# Alvis-Curtis Duality for Representations of Reductive Groups with Frobenius Maps

## TL;DR

This paper extends the Alvis-Curtis duality concept to abstract representations of reductive groups with Frobenius maps, showing it interchanges irreducible representations similarly to finite cases.

## Contribution

It introduces a generalized Alvis-Curtis duality for infinite-type representations of reductive groups with Frobenius maps, expanding its applicability.

## Key findings

- Duality interchanges irreducible representations in principal categories
- Generalization applies to infinite-type representations
- Aligns with known finite reductive group results

## Abstract

We generalize the Alvis-Curtis duality to the abstract representations of reductive groups with Frobenius maps. Similar to the case of representations of finite reductive groups, we show that the Alvis-Curtis duality of infinite type which we define in this paper also interchanges the irreducible representations in the principal representation category.