Bruhat Tits Building

Bruhat Tits Tree for . ${\displaystyle SL(2,\mathbb {Q} _{2})}$

In mathematics , Bruhat Tits buildings are a non-Archimedean variant of symmetrical spaces . They are named after François Bruhat and Jacques Tits .

Bruhat Tits building for SL (n, K)

Be one body and be a discreet review . The evaluation ring is defined by . ${\displaystyle K}$ ${\displaystyle v}$ ${\displaystyle {\mathcal {O}}_{v}}$ ${\displaystyle {\mathcal {O}}_{v}=\left\{x\in K\colon v(x)\geq 0\right\}}$

The Bruhat-Tits building for the special linear group is an (n-1) -dimensional simplicial complex . ${\displaystyle G=SL(n,K)}$

Corners : Its corners (0-simplices) are the homothesis classes of lattices in . (A lattice is a module of rank , two lattices belong to the same homothetic class if for one .) ${\displaystyle K^{n}}$ ${\displaystyle {\mathcal {O}}_{v}}$ ${\displaystyle n}$ ${\displaystyle \Lambda ,\Lambda ^{\prime }}$ ${\displaystyle \Lambda ^{\prime }=\alpha \Lambda }$ ${\displaystyle \alpha \in K}$

Simplizes : Corners form a -dimensional simplex if and only if they are interspersed with ${\displaystyle m+1}$ ${\displaystyle s_{0},\ldots ,s_{m}}$ ${\displaystyle m}$ ${\displaystyle \Lambda _{0},\ldots ,\Lambda _{m}}$

${\displaystyle \omega \Lambda _{m}\subset \Lambda _{0}\subset \Lambda _{1}\subset \ldots \subset \Lambda _{m}}$

be represented with an irreducible element . ${\displaystyle \omega \in {\mathcal {O}}_{v}}$

In particular, the Bruhat-Tits building is from an infinite tree whose nodes have the valence , where the remainder class field to be associated is. In this case one speaks of a Bruhat Tits tree . ${\displaystyle SL(2,K)}$ ${\displaystyle char(k)+1}$ ${\displaystyle k}$ ${\displaystyle K}$

In general, a Bruhat-Tits building can be defined for each reductive group over a local body . ${\displaystyle G}$ ${\displaystyle K}$

properties

The Bruhat Tits Building is a Euclidean building, and in particular a CAT (0) room . The link of each corner is a spherical Tits building and in particular a CAT (1) room .

The group acts properly discontinuously by simplicial automorphisms on its Bruhat-Tits buildings. ${\displaystyle G}$

The Bruhat Tits building is contractible , finite-dimensional and locally finite; the latter means that each simplex is only adjacent to a finite number of simplices .

literature

• Jean-Pierre Serre : Trees (= Springer Monographs in Mathematics. ). Translated from the French original by John Stillwell . Corrected 2nd printing of the 1980 English translation. Springer, Berlin et al. 2003, ISBN 3-540-44237-5 .
• Ian G. MacDonald: Spherical functions on a group of p-adic type (= Publications of the Ramanujan Institute. 2, ISSN  0304-9965 ). University of Madras - Ramanujan Institute, Madras 1971.

Web links

• Witte Morris: Introduction to Bruhat-Tits buildings
• Rabinoff: The Bruhat-Tits building of a p-adic Chevalley group and an application to representation theory
• Remy-Thuillier-Werner: Bruhat-Tits buildings and analytic geometry

Individual evidence

1. ↑ Section 3.2 in Remy-Thuillier-Werner, op. Cit.