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Edifices: building-like spaces associated to linear algebraic groups

Michael Bate, Benjamin Martin and Gerhard Röhrle

Vol. 20 (2023), No. 2-3, 79–134
Abstract

Given a semisimple linear algebraic k-group G, one has a spherical building ΔG, and one can interpret the geometric realisation ΔG() of ΔG in terms of cocharacters of G. The aim of this paper is to extend this construction to the case when G is an arbitrary connected linear algebraic group; we call the resulting object ΔG() the spherical edifice of G. We also define an object V G() which is an analogue of the vector building for a semisimple group; we call V G() the vector edifice. The notions of linear map and of isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations. We also devise a family of metrics on V G() and show they are all bi-Lipschitz equivalent to each other; with this extra structure, V G() becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits centre conjecture.

In memory of Jacques Tits

Keywords
spherical buildings, edifices, Tits centre conjecture, geometric invariant theory
Mathematical Subject Classification
Primary: 20E42, 20G15, 51E24
Milestones
Received: 5 September 2022
Revised: 18 April 2023
Accepted: 16 May 2023
Published: 13 September 2023
Authors
Michael Bate
Department of Mathematics
University of York
York
United Kingdom
Benjamin Martin
Department of Mathematics
University of Aberdeen, King’s College
Aberdeen
United Kingdom
Gerhard Röhrle
Fakultät für Mathematik
Ruhr-Universität Bochum
Bochum
Germany