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

TL;DR

This paper introduces a new abelian category for studying complex representations of reductive algebraic groups with Frobenius maps, classifies simple objects, and establishes its highest weight structure.

Contribution

It constructs the category al X(G), classifies its simple objects, and proves it has a highest weight category structure, advancing understanding of these representations.

Findings

Classification of all simple objects in al X(G)

Proof that al X(G) is a highest weight category

Framework for analyzing complex representations with Frobenius maps

Abstract

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