Questions tagged [buildings]
The buildings tag has no usage guidance.
90
questions
2
votes
0
answers
104
views
Bruhat-Tits theory and Jordan-Chevalley decomposition
The theory of Bruhat-Tits buildings is known to be able to unify some Lie group decompositions. Is there a sense in which an appropriate choice of building can unify the Jordan-Chevalley with the ...
- 4,511
2
votes
0
answers
66
views
Decompositions of groups and the existence of apartments
Let $X$ be an affine building and $G$ a group with isometric action on $X$. For any non-empty subset $\Omega$ of $X$, we denote by $P_{\Omega}$ the fixer of $\Omega$. Similarly, for any sector $\...
- 479
0
votes
1
answer
110
views
Does an affine building associated to a group satisfy the axioms of building?
Now I am reading Groupes réductifs sur un corps local : I. Données radicielles valuées written by Bruhat and Tits in 1972. Let $\Phi$ be a root system, $G$ a group with root data $(T,(U_{a},M_{a})_{a\...
- 479
7
votes
1
answer
225
views
Is there a "spherical building" for a reductive group over a Henselian local ring?
Let $A$ be a Henselian local ring and let $G$ be a split reductive $A$-group. I'm interested in some notion of a "building of parabolic subgroups" for the group scheme $G$.
In my specific ...
- 437
4
votes
0
answers
156
views
What are the good maximal compact subgroups in $p$-adic unitary groups?
Let $E/\mathbb Q_{p}$ be a quadratic extension and let $V$ be an $n$-dimensional $E$-hermitian space. Denote the hermitian form by $(\cdot,\cdot):V\times V \rightarrow E$. Let $G := \mathrm{U}(V)$ be ...
- 465
2
votes
0
answers
99
views
Affine Bruhat-Tits building associated to $\mathrm{SU}_3(\mathbb Q_p)$
I saw the following results on affine Bruhat-Tits building associated to $\mathrm{SU}_3(\mathbb Q_p)$ without giving any references, where $\mathrm{SU}_3$ is the quasi-split inner form of special ...
- 73
4
votes
2
answers
225
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 ...
- 9,937
4
votes
1
answer
176
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 ...
- 9,937
2
votes
1
answer
184
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 ...
- 479
3
votes
0
answers
109
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 $B$...
- 479
3
votes
1
answer
183
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 $F$...
- 465
2
votes
0
answers
164
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/...
- 465
6
votes
0
answers
72
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/...
- 873
5
votes
0
answers
100
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 $K$ ...
- 71
9
votes
1
answer
207
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 (...
- 1,950
2
votes
0
answers
287
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(...
user100742
6
votes
0
answers
228
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 ...
- 423
3
votes
0
answers
262
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 ...
- 821
9
votes
0
answers
447
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 ...
- 821
2
votes
0
answers
115
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 ...
- 5,898
2
votes
0
answers
160
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 ...
- 1,865
5
votes
0
answers
161
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{...
- 1,950
7
votes
0
answers
137
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 ...
- 5,607
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-...
- 5,607
2
votes
0
answers
86
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 ...
- 1,865
3
votes
0
answers
572
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 ...
- 409
3
votes
0
answers
60
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 $X$...
- 913
7
votes
1
answer
217
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 $i$-...
- 5,607
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 ...
- 5,607
9
votes
2
answers
320
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-...
- 93
8
votes
0
answers
179
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 ...
- 1,546
3
votes
1
answer
83
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 ...
- 5,898
5
votes
0
answers
194
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, ...
- 848
9
votes
0
answers
229
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 ...
- 5,607
2
votes
0
answers
99
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 ...
- 121
4
votes
1
answer
158
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 ...
- 41
5
votes
2
answers
858
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 ...
- 873
6
votes
1
answer
304
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 $CAT(0)$ ...
- 409
6
votes
1
answer
236
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 $P$...
- 2,359
4
votes
0
answers
141
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{...
- 2,359
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 ...
- 1,584
3
votes
0
answers
250
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 ...
- 121
8
votes
1
answer
404
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 ...
- 17.9k
2
votes
1
answer
299
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 ...
- 121
3
votes
0
answers
126
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 ...
- 24.7k
2
votes
0
answers
327
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/~...
- 685
4
votes
1
answer
352
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 $\...
- 339
6
votes
2
answers
107
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 ...
- 161
3
votes
1
answer
363
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 $...
- 339
9
votes
2
answers
602
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 ...
- 1,433