Gerbal central extensions of reductive groups by $K_3$

TL;DR

This paper classifies certain central extensions of reductive groups by sheaves related to third Quillen K-theory, revealing they are parametrized by Weyl-invariant quadratic and cubic forms on the cocharacter lattice.

Contribution

It provides a complete classification of central extensions of reductive groups by $\\mathcal{K}_3$ and $B\mathcal{K}_3$, linking them to Weyl-invariant forms.

Findings

Extensions parametrized by Weyl-invariant quadratic forms

Extensions parametrized by Weyl-invariant cubic forms

Classification results for sheaves related to K-theory

Abstract

We classify central extensions of a reductive group $G$ by $\mathcal{K}_3$ and $B\mathcal{K}_3$, the sheaf of third Quillen $K$-theory groups and its classifying stack. These turn out to be parametrized by the group of Weyl-invariant quadratic forms on the cocharacter lattice valued in $k^\times$ and the group of integral Weyl-invariant cubic forms on the cocharacter lattice respectively.