The centralizer of a classical group and Bruhat-Tits buildings

Daniel Skodlerack[1]

  • [1] Universität Münster Mathematisches Institut Einsteinstrasse 62 48149 Münster (Germany)

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 2, page 515-546
  • ISSN: 0373-0956

Abstract

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Let G be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let H be the centralizer of a semisimple rational Lie algebra element of G . We prove that the Bruhat-Tits building 𝔅 1 ( H ) of H can be affinely and G -equivariantly embedded in the Bruhat-Tits building 𝔅 1 ( G ) of G so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let j and j ′ be maps from 𝔅 1 ( H ) to 𝔅 1 ( G ) which preserve the Moy–Prasad filtrations. We prove that if there is no split torus in the center of the connected component of H then j and j ′ are equal, and in general if both maps are affine and satisfy a mild equivariance condition they differ up to a translation of 𝔅 1 ( H ) .

How to cite

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Skodlerack, Daniel. "The centralizer of a classical group and Bruhat-Tits buildings." Annales de l’institut Fourier 63.2 (2013): 515-546. <http://eudml.org/doc/275450>.

@article{Skodlerack2013,
abstract = {Let $G$ be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let $H$ be the centralizer of a semisimple rational Lie algebra element of $G.$ We prove that the Bruhat-Tits building $\mathfrak\{B\}^1(H)$ of $H$ can be affinely and $G$-equivariantly embedded in the Bruhat-Tits building $\mathfrak\{B\}^1(G)$ of $G$ so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let $j$ and $j^\{\prime\}$ be maps from $\mathfrak\{B\}^1(H)$ to $\mathfrak\{B\}^1(G)$ which preserve the Moy–Prasad filtrations. We prove that if there is no split torus in the center of the connected component of $H$ then $j$ and $j^\{\prime\}$ are equal, and in general if both maps are affine and satisfy a mild equivariance condition they differ up to a translation of $\mathfrak\{B\}^1(H).$},
affiliation = {Universität Münster Mathematisches Institut Einsteinstrasse 62 48149 Münster (Germany)},
author = {Skodlerack, Daniel},
journal = {Annales de l’institut Fourier},
keywords = {Building; classical group over a local field; centralizer; Bruhat-Tits buildings; classical groups over local fields; centralizers; Moy-Prasad filtrations; equivariant embeddings},
language = {eng},
number = {2},
pages = {515-546},
publisher = {Association des Annales de l’institut Fourier},
title = {The centralizer of a classical group and Bruhat-Tits buildings},
url = {http://eudml.org/doc/275450},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Skodlerack, Daniel
TI - The centralizer of a classical group and Bruhat-Tits buildings
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 2
SP - 515
EP - 546
AB - Let $G$ be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let $H$ be the centralizer of a semisimple rational Lie algebra element of $G.$ We prove that the Bruhat-Tits building $\mathfrak{B}^1(H)$ of $H$ can be affinely and $G$-equivariantly embedded in the Bruhat-Tits building $\mathfrak{B}^1(G)$ of $G$ so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let $j$ and $j^{\prime}$ be maps from $\mathfrak{B}^1(H)$ to $\mathfrak{B}^1(G)$ which preserve the Moy–Prasad filtrations. We prove that if there is no split torus in the center of the connected component of $H$ then $j$ and $j^{\prime}$ are equal, and in general if both maps are affine and satisfy a mild equivariance condition they differ up to a translation of $\mathfrak{B}^1(H).$
LA - eng
KW - Building; classical group over a local field; centralizer; Bruhat-Tits buildings; classical groups over local fields; centralizers; Moy-Prasad filtrations; equivariant embeddings
UR - http://eudml.org/doc/275450
ER -

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