  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 ${\bf
G}=SL_2(\bar{\mathbb{F}}_q)$.
