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Isotropic random walks on affine buildings

James Parkinson (2007)

Annales de l’institut Fourier

In this paper we apply techniques of spherical harmonic analysis to prove a local limit theorem, a rate of escape theorem, and a central limit theorem for isotropic random walks on arbitrary thick regular affine buildings of irreducible type. This generalises results of Cartwright and Woess where A ˜ n buildings are studied, Lindlbauer and Voit where A ˜ 2 buildings are studied, and Sawyer where homogeneous trees are studied (these are A ˜ 1 buildings).

Kac-Moody groups, hovels and Littelmann paths

Stéphane Gaussent, Guy Rousseau (2008)

Annales de l’institut Fourier

We give the definition of a kind of building for a symmetrizable Kac-Moody group over a field K endowed with a discrete valuation and with a residue field containing . Due to the lack of some important property of buildings, we call it a hovel. Nevertheless, some good ones remain, for example, the existence of retractions with center a sector-germ. This enables us to generalize many results proved in the semisimple case by S. Gaussent and P. Littelmann. In particular, if K = ( ( t ) ) , the geodesic segments...

Macdonald formula for spherical functions on affine buildings

A. M. Mantero, A. Zappa (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper we explicitly determine the Macdonald formula for spherical functions on any locally finite, regular and affine Bruhat-Tits building, by constructing the finite difference equations that must be satisfied and explaining how they arise, by only using the geometric properties of the building.

Opérateurs invariants sur certains immeubles affines de rang 2

Ferdaous Kellil, Guy Rousseau (2007)

Annales de la faculté des sciences de Toulouse Mathématiques

On considère un immeuble Δ de type A 2 ˜ ou B 2 ˜ , différents sous-ensembles 𝒮 de l’ensemble 𝒮 des sommets de Δ et différents groupes G d’automorphismes de Δ , très fortement transitifs sur Δ . On montre que l’algèbre des opérateurs G -invariants agissant sur l’espace des fonctions sur 𝒮 est souvent non commutative (contrairement aux résultats classiques). Dans certains cas on décrit sa structure et on détermine ses fonctions radiales propres. On en déduit que la conjecture d’Helgason n’est pas toujours vérifiée...

Polyèdres finis de dimension 2 à courbure 0 et de rang 2

Sylvain Barré (1995)

Annales de l'institut Fourier

On définit localement la notion de polyèdre de rang deux pour un polyèdre fini de dimension deux à courbure négative ou nulle. On montre que le revêtement universel d’un tel espace est soit le produit de deux arbres, soit un immeuble de Tits euclidien de rang deux.

Receding polar regions of a spherical building and the center conjecture

Bernhard Mühlherr, Richard M. Weiss (2013)

Annales de l’institut Fourier

We introduce the notion of a polar region of a spherical building and use some simple observations about polar regions to give elementary proofs of various fundamental properties of root groups. We combine some of these observations with results of Timmesfeld, Balser and Lytchak to give a new proof of the center conjecture for convex chamber subcomplexes of thick spherical buildings.

Representations of PGL ( 2 ) of a local field and harmonic cochains on graph

Paul Broussous (2009)

Annales de la faculté des sciences de Toulouse Mathématiques

We give combinatorial models for non-spherical, generic, smooth, complex representations of the group G = PGL ( 2 , F ) , where F is a non-Archimedean locally compact field. More precisely we carry on studying the graphs ( X ˜ k ) k 0 defined in a previous work. We show that such representations may be obtained as quotients of the cohomology of a graph X ˜ k , for a suitable integer k , or equivalently as subspaces of the space of discrete harmonic cochains on such a graph. Moreover, for supercuspidal representations, these models...

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