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In this paper we introduce an abelian category $\mathscr{X}(\bf G)$ to studythe complex representations of a reductive algebraic groups $\bf G$ withFrobenius map. We classify all the simple objects in $\mathscr{X}(\bf G)$ andshow that this category is a highest weight category.

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