# Questions tagged [buildings]

The buildings tag has no usage guidance.

100
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### Arbitrary base change of a parahoric subgroup in split case

Assume $R\subset R'$ are henselien discretly valued rings with fraction field $K$ and $K'$, $G$ is a semisimple split group over $K$. Consider the parahoric group scheme $\mathcal{P}_F$ over $R$ ...

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### Proof involving retractions onto apartments

Let $\Delta$ be a (thick) building and let $\Sigma$ be an apartment. Let $C$ and $C'$ be adjacent chambers of $\Sigma$. Then $C$ and $C'$ have common wall $B \in \Sigma$. Since $\Delta$ is thick, ...

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### Weyl groups are Coxeter groups proof

I'm reading part of a proof that says that Weyl groups of apartments of buildings are Coxeter groups.
Let $\Delta$ be a building and let $\Sigma$ be a fixed apartment of $\Delta$. Let $C$ be a fixed ...

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### Reflections of an apartment in building — Weyl groups

I will give the definition of what I mean by reflection which is in Suzuki's group theory I.
Let $\Sigma $ be an apartment of a building that contains adjacent chambers $C$ and $C'$. Then there are ...

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101
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### Structure theory of Schubert varieties (extend results from semisimple groups to reductive)

The lecture notes Borel–Weil–Bott theorem and geometry of Schubert varieties by Shrawan Kumar present a concise summary of major results on cohomology of flag varieties $G/B$ for $G$ semisimple, ...

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### Tempered representations and unramified principal series

For $V$ a tempered representation of connected reductive group over a local field of characteristic zero. I want to show that for an Iwahori subgroup $B$, the set of fixed points $V^B\neq 0$, thereby ...

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### Is Landvogt's thesis "The functorial properties of the Bruhat–Tits building" available online?

Universität Münster publishes theses online through "miami", but "miami" doesn't have Erasmus Landvogt's thesis (search).
ProQuest (predatorily) provides many theses, but they don'...

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### Weyl group is the set of type preserving automorphisms

So in Suzuki’s group theory I we have a proposition 3.20 (ii) which says if we have a building $ \Delta $ and $\Sigma $ an apartment with $C \in \Sigma $ a chamber. Also we have $B \in \Sigma $ is any ...

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351
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### Buildings as generalizations of symmetric spaces

In almost every introductory notes on Tits buildings these are motivated
as structures capturing/ sharing several features of symmetric spaces. Could somebody elaborate what are precisely the main ...

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### Group theory - buildings/complexes, retractions Suzuki

I’m reading this book called group theory I by Michio Suzuki and part of a proof I just cannot figure out. It’s on page 322 proposition 3.18. I will include it here:
(3.18) Let $\Delta$ be a building,...

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### 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 ...

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### 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 $\...

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135
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### 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\...

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282
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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $B$...

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### 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 $F$...

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### 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/...

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### 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/...

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### 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 $K$ ...

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### 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 (...

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### 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(...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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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### 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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### 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 $X$...

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### 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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### 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 ...

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### 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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### 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 ...

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### 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 ...

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### 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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $CAT(0)$ ...

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### 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{...