Certain complex representations of $S{L}_2({\bar{\mathbb{F}}}_q)$

We introduce the representation category $\mathcal{C}(\mathbf{G})$ for a connected reductive algebraic group $\mathbf{G}$ which is defined over a finite field ${\mathbb{F}}_q$ of $q$ elements. We show that this category has many good properties for $\mathbf{G}=S{L}_2({\bar{\mathbb{F}}}_q)$. In particular, it is an abelian category and a highest weight category. Moreover, we classify the simple objects in $\mathcal{C}(\mathbf{G})$ for ${\bf G}=SL_2(\bar{$\mathbb{F}$}_q)$.