# The Principal Representations of Reductive Algebraic Groups with Frobenius Maps

## TL;DR

This paper introduces a new principal representation category for reductive algebraic groups with Frobenius maps, conjectures it is a highest weight category, and provides evidence and related algebraic structures.

## Contribution

It proposes the principal representation category $ ext{O}(f G)$ for reductive algebraic groups with Frobenius maps and conjectures its highest weight structure, supported by initial evidence.

## Key findings

- Conjecture that $ ext{O}(f G)$ is a highest weight category.
- Evidence provided for the conjecture in the case of complex fields.
- Study of bound quiver algebras related to the principal representation category.

## Abstract

We introduce the principal representation category $\mathscr{O}({\bf G})$ of reductive algebraic groups with Frobenius maps and put forward a conjecture that this category is a highest weight category. When $\Bbbk$ is complex field $\mathbb{C}$, we provide some evidences of this conjecture. We also study certain kind of bound quiver algebras whose representations are related to the principal representation category $\mathscr{O}({\bf G})$ .