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3
votes
2answers
141 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 ...
1
vote
0answers
88 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 $U_a(K)$ ...
5
votes
1answer
177 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 ...
3
votes
1answer
117 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 ...
6
votes
1answer
156 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 ...
8
votes
3answers
557 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
1answer
199 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
vote
2answers
168 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 ...
7
votes
2answers
552 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
votes
2answers
383 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
vote
1answer
188 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
votes
2answers
373 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 ...
6
votes
2answers
478 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
votes
2answers
476 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 ...
2
votes
3answers
303 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
1answer
289 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
votes
1answer
742 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
votes
2answers
652 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). ...
12
votes
1answer
582 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
3answers
929 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, ...