Certain complex representations of $SL_2(\bar{\mathbb{F}}_q)$

We introduce the representation category $\mathscr{C}({\bf G})$ for a connected reductive algebraic group ${\bf G}$ which is defined over a finite field $\mathbb{F}_q$ of $q$ elements. We show that this category has many good properties for ${\bf G}=SL_2(\bar{\mathbb{F}}_q)$. In particular, it is an abelian category and a highest weight category. Moreover, we classify the simple objects in $\mathscr{C}({\bf G})$ for ${\bf G}=SL_2(\bar{\mathbb{F}}_q)$.