{F}_q$ where $q$ is a power of a prime $p$ and let $\sigma$ a nontrivial field automorphism of $S$. Set $\mathbb{F}_{q_0}$ is the fixed field of $\sigma$.

Question 1. Is there a formula for $|\mathrm{Irr}(S)|$?

Question 2. Write $S={}^d\Sigma(q)$. Is it true that $\mathrm{C}_{S}(\sigma)={}^d\Sigma(q_0)$? Is it true that $|\mathrm{Irr}(\mathrm{C}_{S}(\sigma))|<|\mathrm{Irr}(S)|$?