# Project B4: Reductive groups and combinatorial structures

## Principal investigator

## Participating scientists

- Nils Leder
- Rupert McCallum
- Prof. Dr. Peter Schneider
- Dr. Petra Schwer
- Dr. Daniel Skodlerack
- Prof. Dr. Dr. Katrin Tent
- Olga Varghese
- Cora Welsch
- Dr. Stefan Witzel

## Summary

The project is concerned with the interplay of combinatorial structures and continuous groups. We study buildings, which are spaces of nonpositive curvature, and groups acting on them. There are natural questions about the large-scale geometry of buildings, such as rigidity under coarse equivalences. On the other hand, the rich local structure of buildings leads to coefficient systems which may be used to study the representation theory of reductive groups over local fields.Buildings were introduced by Tits as certain metric cell complexes. They admit a covering by a family of subcomplexes which are called apartments. These apartments are in turn isomorphic to a fixed Coxeter complex. The buildings we are concerned with in this project are spherical buildings, where the Coxeter complex is a triangulated sphere, and affine buildings, where the Coxeter complex is a triangulated euclidean space. Typical examples of such buildings arise from isotropic reductive groups

__G__over a field F. The group of F-points

__G__(F) 'is' then the automorphism group of a spherical building D(

__G__, F). In grouptheoretic terms, D(

__G__, F) is the set of all F-parabolics in

__G__, partially ordered by reversed inclusion.

If F is a valued field then a second, affine building can be constructed. This affine building is called the Bruhat-Tits building of

__G__. Its vertices are the maximal bounded subgroups of

__G__(F). The local structure around a point of the Bruhat-Tits building corresponds to the local structure of the field and the group, i.e. suitable reductive groups over the residue field of F. The affine building itself encodes how these local data are patched together. Bruhat-Tits buildings are important tools in the structure theory of reductive groups over local fields.

Bruhat-Tits buildings are special cases of euclidean buildings. A euclidean building is a metric space of nonpositive curvature which is glued together from copies of a fixed euclidean space R

^{n}. The transition maps between the different copies of R

^{n}lie in a euclidean reflection groupW (which is not necessarily discrete). For generalized affine buildings, R

^{n}is replaced by Λ

^{n}, where Λ is an ordered abelian group.

Together with the noncompact Riemannian symmetric spaces, euclidean buildings form an interesting class of CAT(0) spaces. In fact, they may be viewed as nice 'model spaces'. A natural question is how this subclass of CAT(0) spaces can be characterized by metric, group theoretic or topological properties.

We plan to investigate the following projects.

**Coarse rigidity of affine buildings.** Here we want to study the coarse or
asymptotic behavior of euclidean buildings and, more generally, of
generalized affine buildings. The tools are ultrapowers and asymptotic
cones, sheaf-theoretic homology and CAT(0) geometry. The overall
goal is to prove metric rigidity in the coarse category.

**Coefficient systems and sheaves on Bruhat-Tits buildings.** The rich combinatorial
structure of p-adic Bruhat-Tits buildings yields interesting
equivariant sheaves and coefficient systems. These sheaves can be
used to study the representation theory of the reductive group acting
on the building.

**Structure of affine buildings and dynamics of automorphisms.** The curvature
properties of euclidean buildings can be used to study the dynamics
and fixed points of automorphisms. This leads to decomposition theorems
&aecute; la Kostant both for individual automorphisms and for the whole
automorphism group.

**Characterizations of buildings and simple Lie groups.** If the field of definition
F is a local field, then the spherical building of a reductive
group over F carries a natural compact topology. Geometrically, these
compact buildings are boundaries of symmetric spaces or locally finite
Bruhat-Tits buildings. Our aim is to classify these buildings both topologically
and geometrically.