Complex representations of reductive algebraic groups with Frobenius maps in the category $\mathcal{X}$

In this paper we introduce an abelian category $\mathcal{X}(\mathbf{G})$ to study the complex representations of a reductive algebraic groups $\mathbf{G}$ with Frobenius map. We classify all the simple objects in $\mathcal{X}(\mathbf{G})$ and show that this category is a highest weight category.