Grothendieck group of the stack of G-Zips

TL;DR

This paper investigates the Grothendieck group of the algebraic stack of G-Zips associated with a reductive group over a finite field, providing a description under certain conditions.

Contribution

It offers a new description of the Grothendieck group of the G-Zip stack as a quotient of the Levi subgroup’s representation ring when the derived group is simply connected.

Findings

Grothendieck group described as a quotient of the Levi subgroup's representation ring

Provides explicit algebraic description under the simply connected derived group assumption

Advances understanding of K-theory of G-Zip stacks

Abstract

Given a connected reductive group G over the finite field of order p and a cocharacter of G over the algebraic closure of the finite field, we can define G-Zips. The collection of these G-Zips form an algebraic stack which is a stack quotient of G. In this paper we study the K-theory rings of this quotient stack, focusing on the Grothendieck group. Under the additional assumption that the derived group is simply connected, the Grothendieck group is described as a quotient of the representation ring of the Levi subgroup centralising the cocharacter.