scholarly journals Rigid connections on P1 via the Bruhat–Tits building

2020 ◽  
Author(s):  
Masoud Kamgarpour ◽  
Daniel S. Sage
Keyword(s):  
Tits Building

The Standard Metric

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Affine Transformation ◽  
Convex Sets ◽  
Finite Dimension ◽  
Affine Subspace ◽  
Affine Transformations ◽  
Real Vector ◽  
Euclidean Metric ◽  
Affine Hyperplane ◽  
Tits Building ◽  
Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

scholarly journals Sur les -blocs de niveau zéro des groupes -adiques

2018 ◽  
Vol 154 (7) ◽  
pp. 1473-1507
Author(s):  
Thomas Lanard
Keyword(s):  
Abelian Category ◽  
Langlands Correspondence ◽  
Parabolic Induction ◽  
Langlands Parameters ◽  
Tits Building ◽  
Local Langlands Correspondence

Let $G$ be a $p$-adic group that splits over an unramified extension. We decompose $\text{Rep}_{\unicode[STIX]{x1D6EC}}^{0}(G)$, the abelian category of smooth level $0$ representations of $G$ with coefficients in $\unicode[STIX]{x1D6EC}=\overline{\mathbb{Q}}_{\ell }$ or $\overline{\mathbb{Z}}_{\ell }$, into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat–Tits building and Deligne–Lusztig theory. Then, we prove compatibilities with parabolic induction and restriction functors and the local Langlands correspondence.

scholarly journals Stable vectors in Moy–Prasad filtrations

2017 ◽  
Vol 153 (2) ◽  
pp. 358-372 ◽  
Author(s):  
Jessica Fintzen ◽  
Beth Romano
Keyword(s):  
Sufficient Conditions ◽  
Reductive Group ◽  
Finite Extension ◽  
Necessary And Sufficient Conditions ◽  
Supercuspidal Representations ◽  
Tits Building ◽  
Stable Vectors ◽  
Necessary And Sufficient

Let $k$ be a finite extension of $\mathbb{Q}_{p}$, let ${\mathcal{G}}$ be an absolutely simple split reductive group over $k$, and let $K$ be a maximal unramified extension of $k$. To each point in the Bruhat–Tits building of ${\mathcal{G}}_{K}$, Moy and Prasad have attached a filtration of ${\mathcal{G}}(K)$ by bounded subgroups. In this paper we give necessary and sufficient conditions for the dual of the first Moy–Prasad filtration quotient to contain stable vectors for the action of the reductive quotient. Our work extends earlier results by Reeder and Yu, who gave a classification in the case when $p$ is sufficiently large. By passing to a finite unramified extension of $k$ if necessary, we obtain new supercuspidal representations of ${\mathcal{G}}(k)$.

Discrete subgroups acting transitively on vertices of a Bruhat–Tits building

2012 ◽  
Vol 161 (3) ◽  
pp. 483-544 ◽  
Author(s):  
Amir Mohammadi ◽  
Alireza Salehi Golsefidy
Keyword(s):  
Discrete Subgroups ◽  
Tits Building

Ramified Quadrangles of Type E6, E7 and E8

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Basic Proposition ◽  
Tits Building ◽  
Quaternion Division Algebra

This chapter deals with the case that the building at infinity of the Bruhat-Tits building Ξ‎ is a Moufang ramified quadrangle of type E⁶, E₇ and E₈. The basic proposition is that Ξ‎ is a ramified quadrangle if δ‎Λ‎ = δ‎Ψ‎ = 1 holds. The chapter proves the theorem that if δ‎Ψ‎ = 1 and the Moufang residues R₀ and R₁ are not both indifferent, there exists an involutory set. It also discusses the cases ℓ = 6, ℓ = 7, and ℓ = 8, in which D is a quaternion division algebra.

scholarly journals Characters and growth of admissible representations of reductive p-adic groups

2011 ◽  
Vol 11 (2) ◽  
pp. 289-331 ◽  
Author(s):  
Ralf Meyer ◽  
Maarten Solleveld
Keyword(s):  
Tits Building

AbstractWe use coefficient systems on the affine Bruhat–Tits building to study admissible representations of reductive p-adic groups in characteristic not equal to p. We show that the character function is locally constant and provide explicit neighbourhoods of constancy. We estimate the growth of the subspaces of invariants for compact open subgroups.

scholarly journals The Supersingular Locus of the Shimura Variety for GU(1, s)

2010 ◽  
Vol 62 (3) ◽  
pp. 668-720 ◽  
Author(s):  
Inken Vollaard
Keyword(s):  
Complete Intersection ◽  
Moduli Space ◽  
Unitary Group ◽  
Shimura Variety ◽  
Incidence Relation ◽  
Irreducible Components ◽  
Reduction Modulo ◽  
Tits Building ◽  
Divisible Groups ◽  
Dieudonne Theory

AbstractIn this paper we study the supersingular locus of the reduction modulopof the Shimura variety for GU(1,s) in the case of an inert primep. Using Dieudonné theory we define a stratification of the corresponding moduli space ofp-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat–Tits building of a unitary group.In the case of GU(1, 2), we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.

scholarly journals Half-dimensional collapse of ends of manifolds of nonpositive curvature

2019 ◽  
Vol 29 (6) ◽  
pp. 1638-1702
Author(s):  
Grigori Avramidi ◽  
T. Tâm Nguyễn-Phan
Keyword(s):  
Finite Volume ◽  
Nonpositive Curvature ◽  
Tits Building

AbstractThis paper accomplishes two things. First, we construct a geometric analog of the rational Tits building for general noncompact, complete, finite volume n-manifolds M of bounded nonpositive curvature. Second, we prove that this analog has dimension less than $$\lfloor n/2\rfloor $$⌊n/2⌋.

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