# Linear representations of twin cities

###### Abstract

For spherical Tits buildings of the classical types there are well-known explicit descriptions as flag complexes. Similarly for affine buildings of the classical types there are explicit constructions in terms of lattices. In this article we generalize the flag complex description to twin cities, a generalization of twin buildings adapted to analytic completions of Kac-Moody groups as they appear for example in Kac-Moody geometry.
We construct linear representations of twin cities as flag complexes of certain subspaces in Hilbert spaces.

Key words: twin building, twin city, Kac-Moody group, loop group, flag complex, periodic flag

MSC(2010): 20F42, 20Gxx, 22E67, 14M15

## 1 Introduction

A priori Tits buildings are abstract objects associated to groups with a BN-pair, for example Lie groups. They are defined either as simplicial complexes or as metric graphs satisfying some lists of axioms [AbramenkoBrown08]. From this abstract point of view Lie groups are smooth manifolds with a group structure. Complementary to this abstract approach is the construction of explicit realizations. Explicit realizations of Lie groups are linear representations, that is homomorphisms of a Lie group into the general linear group of a suitable vector space . The corresponding realizations of buildings are flag complexes. A flag complex is a simplicial complexes constructed from a set of flags (chains of subspaces such that the spaces are subspaces of ) satisfying certain technical requirements. Furthermore we want the representation of the building to be equivariant with respect to the action of the representation of its isometry group. To be more precise let be a Tits building, its isometry group. Let furthermore be a representation of and define to be the space of “admissible” subspaces of and its power set. The representation is a map such that the following diagram commutes.