Maximal Levi Subgroups Acting on the Building of GL_n(F)

TL;DR

This paper characterizes the action of maximal Levi subgroups on the Euclidean building of GL_n(F), providing invariants and metrics for understanding the quotient space, with applications to vertex calculations in specific cases.

Contribution

It introduces a complete invariant for the action of certain subgroups on the building and develops a natural metric on the quotient space, advancing understanding of these geometric actions.

Findings

Complete invariant of subgroup actions on the building

A natural metric on the quotient space

Method for calculating vertices to apartments

Abstract

In this paper we give a complete invariant of the action of GL_n(F)\times GL_m(F) acting on the Euclidean building B[GL_n(F)], where F is a non-archimedian field. We then use this invariant to give a natural metric on the resulting quotient space. In the special case of the torus acting on B[GL_2(F)] this gives a method of calculating of any vertex to any fixed apartment.