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A tropical view on Bruhat-Tits buildings and their compactifications

Annette Werner (2011)

Open Mathematics

We relate some features of Bruhat-Tits buildings and their compactifications to tropical geometry. If G is a semisimple group over a suitable non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of G and in some of its compactifications are described by tropical linear algebra. The compactifications we consider arise from algebraic representations of G. We show that the fan which is used to compactify an apartment in this theory is given by the weight polytope of the representation...

Base change for Bernstein centers of depth zero principal series blocks

Thomas J. Haines (2012)

Annales scientifiques de l'École Normale Supérieure

Let  G be an unramified group over a p -adic field. This article introduces a base change homomorphism for Bernstein centers of depth-zero principal series blocks for  G and proves the corresponding base change fundamental lemma. This result is used in the approach to Shimura varieties with Γ 1 ( p ) -level structure initiated by M. Rapoport and the author in [15].

Combinatorial and group-theoretic compactifications of buildings

Pierre-Emmanuel Caprace, Jean Lécureux (2011)

Annales de l’institut Fourier

Let X be a building of arbitrary type. A compactification 𝒞 sph ( X ) of the set Res sph ( X ) of spherical residues of X is introduced. We prove that it coincides with the horofunction compactification of Res sph ( X ) endowed with a natural combinatorial distance which we call the root-distance. Points of 𝒞 sph ( X ) admit amenable stabilisers in Aut ( X ) and conversely, any amenable subgroup virtually fixes a point in 𝒞 sph ( X ) . In addition, it is shown that, provided Aut ( X ) is transitive enough, this compactification also coincides with the group-theoretic...

Counting arithmetic subgroups and subgroup growth of virtually free groups

Amichai Eisenmann (2015)

Journal of the European Mathematical Society

Let K be a p -adic field, and let H = P S L 2 ( K ) endowed with the Haar measure determined by giving a maximal compact subgroup measure 1 . Let A L H ( x ) denote the number of conjugacy classes of arithmetic lattices in H with co-volume bounded by x . We show that under the assumption that K does not contain the element ζ + ζ - 1 , where ζ denotes the p -th root of unity over p , we have lim x log A L H ( x ) x log x = q - 1 where q denotes the order of the residue field of K .

Dual Blobs and Plancherel Formulas

Ju-Lee Kim (2004)

Bulletin de la Société Mathématique de France

Let k be a p -adic field. Let G be the group of k -rational points of a connected reductive group 𝖦 defined over k , and let 𝔤 be its Lie algebra. Under certain hypotheses on 𝖦 and k , wequantifythe tempered dual G ^ of G via the Plancherel formula on 𝔤 , using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on 𝔤 and G . As a consequence, we prove that any tempered representation contains a good minimal 𝖪 -type; we extend this result to irreducible...

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