Kac-Moody groups, hovels and Littelmann paths

Stéphane Gaussent[1]; Guy Rousseau[1]

  • [1] Institut Élie Cartan Unité Mixte de Recherche 7502 Nancy-Université, CNRS, INRIA Boulevard des Aiguillettes BP 239 54506 Vandœuvre-lès-Nancy cedex (France)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 7, page 2605-2657
  • ISSN: 0373-0956

Abstract

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We give the definition of a kind of building ℐ for a symmetrizable Kac-Moody group over a field K endowed with a discrete valuation and with a residue field containing ℂ . Due to the lack of some important property of buildings, we call it a hovel. Nevertheless, some good ones remain, for example, the existence of retractions with center a sector-germ. This enables us to generalize many results proved in the semisimple case by S. Gaussent and P. Littelmann. In particular, if K = ℂ ( ( t ) ) , the geodesic segments in ℐ , ending in a special vertex and retracting onto a given path π , are parametrized by a Zariski open subset P of ℂ N . This dimension N is maximal when π is a LS path and then P is closely related to some Mirković-Vilonen cycle.

How to cite

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Gaussent, Stéphane, and Rousseau, Guy. "Kac-Moody groups, hovels and Littelmann paths." Annales de l’institut Fourier 58.7 (2008): 2605-2657. <http://eudml.org/doc/10387>.

@article{Gaussent2008,
abstract = {We give the definition of a kind of building $\mathcal\{I\}$ for a symmetrizable Kac-Moody group over a field $K$ endowed with a discrete valuation and with a residue field containing $\mathbb\{C\}$. Due to the lack of some important property of buildings, we call it a hovel. Nevertheless, some good ones remain, for example, the existence of retractions with center a sector-germ. This enables us to generalize many results proved in the semisimple case by S. Gaussent and P. Littelmann. In particular, if $K=\mathbb\{C\}(\!(t)\!)$, the geodesic segments in $\mathcal\{I\}$, ending in a special vertex and retracting onto a given path $\pi $, are parametrized by a Zariski open subset $P$ of $\mathbb\{C\}^N$. This dimension $N$ is maximal when $\pi $ is a LS path and then $P$ is closely related to some Mirković-Vilonen cycle.},
affiliation = {Institut Élie Cartan Unité Mixte de Recherche 7502 Nancy-Université, CNRS, INRIA Boulevard des Aiguillettes BP 239 54506 Vandœuvre-lès-Nancy cedex (France); Institut Élie Cartan Unité Mixte de Recherche 7502 Nancy-Université, CNRS, INRIA Boulevard des Aiguillettes BP 239 54506 Vandœuvre-lès-Nancy cedex (France)},
author = {Gaussent, Stéphane, Rousseau, Guy},
journal = {Annales de l’institut Fourier},
keywords = {Kac-Moody group; valuated field; building; path},
language = {eng},
number = {7},
pages = {2605-2657},
publisher = {Association des Annales de l’institut Fourier},
title = {Kac-Moody groups, hovels and Littelmann paths},
url = {http://eudml.org/doc/10387},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Gaussent, Stéphane
AU - Rousseau, Guy
TI - Kac-Moody groups, hovels and Littelmann paths
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 7
SP - 2605
EP - 2657
AB - We give the definition of a kind of building $\mathcal{I}$ for a symmetrizable Kac-Moody group over a field $K$ endowed with a discrete valuation and with a residue field containing $\mathbb{C}$. Due to the lack of some important property of buildings, we call it a hovel. Nevertheless, some good ones remain, for example, the existence of retractions with center a sector-germ. This enables us to generalize many results proved in the semisimple case by S. Gaussent and P. Littelmann. In particular, if $K=\mathbb{C}(\!(t)\!)$, the geodesic segments in $\mathcal{I}$, ending in a special vertex and retracting onto a given path $\pi $, are parametrized by a Zariski open subset $P$ of $\mathbb{C}^N$. This dimension $N$ is maximal when $\pi $ is a LS path and then $P$ is closely related to some Mirković-Vilonen cycle.
LA - eng
KW - Kac-Moody group; valuated field; building; path
UR - http://eudml.org/doc/10387
ER -

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