We prove an analogue of Kostants convexity theorem for thick affine buildings and give an applica... more We prove an analogue of Kostants convexity theorem for thick affine buildings and give an application for groups with affine BN-pair. Recall that there are two natural retractions of the affine building onto a fixed apartment A: The retraction r centered at an alcove in A and the retraction ρ centered at a chamber in the spherical building at infinity. We prove that for each special vertex x ∈ A the set ρ(r −1 (W.x)) is a certain convex hull of W.x. The proof can be reduced to a statement about Coxeter complexes and heavily relies on a character formula for highest weight representations of algebraic groups.
We describe all metric spaces that have sufficiently many affine functions. As an application we ... more We describe all metric spaces that have sufficiently many affine functions. As an application we obtain a metric characterization of linear-convex subsets of Banach spaces.
This is joint work with Julia Heller
For any finite Coxeter group W of rank n we show that the... more This is joint work with Julia Heller
For any finite Coxeter group W of rank n we show that the order complex of the lattice of non-crossing partitions NC(W) embeds as a connected chamber subcomplex into a spherical building of type An−1. In types A and B explicit constructions of such an embedding are given in terms of the pictorial descriptions of the non-crossing partitions. We use this embedding to give a lower bound on the radius of the Hurwitz graph H(W) in all types and reprove that in type An the radius is n 2 .
It is shown that the folding operators introduced by Gaussent and Littelmann can be expressed pur... more It is shown that the folding operators introduced by Gaussent and Littelmann can be expressed purely in terms of retractions of the building which in turn are tightly connected to root groups and root data with valuations. The relation shown here highlights the deep connection between folding operators and to certain kinds of double coset intersections of subgroups of a semi-simple algebraic group.
This is joint work with Elisabeth Milicevic and Anne Thomas
Let G be a reductive group over the ... more This is joint work with Elisabeth Milicevic and Anne Thomas
Let G be a reductive group over the field F = k((t)), where k is an algebraic closure of a finite field, and let W be the (extended) affine Weyl group of G. The associated affine Deligne–Lusztig varieties Xx(b), which are indexed by elements b ∈ G(F) and x ∈ W , were introduced by Rapoport [Rap00]. Basic questions about the varieties Xx(b) which have remained largely open include when they are nonempty, and if nonempty, their dimension. We use techniques inspired by geometric group theory and combinatorial representation theory to address these questions in the case that b is a pure translation, and so prove much of a sharpened version of Conjecture 9.5.1 of Görtz, Haines, Kottwitz, and Reuman [GHKR10]. Our approach is constructive and type-free, sheds new light on the reasons for existing results in the case that b is basic, and reveals new patterns. Since we work only in the standard apartment of the building for G(F), our results also hold in the p-adic context, where we formulate a definition of the dimension of a p-adic Deligne–Lusztig set. We present two immediate applications of our main results, to class polynomials of affine Hecke algebras and to affine reflection length.
This is joint work with Piotr Przytycki
We introduce a construction turning some Coxeter and Da... more This is joint work with Piotr Przytycki
We introduce a construction turning some Coxeter and Davis realizations of buildings into systolic complexes. Consequently groups acting geometrically on buildings of triangle types distinct from (2, 4, 4), (2, 4, 5), (2, 5, 5), and various rank 4 types are systolic.
This is joint work with Dawid Kielak and Thomas Haettel
We show that braid groups with at most 6... more This is joint work with Dawid Kielak and Thomas Haettel
We show that braid groups with at most 6 strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes, and the em-beddability of their diagonal links into spherical buildings of type A. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCam-mond.
We classify the epimorphisms of the buildings BC l (K, K0, σ, L, q0), l ≥ 2, of pseudo-quadratic ... more We classify the epimorphisms of the buildings BC l (K, K0, σ, L, q0), l ≥ 2, of pseudo-quadratic form type. This completes the classification of epimorphisms of irreducible spherical Moufang buildings of rank at least two.
In this paper we prove equivalence of sets of axioms for non-discrete affine buildings, by provid... more In this paper we prove equivalence of sets of axioms for non-discrete affine buildings, by providing different types of metric, exchange and atlas conditions. We apply our result to show that the definition of a Euclidean building depends only on the topological equivalence class of the metric on the model space. The sharpness of the axioms dealing with metric conditions is illustrated in an appendix. There it is shown that a space X defined over a model space with metric d is possibly a building only if the induced distance function on X satisfies the triangle inequality.
We prove an analog of the base change functor of Λ–trees in the setting of generalized affine bui... more We prove an analog of the base change functor of Λ–trees in the setting of generalized affine buildings. The proof is mainly based on local and global combinatorics of the associated spherical buildings. As an application we obtain that the class of generalized affine buildings is closed under taking ultracones and asymptotic cones. Other applications involve a complex of groups decompositions and fixed point theorems for certain classes of generalized affine buildings.
Ane buildings are in a certain sense analogs of symmetric spaces. It is therefore natural to try ... more Ane buildings are in a certain sense analogs of symmetric spaces. It is therefore natural to try to nd analogs of results for symmetric spaces in the theory of buildings. In this paper we prove a version of Kostant's convexity theorem for thick non-discrete ane buildings. Kostant proves that the image of a certain orbit of a point x in the symmetric space under a projection onto a maximal at is the convex hull of the Weyl group orbit of x. We obtain the same result for a projection of a certain orbit of a point in an ane building to an apartment. The methods we use are mostly borrowed from metric geometry. Our proof makes no appeal to the automorphism group of the building. However the nal result has an interesting application for groups acting nicely on non-discrete buildings, such as groups admitting a root datum with non-discrete valuation. Along the proofs we obtain that segments are contained in apartments and that certain retractions onto apartments are distance diminishing.
Ein Gebäude steht da von uralten Zeiten, Es ist kein Tempel, es ist kein Haus; Ein Reiter kann hu... more Ein Gebäude steht da von uralten Zeiten, Es ist kein Tempel, es ist kein Haus; Ein Reiter kann hundert Tage reiten, Er umwandert es nicht, er reitet's nicht aus. Jahrhunderte sind vorüber geogen, Es trotzte der Zeit und der Stürme Heer; Frei steht es unter dem himmlischen Bogen, Es reicht in die Wolken, es netzt sich im Meer. Nicht eitle Prahlsucht hat es gethürmet, Es dienet zum Heil, es rettet und schirmet; Seines Gleichen ist nicht auf Erden bekannt, Und doch ist's ein Werk von Menschenhand.
This is Joint work with Linus Kramer and Richard Weiss
We call a non-discrete Euclidean buildin... more This is Joint work with Linus Kramer and Richard Weiss
We call a non-discrete Euclidean building a Bruhat-Tits space if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root datum of the building at infinity. This is the class of non-discrete Euclidean buildings introduced and studied by Bruhat and Tits in [2]. We give the complete classication of Bruhat-Tits spaces whose building at infinity is the fixed point set of a polarity of an ambient building of type B2, F4 or G2 associated with a Ree or Suzuki group endowed with the usual root datum. We also show that each of these Bruhat-Tits spaces has a natural embedding in the unique Bruhat-Tits space whose building at infinity is the corresponding ambient building.
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Papers by Petra Schwer
For any finite Coxeter group W of rank n we show that the order complex of the lattice of non-crossing partitions NC(W) embeds as a connected chamber subcomplex into a spherical building of type An−1. In types A and B explicit constructions of such an embedding are given in terms of the pictorial descriptions of the non-crossing partitions. We use this embedding to give a lower bound on the radius of the Hurwitz graph H(W) in all types and reprove that in type An the radius is n 2 .
Let G be a reductive group over the field F = k((t)), where k is an algebraic closure of a finite field, and let W be the (extended) affine Weyl group of G. The associated affine Deligne–Lusztig varieties Xx(b), which are indexed by elements b ∈ G(F) and x ∈ W , were introduced by Rapoport [Rap00]. Basic questions about the varieties Xx(b) which have remained largely open include when they are nonempty, and if nonempty, their dimension. We use techniques inspired by geometric group theory and combinatorial representation theory to address these questions in the case that b is a pure translation, and so prove much of a sharpened version of Conjecture 9.5.1 of Görtz, Haines, Kottwitz, and Reuman [GHKR10]. Our approach is constructive and type-free, sheds new light on the reasons for existing results in the case that b is basic, and reveals new patterns. Since we work only in the standard apartment of the building for G(F), our results also hold in the p-adic context, where we formulate a definition of the dimension of a p-adic Deligne–Lusztig set. We present two immediate applications of our main results, to class polynomials of affine Hecke algebras and to affine reflection length.
We introduce a construction turning some Coxeter and Davis realizations of buildings into systolic complexes. Consequently groups acting geometrically on buildings of triangle types distinct from (2, 4, 4), (2, 4, 5), (2, 5, 5), and various rank 4 types are systolic.
We show that braid groups with at most 6 strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes, and the em-beddability of their diagonal links into spherical buildings of type A. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCam-mond.
We call a non-discrete Euclidean building a Bruhat-Tits space if its automorphism group contains a subgroup that induces the subgroup generated by all the root groups of a root datum of the building at infinity. This is the class of non-discrete Euclidean buildings introduced and studied by Bruhat and Tits in [2]. We give the complete classication of Bruhat-Tits spaces whose building at infinity is the fixed point set of a polarity of an ambient building of type B2, F4 or G2 associated with a Ree or Suzuki group endowed with the usual root datum. We also show that each of these Bruhat-Tits spaces has a natural embedding in the unique Bruhat-Tits space whose building at infinity is the corresponding ambient building.