The buildings tag has no usage guidance.

**12**

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397 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 ...

**3**

votes

**1**answer

80 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 ...

**0**

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**0**answers

120 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**

vote

**1**answer

84 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 ...

**2**

votes

**1**answer

87 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 ...

**6**

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**2**answers

427 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 ...

**3**

votes

**1**answer

191 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**

votes

**1**answer

209 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 ...

**2**

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**0**answers

215 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 ...

**6**

votes

**1**answer

237 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**

votes

**1**answer

280 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 http://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer)
Let $T$ be the diagonal torus ...

**6**

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**1**answer

231 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 ...

**1**

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**2**answers

126 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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**1**answer

145 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 ...

**2**

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**0**answers

111 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 ...

**4**

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149 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 ...

**8**

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**2**answers

307 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 ...

**4**

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**0**answers

91 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**

votes

**2**answers

207 views

### Does a spherical building embeds 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 ...

**4**

votes

**1**answer

301 views

### Clarification about Tits' article in the Corvallis

I am studying Tits' article in the Corvallis wherein he defines the apartment in the general case (not necessarily split). I wish to know what he means about the filtration of the groups $U_a(K)$ ...

**7**

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**1**answer

213 views

### spherical buildings for non-split groups

I am looking for references to explicit descriptions of Tits buildings for semisimple (classical) Lie groups via language of incidence geometry. Such descriptions are well-documented in the case of ...

**4**

votes

**1**answer

187 views

### Density/Thickness of rank 3 spherical buildings

I am trying to study (finite) spherical buildings from a very combinatorial point of view : Every rank 3 spherical building is a finite simplicial complex of dimension 3, so one can define its density ...

**7**

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**1**answer

285 views

### When is a Moy-Prasad filtration subgroup the stabilizer of a subset of the building (up to center)?

Let $G$ be a connected, simply connected, semi-simple algebraic group defined and split over a local non-arch field $k$ with integer ring $R$. Let $B$ be the corresponding reduced building. Fix an ...

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**3**answers

743 views

### What is a (generalized) BN-pair?

Let us consider $GL_n(K)$ over a local field $K$. It has standard subgroups $N$ and $B$. $B$ is Iwahori subgroup, $N$ consists of monomial matrices. The pair comes close to a romantic ending, i.e. ...

**1**

vote

**1**answer

211 views

### Heights in reductive groups

Let $G$ be a reductive group over a local non-archimedean field $F$, and let $B$ a Borel subgroup. For my purposes, the case $G = GL_2(\mathbb{Q}_p)$ will be sufficient with $B$ upper triangular ...

**1**

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**2**answers

177 views

### What is this measure on the affine Weyl group?

Let $G$ be $SL(n, F)$ for a non-archimedean local field $F$ with Iwahori subgroup $I$. Let $\mu$ be the Haar measure of $G$.
What are the properties of the function $w\mapsto \mu(IwI)/\mu(I)$ for ...

**8**

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**2**answers

597 views

### Conjugation in GL(n) (p-adic setting)

In $GL(n, \mathbb{Q}_p)$, what are the orbits under conjugation of $GL(n, \mathbb{Z}_p)$?

**2**

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461 views

### Parabolic subgroups and BN-pairs

We note $G$ a connected algebraic group, $T$ a maximal torus in $G$, $B$ a Borel subgroup containing $T$. We put also, $N=N_{G}(T)$ the normalizer of $T$ in $G$. We know that $(B,N)$ is a BN-pair of ...

**1**

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**1**answer

198 views

### Algorithm for the cell multiplication rule for GL(n,F)

Consider $F$ a non archimedean field and let $o$ be its ring of integer
Let $B$ be the Iwahori subgroup of $GL_n(F)$ (resp. $GL_n(o)$) and let $N$ be the normalizer of the diagonal matrices ...

**2**

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**2**answers

457 views

### What is a canonical set of representatives in $GL(n,F)$ for the vertices in the Bruhat Tits building?

$F$ is a non archimedean field here. To be more precise, I would actually prefer a set of representative in $B(F)$ for the discrete space $B(F) / B(o)Z(F)$?
This can be phrased also as question about ...

**7**

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**2**answers

518 views

### Spherical building of an exceptional group of Lie type

I've read that one of Tits' original motivations for studying buildings was that he wanted to give a unified description of algebraic groups that would allow the definition of exceptional groups such ...

**8**

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**2**answers

577 views

### Subexpressions of reduced words in Coxeter groups

Let $\underline{w} = [s_1, s_2, \dots ,s_n]$ be a reduced expression in a Coxeter group $W$. Given $x$ in $W$ one can consider the set $\Pi(\underline{w},x)$ consisting of all subexpressions of ...

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**6**answers

4k views

**3**

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**3**answers

350 views

### Is this the CAT(0) metric on an affine building?

Let $R$ be a discrete valuation ring qith quotient field $Q$ and let $t\in R$ be a generator of the unique maximal ideal in $R$. Let $V$ be a finite-dimensional $Q$-vector space. Then one can consider ...

**8**

votes

**1**answer

296 views

### On which space does $GL_n(F_p[X])$ act nicely?

The group $GL_n(\mathbb{Z})$ acts properly and isometrically on the space of homothety classes of scalar products on $\mathbb{R}^n$. This is a Riemannian manifold of nonpositive sectional curvature.
...

**5**

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**1**answer

778 views

### Proof of an 'easy' exercise in a book of Tits

In 'Buildings and Finite $BN$-Pairs', Jacques Tits gives the following statement which is left as an easy exercise.
Let $G_1,G_2,G_3$ be three subgroups of a group $G$. Then the following conditions ...

**5**

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**2**answers

691 views

### Geometric interpretation of $BN$-pairs

My question is relative to a geometric interpretation of the $BN$-pairs that arise in Tits' theory of buildings. Here is a definition that comes from an article by G. Stroth (Nonspherical spheres).
...

**13**

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**1**answer

644 views

### Distance to an apartment of the affine building of GL(N)

Here $F$ is a locally compact non-archimedean non-discrete field.
Let $X$ be the reduced (affine) Bruhat-Tits building of ${\rm GL}(n,F)$. Fix a maximal split torus $T$. Let $B$ be a Borel subgroup ...

**10**

votes

**3**answers

989 views

### Looking for figure of part of an A2 affine building

Somewhere, I don't remember where, I saw a beautiful 3D figure of part a CAT(0) simplicial complex. I am thinking and hoping that this was some finite piece of an affine building of type A2, ...