INTRODUCTION TO SPHERICAL
AND AFFINE BUILDINGS
by RICHARD M. WEISS
Syllabus: The course
has four parts:
A. Introduction to
Coxeter groups. A few basic facts about Coxeter groups are needed before
we can talk about buildings.
B. Introduction to
buildings. We will introduce buildings as edge-colored graphs satisfying
certain properties and study the basic substructures of a building: residues,
roots, apartments, the projection map and the ''gated property''.
C. Spherical buildings.
These are the buildings whose apartments are finite. We will study root
groups, generalized polygons, the opposite map, the Moufang property, root
data and related algebraic structures. We will look at examples coming
from classical and exceptional groups.
D. Affine buildings.
These are the buildings whose apartments have a natural representation
as a tiling of a Euclidean space: We will study sectors, the building at
infinity (which is spherical) and root data with valuation.
Two of Jacques Tits'
most remarkable accomplishments are his classification of spherical buildings
and his classification (some of it in collaboration with F. Bruhat) of
affine buildings (in sufficiently high rank). One goal of this course is
to leave the students with some understanding of these two results.
References:
[1] K. Brown, Buildings,
Springer, 1989.
[2] M. Ronan, Lectures
on Buildings, Academic Press, New York 1989.
[3] J.Tits, Buildings
of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics
386,
Springer, 1974.
[4] R. M. Weiss, The
Structure of Spherical Buildings, Princeton University Press, 2003.
[5] R. M. Weiss, The
Structure of Affine Buildings, notes in preparation.
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