# Questions tagged [buildings]

The buildings tag has no usage guidance.

84
questions

5
votes

2
answers

187
views

### Fixed points on spherical buildings

A crucial aspect of the Bruhat–Tits theory of affine buildings is the Bruhat–Tits fixed-point theorem, which, in one of many formulations, states that, if $\Gamma$ is a group of isometries of an ...

4
votes

1
answer

143
views

### Does the "building of parabolics" of a semisimple group have a simplex corresponding to the entire group?

Let $G$ be a semisimple (not just reductive) group over a field $k$. I believe that the question I am asking is what was meant in the second paragraph of Tits building of a linear algebraic group.
I ...

2
votes

1
answer

149
views

### Parahoric subgroup over a local field

$\DeclareMathOperator\SL{SL}$Let $F$ be a local field and $\mathcal{O}_{F}$ its valuation ring. Let $\pi\in \mathcal{O}_{F}$ be a uniformizer and $\mathfrak{p}=\pi\mathcal{O}_{F}$. Let $G$ be a split ...

3
votes

0
answers

102
views

### Why inherit the Tits systems structure by a $B$-adapted homomorphism?

Let $(G,B,N,S)$ be a Tits system and $\phi\colon G\longrightarrow \hat{G}$ a $B$-adapted in the sense of the paper Groupes réductifs sur un corps local: I of Bruhat–Tits. They said that $\phi$ is a $B$...

3
votes

1
answer

113
views

### Is it possible to detect when a maximal parahoric subgroup is (hyper)special from its finite reductive quotient?

Let $F$ be a $p$-adic field with residue field $k$ and let $G$ be a connected reductive group over $F$. Let us assume that $G$ is simply connected as an algebraic group over an algebraic closure of $F$...

1
vote

0
answers

93
views

### Confusion regarding special parahoric subgroups of the unitary group

This question is to clarify some confusion about special parahoric subgroups of a unitary group $G = \mathrm U_n(F)$ in an odd number of variables, with respect to an unramified quadratic extension $E/...

6
votes

0
answers

67
views

### Criterion for a collection of simplexes to lie in a common apartment (in a spherical building)

In Klyachko's paper "Equivariant vector bundles on toral varieties" I saw a statement about the Tits building of $\operatorname{GL}(n, \mathbb{C})$. I was wondering if this statement/...

5
votes

0
answers

93
views

### Problem with affine root subgroups of $SU_3$ ramified, residue characteristic $p=2$

Let $L/K$ be ramified quadratic extension of local fields, and let characteristic of the residue field of $K$ be $2$. Let $\mathbb{G}=SU_3$, $G=\mathbb{G}(K)$. Let $\text{val}$ be a valuation on $K$ ...

9
votes

1
answer

183
views

### Kazhdan's property (T) for $\tilde{C}_2$-lattices

It is known that higher rank lattices have property (T) and also that lattices on 2-dimensional Euclidean buildings have property (T) provided the thickness $q+1$ of the building is large enough (...

2
votes

0
answers

274
views

### Volume of double cosets $BwB$

In Macdonald's book "Spherical functions on a group of $p$-adic type", Prop. (3.1.7), it is stated that if $w=w_1\dots w_r$ is a reduced word for $w\in W$ (the affine Weyl group), and if $q(...

6
votes

0
answers

159
views

### The space of $p$-adic norms

The 1963 paper by Goldman and Iwahori The space of $p$-adic norms deals with the space of norms on a finite dimensional vector space $E$ over a locally compact complete discrete valuation field $K$. I ...

3
votes

0
answers

189
views

### Tits Reductive Groups over Local Fields Example 1.15 (Quasi-split special unitary groups in odd dimension)

I hope this question about Tits's paper "Reductive groups over local fields" in Algebraic groups and discontinuous subgroups ends up having an easy answer, but I'm a little stuck on the ...

9
votes

0
answers

423
views

### What is wrong with $A^{(2)}_{2n}$?

When dealing with affine Kac-Moody groups, especially geometrically (e.g. by examining their affine flag varieties or affine Grassmannians) I've been taught that time and time again, issues arise in ...

2
votes

0
answers

95
views

### Bruhat-Tits theory: how does the normalizer act on an apartment?

Let $G$ be the points of a split, simply connected, semisimple algebraic group over a $p$-adic field $k$. Let $T$ be a maximal torus of $G$, $T_c$ its unique maximal compact subgroup, and $N$ the ...

2
votes

0
answers

131
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 ...

5
votes

0
answers

151
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{...

7
votes

0
answers

124
views

### 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 ...

1
vote

0
answers

73
views

### 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-...

2
votes

0
answers

78
views

### 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 ...

3
votes

0
answers

544
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 ...

3
votes

0
answers

55
views

### 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
votes

1
answer

203
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 $i$-...

2
votes

0
answers

55
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
votes

2
answers

282
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-...

8
votes

0
answers

167
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
votes

1
answer

73
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
votes

0
answers

186
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, ...

9
votes

0
answers

217
views

### 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 ...

2
votes

0
answers

96
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
votes

1
answer

151
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 ...

5
votes

2
answers

767
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
votes

1
answer

270
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
votes

1
answer

223
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 $P$...

4
votes

0
answers

140
views

### 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{...

10
votes

1
answer

2k
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 ...

3
votes

0
answers

240
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
votes

1
answer

396
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
votes

1
answer

250
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 ...

3
votes

0
answers

125
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 ...

2
votes

0
answers

311
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
votes

1
answer

330
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
votes

2
answers

102
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 ...

3
votes

1
answer

286
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 $...

8
votes

2
answers

554
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 ...

5
votes

0
answers

505
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),...

22
votes

3
answers

2k
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
votes

1
answer

194
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
votes

0
answers

148
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

101
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
votes

1
answer

125
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 ...