Normed Fiber Functors

TL;DR

This paper generalizes the description of Bruhat-Tits buildings from GL_n to unramified reductive groups using a Tannakian formalism, linking norms, fiber functors, and group schemes.

Contribution

It introduces a Tannakian framework to describe Bruhat-Tits buildings of unramified reductive groups via norms on fiber functors, extending classical descriptions.

Findings

Bruhat-Tits building of G described as set of norms on fiber functor

Moduli-theoretic description of parahoric group schemes

Extension of classical GL_n results to general unramified reductive groups

Abstract

By Goldman-Iwahori, the Bruhat-Tits building of the general linear group $\operatorname{GL}_n$ over a local field $\ell$ can be described as the set of non-archimedean norms on $\ell^n$. Via a Tannakian formalism, we generalize this picture to a description of the Bruhat-Tits building of an unramified reductive group $G$ over $\ell$ as the set of norms on the standard fiber functor of a special parahoric integral model of $G$. We also give a moduli-theoretic description of the parahoric group scheme associated to a point of the building as the group scheme of tensor automorphisms of the lattice chains defined by the corresponding norm.