Automorphisms of buildings

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2015

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Wermer, Markus-Ludwig

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DOI:
http://dx.doi.org/10.22029/jlupub-9700

Abstract

This thesis discusses several aspects on automorphisms on (non-spherical) buildings. In particular, given a building automorphism theta, the set of Weyl displacements W theta of theta is of major interest. The Weyl displacements are the elements w of the underlying Weyl group W of a building beta which are the Weyl distance some chamber to its image. One motivation to work with this set is the Deligne-Lusztig theory, where the studied affine Deligne-Lusztig varieties are (chamber wise) level sets of a given displacement.As an introduction, we show W theta unequal W for any (non-type preserving) automorphism of an affine building. The same holds for any type-preserving automorphism of any non-spherical building. The general concept is to find suitable small subsets of chambers of a building on which theta gives us enough information to compute W theta}.The first result uses the Davis realization of buildings and and the induced action of theta on this CAT(0) space. We show that under certain conditions all displacements can be attained from the displacements of chambers which contain an element with minimal displacement. Examples are all automorphisms on Coxeter systems themselves, all automorphisms of buildings whose underlying Coxeter group is universal, as well as automorphisms of affine building fixing exactly one wall. We introduce the concept of tie trees which are tree structures on buildings. Given a tie tree T for which theta induces a tree automorphism, we obtain W theta once we have the displacements of all chambers associated to the vertices of T a minimal displacement. We only have to extend the computed displacements to all their theta-conjugates. A pretty large class of buildings admitting a non-trivial tie tree are all non-2-spherical buildings. A slightly weaker version of the result for tie trees allows us to study automorphisms fixing exactly one apartment and automorphisms of affine buildings preserving a wall tree.The last part of the thesis deals with the affine building associated to SLn(K) for a field K with discrete valuation. We discuss the action of GLn(K) on this building and derive a formula for computing delta(C,gC) for any chamber C and any element g in GLn(K). Using this result we present the code of a SAGE-program for exactly those computations.

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