[Sur la 1-cohomologie des groupes unitaires de dimension infinie]
Parmi les représentations unitaires irréductibles du groupe , la limite directe dénombrable des groupes unitaires de dimension finie , qui admettent un plus haut poids, nous déterminons précisément celles qui n’ont pas une 1-cohomologie triviale. Cela se produit en particulier si un plus haut poids, considéré comme une fonction de valeur entière sur , est une fonction à support fini. De plus, nous étendons les représentations admettant un plus haut poids à support fini en des représentations unitaires irréductibles des complétés de Banach de la limite directe par rapport à la norme de Schatten pour . Si , alors la 1-cohomologie n’est pas triviale non plus avec l’exception de trois cas particuliers. Nous en déduisons que ces groupes n’ont pas la propriété (T) de Kazhdan. De l’autre part, en cas de , la 1-cohomologie est triviale puisque le groupe topologique possède la propriété (FH).
We determine precisely for which irreducible unitary highest weight representation of the group , the countable direct limit of the finite-dimensional unitary groups , the corresponding 1-cohomology space does not vanish. This occurs in particular if a highest weight, viewed as an integer-valued function on , is finitely supported. In a second step, we extend the finitely supported highest weight representations to norm-continuous unitary representations of the Banach-completions of the direct limit with respect to the th Schatten norm for . For , the corresponding 1-cohomology spaces do not vanish either, except in three cases. We conclude that these groups do not have Kazhdan’s Property (T). On the other hand, for , the first cohomology spaces all vanish because has property (FH) as a bounded topological group.
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Keywords: First order group cohomology, unitary representation, (Banach–)Lie group, Lie algebra, direct limit group, Kazhdan’s property (T)
Mot clés : 1-Cohomologie des groupes, représentations unitaires, groupes de (Banach–) Lie, algèbres de Lie, limite inductive des groupes, la proprété (T) de Kazhdan
Herbst, Manuel 1 ; Neeb, Karl-Hermann 1
CC-BY-ND 4.0
@article{AIF_2018__68_5_2149_0,
author = {Herbst, Manuel and Neeb, Karl-Hermann},
title = {On the {First} {Order} {Cohomology} of {Infinite-Dimensional} {Unitary} {Groups}},
journal = {Annales de l'Institut Fourier},
pages = {2149--2176},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {68},
number = {5},
year = {2018},
doi = {10.5802/aif.3205},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3205/}
}
TY - JOUR AU - Herbst, Manuel AU - Neeb, Karl-Hermann TI - On the First Order Cohomology of Infinite-Dimensional Unitary Groups JO - Annales de l'Institut Fourier PY - 2018 SP - 2149 EP - 2176 VL - 68 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3205/ DO - 10.5802/aif.3205 LA - en ID - AIF_2018__68_5_2149_0 ER -
%0 Journal Article %A Herbst, Manuel %A Neeb, Karl-Hermann %T On the First Order Cohomology of Infinite-Dimensional Unitary Groups %J Annales de l'Institut Fourier %D 2018 %P 2149-2176 %V 68 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3205/ %R 10.5802/aif.3205 %G en %F AIF_2018__68_5_2149_0
Herbst, Manuel; Neeb, Karl-Hermann. On the First Order Cohomology of Infinite-Dimensional Unitary Groups. Annales de l'Institut Fourier, Tome 68 (2018) no. 5, pp. 2149-2176. doi : 10.5802/aif.3205. https://aif.centre-mersenne.org/articles/10.5802/aif.3205/
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