Questions tagged [buildings]

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86 views

Full automorphism group of a Bruhat-Tits building

If we start with a semisimple algebraic group $G$ defined over a non-archimedean local field and want to understand the relationship of this group with the full type-preserving automorphism group of ...
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134 views

Homology of a semisimplicial scheme

This is a question about the homology of a complex made of algebraic varieties. Consider the following subgroups of $\mathrm{SL}_3$ (defined over $\mathbb{Z}$). $$ P_{1,2} = \left\{\left(\begin{...
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Cycles in Tits building

Tits building for an $n$-dimensional vector space $V$ is defined to be the simplicial complex corresponding to the poset of proper and non-zero subspaces of $V$. It is denoted by $T(V)$. This is known ...
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Poset of degree zero bundles

Let’s assume we are working on a smooth projective curve $X$. For any vector bundle $E$ on $X$, the poset of non-trivial proper sub-bundles of $E$ is in bijection with the poset of non-zero proper sub-...
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Query about Bruhat-Tits buildings over completions of fields with respect to a valuation, but the residue class field is not necessarily finite

I'm reading Soulé's article "Chevalley groups over polynomial rings", and he has a situation where $k$ is an arbitrary field, not necessarily finite, then you take a simple transcendental extension of ...
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515 views

Pointwise stabilizer of an apartment of the Bruhat-Tits building of $\mathrm{SL}_n(\mathbb{Q}_p)$

Denote by $X$ the Bruhat-Tits building of $\mathrm{SL}_n(\mathbb{Q}_p)$. Let $\Sigma$ be the fundamental apartment of $X$. Let $\Gamma=\mathrm{SL}_n(\mathbb{Z}[\frac{1}{p}])$. We can prove that the ...
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Directed galleries of the building of type $\widetilde{A}_{n}$

Let $X$ be the affine building of $GL_n(\mathbb{Q}_{p})$. We call oreinetd chamber of $X$ every sequence $\overrightarrow{C}=(s_1,...,s_n)$ of vertices such that $C=\{s_1,...,s_n\}$ is a chamber of $X$...
7
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1answer
176 views

Tits building of free modules

The Tits building for a vector space $V$ denoted by $T(V)$ is defined as a simplicial complex whose vertices are non-zero proper sub-vector spaces and edges are inclusion of subspaces and $i$-...
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53 views

Classifying some specific type of vector bundles by buildings

Assume $C$ is a curve and $p={\infty}$ a point at infinity such that $C\setminus p$ is an affine curve $U=Spec(A)$ and let $R$ be the local ring at $p$ which is a DVR. Let $F$ be a fixed free module ...
9
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2answers
234 views

Is the poset of affine subspaces of a vector space highly connected?

The question is in the title. Fix a field $k$. Let $P_n$ be the poset of proper nonempty affine subspaces of $k^n$ under inclusion. The geometric realization $|P_n|$ is $n$-dimensional. Is it $(n-...
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136 views

Relationship between the p-radical subgroups and the parabolics in a BN-pair generality

A theorem of Quillen says that if $G$ is a finite Chevalley group over characteristic $p$, then the poset $\mathcal{A}_p(G)$ of nontrivial elementary abelian subgroups of $G$ is homotopy equivalent (I ...
3
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1answer
58 views

Reference request: existence of a subgroup of $G(\mathcal O_k)$ that is “uniform” across $P \overline{N}$

Let $G$ be a connected, reductive group over a $p$-adic field $k$. Let $P_0$ be a minimal parabolic subgroup of $G$ containing a maximal split torus $A_0$. Let $K$ be a maximal compact open subgroup ...
5
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155 views

Does $G$ act 2-transitively on its Bruhat-Tits building?

Let $k$ be a finite extension of $\mathbb{Q}_p$ and let $G$ be a semisimple Lie group over $k$. We consider the action of $G$ on its Bruhat-Tits building $X$. Question: If $x,y,x',y'$ are vertices, ...
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Decomposition of linear groups into free products

I recently learnt about Nagao's theorem which states $SL_2(k[t])\cong SL_2(k)\ast_{B(k)}B(k[t])$ for a field $k$. I read in "A.W. Mason, Serre's generalization of Nagao's theorem" that a theorem of ...
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85 views

Valuations of root group elements appearing in the intersection of Iwasawa and Cartan double cosets

$\newcommand{\GL}{\operatorname{GL}} \newcommand{\diag}{\operatorname{diag}} \newcommand{\val}{\mathit{val}}$Let $F$ be a local non-Archimedean field with valuation $\val$ and $G$ be (the $F$-points ...
4
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1answer
131 views

Smith normal form and affine buildings

In Smith Normal Form of powers of a matrix someone has commented saying that one can reformulate many questions about Smith normal forms in the language of affine buildings. I wanted to know of a ...
3
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1answer
450 views

Tits building of a linear algebraic group

I have a basic (probably naive) question about Tits buildings. Let $G$ be a (connected) linear algebraic group over a field $k$ (I am interested in the case where $k$ is algebraically closed but I ...
6
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1answer
235 views

Bruhat-Tits building of $SL_n(\mathbb{Q})$, hyperbolic isometries and its axis

Consider $G=SL_n(\mathbb{Q})$ and $p$ a prime integer. Associated to $G$ and $p$ we have its Bruhat-Tits building $\Delta$. It is well known that $\Delta$ can be provided with a canonical $CAT(0)$ ...
6
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1answer
191 views

Parahorics in nonsemisimple reductive algebraic groups

If $G$ is a semisimple algebraic group over a local field with finite residue field $K$ and $x$ a point in the Bruhat-Tits building $B(G, K)$ then the parahoric group scheme $P_x$ is a group scheme $P$...
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Functoriality of parahorics

We know from Landvogt's work (Crelle #518, 2000) that if $G$ and $H$ are two reductive groups and $f:G \rightarrow H$ is a morphism that there is a map $\hat{f}$ from $\mathcal{B}(G, K)$ to $\mathcal{...
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1answer
364 views

Does every Coxeter group arise from a BN-Pair? Does $\text{PGL}_2(\Bbb{Z})$?

The question is in the title. Maybe I should explain my interest in it though. To every Coxeter group $(W,S)$ (and even more general groups) and a system of parameters $(a_s,b_s)_{s \in S}$ one can ...
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208 views

abelian subgroups of SL(n,Q_p)

Does anyone know where I can find a classification of abelian subgroups of SL(n,Q_p) isomorphic to (Q_p)^{n-1} up to conjugacy? Even for n=2, it would be very useful. For example, over R, there are ...
8
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1answer
384 views

State of the art knowledge about homology of $SL_2(k[t,t^{-1}])$

What is the current state of knowledge of the group homology of $SL_2(k[t,t^{-1}])$? I am mostly interested in the case $k$ is algebraically closed of characteristic zero. The most recent work I am ...
2
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1answer
210 views

Spherical building at infinity for $SL(n, \mathbb{Q}_p)$

Is there somewhere I can read about the spherical building at infinity for $SL(n, \mathbb{Q}_p)$? I'm looking for something with lots of explicit examples and computations. (I have books on the ...
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116 views

Extension of Tits' theorem on groups with a BN-pair of rank ≥ 3

Tits has proved that a finite simple group $G$ with a BN-pair of rank $n \ge 3$, is of Lie type. Let $B$ be the Borel subgroup and $(W,S)$ the Coxeter system. The subset lattice of the set $S$ is ...
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276 views

Bruhat-Tits building for $PGL_2(F)$ and repressentation theory

I am trying to understand representation theory of $GL_2(F)$ where $F$ is a finite extension of $\mathbb{Q}_p$. I am using the following paper of Colmez to understand it: https://webusers.imj-prg.fr/~...
3
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1answer
295 views

Affine building for SL(n)

Following Garrett's book, we have the usual description of affine buildings of SL(n) in terms of homothety classes of lattices. So let $F$ be a local field, $\mathcal{O}$ be its ring of integers and $\...
6
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2answers
98 views

Describing the action of $^2E_6(q)$

One of the constructions of the group $^2E_6(q)$ was presented by Tits in his paper "Les «formes réelles» des groupes de type $E_6$". It is being constructed by looking at the action of $^2E_6(q)$ on ...
2
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1answer
165 views

When are maximal compacts same as maximal parahorics?

Let $G$ be a reductive algebraic group over a complete non-archimedean field $k$. We know that maximal compacts are exactly the same as maximal parahorics when the Iwahori is open compact subgroup of $...
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2answers
449 views

What are the special parahoric subgroups in unitary groups?

Let $L$ be a $p$-adic field and let $L'/L$ be a quadratic extension. Let $U_{L'/L}(n)$ be a quasi-split unitary group of $n\times n$ matrices with entries in $L'$. I'm curious about what the special ...
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448 views

How do you understand the Moy-Prasad filtration of G_2?

Starting on page 44 of this paper of Reeder and Yu, the authors describe the first graded piece of the Moy-Prasad filtration on $G_2$ at a certain point (in this case it's $GL_2$ of the residue field),...
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2answers
1k views

Is the Steinberg representation always irreducible?

Let $\mathbb{F}$ be a field. The Tits building for $\text{SL}_n(\mathbb{F})$, denoted $T_n(\mathbb{F})$, is the simplicial complex whose $k$-simplices are flags $$0 \subsetneq V_0 \subsetneq \cdots \...
4
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1answer
154 views

Global bound for number of vertices in Bruhat-Tits building

Let $G$ denote a semisimple linear algebraic group over $\mathbb Q$ and let $r$ be its absolute rank. For any prime $p$ let $v_p$ be a vertex of the Bruhat-Tits building of $G({\mathbb Q}_p)$ and let $...
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0answers
145 views

Buildings for Affine groups

Let $G$ denote one of the classical groups over a finite field. Is there a natural way to associate a building to the affine group $V\rtimes G$, and an analog of the Solomon-Tits theorem?
1
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1answer
97 views

Pairs of rays in euclidean buildings

In section 4.1.3 of Kleiner and Leeb's paper on symmetric spaces and euclidean buildings, there's a result about pairs of rays from the same point initially spanning a flat triangle (or being ...
3
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1answer
117 views

Is a cocompact CAT(0) periodic?

Let $X$ be a CAT(0) space and $G$ its group of isometries. Then $X$ is said to be cocompact, if there exists a compact set $K\subset X$ with $X=G.K$. The space $X$ is called periodic, if there exists ...
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2answers
842 views

Understanding how to construct Bruhat-Tits buildings for non-split groups by Galois descent

Is there any way to get on top of the procedure for constructing Bruhat-Tits buildings for non-split groups over a non-archimedean local field $k$, by Galois descent, other than reading both the ...
4
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1answer
421 views

Fixed points in the Bruhat-Tits building

Let $G$ be a connected reductive group over a complete discrete valuation field with perfect residue field (or just a non-arch local field). Let $\mathcal{B}$ be its reduced Bruhat-Tits building, and $...
5
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1answer
396 views

Arithmetic quotients of Bruhat-Tits buildings for groups over local fields of positive characteristic

I have been led to believe that there is a result giving a description of the quotient of a Bruhat-Tits building $\Delta(G,k)$, for a semisimple algebraic group $G$ over a non-archimedean local field ...
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0answers
380 views

Bruhat Tits buiding to visualize closed points of affine flag varieties?

In his survey "affine springer fibers and affine Deligne-Lusztig varieties", Goertz gives us a tutorial session on how to use Bruhat Tits buildings to visualize subsets of affine Grassmannians or of ...
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1answer
298 views

When are toral orbits in buildings the difference of fixed-sets?

Let $L$ be a $p$-adic field, let $G$ be a reductive group over $L$ (I'm even okay assuming semisimplicity for now). Let $T$ be a maximal torus of $G$. Let $B$ be the building for $G(L)$. (Edit 1: "...
6
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1answer
470 views

When do two lattices have the same stabilizer in the diagonal torus?

This is moved from MSE, where I asked and didn't receive an answer (see https://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer) Let $T$ be the diagonal torus ...
7
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1answer
305 views

Embedding Euclidean buildings into products of trees

A Euclidean building has a natural metric space structure. (A definition of Euclidean building can be found on Wikipedia, or, more expansively, in Section 4 of Kleiner-Leeb.) Question: Is it true ...
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2answers
167 views

Question on affine buildings

Let $X$ be an affine building. Assume that $X$ is periodic, by which I mean that there exists a covering $X\to F$ of a finite simplicial complex. Let $\Gamma$ denote the group of deck transformations, ...
3
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1answer
192 views

algebraic groups over non-archimedean local fields acting on buildings

I was wondering could anyone tell me a reference for the fact that an absolutely quasi-simple algebraic group over a non-archimedean local field which is centreless and non-compact acts faithfully and ...
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0answers
189 views

(Co)Homology of groups vs. Lie algebras: polynomial rings

For Lie groups (or algebraic groups over fields) there is a strong relation between the cohomology of the group and the cohomology of its Lie algebra. Some MO-question where this is discussed can be ...
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203 views

Parshin's buildings for higher local fields

What is the status of the theory of buildings for higher local fields? I know that there are some papers of Parshin, in which he describes some examples, like $PGL_2$ and $PGL_3$ over two-...
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2answers
425 views

$G_\mathbb{Z}$-homotopy type of rational Tits building $\Delta_{G, \mathbb{Q}}$

Take $G$ to be a standard semisimple algebraic $\mathbb{Q}$-group, e.g. $Sp_{2g}$ or $SO(h)$ for $h$ a nondegenerate quadratic form over $\mathbb{Q}$. The arithmetic group $\Gamma=G_{\mathbb{Z}}$ has ...
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140 views

Rational cohomology of $S$-arithmetic groups over function fields and Gauss-Bonnet

I have a question on the ranks of rational cohomology groups of $S$-arithmetic groups over function fields. To fix the situation, $G$ is a simple Chevalley group of rank $r$, $k=\mathbb{F}_q$ a finite ...
4
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2answers
292 views

Does a spherical building embed in a building of type $A_n$?

I'm interested in the question in the title: Does a spherical building $B$ always embeds in a building $\tilde B$ of type $A_n$ for some $n$? By embedding I mean an isometric embedding with respect ...