On the First Order Cohomology of Infinite-Dimensional Unitary Groups
Annales de l'Institut Fourier, Volume 68 (2018) no. 5, p. 2149-2176
We determine precisely for which irreducible unitary highest weight representation of the group U(), the countable direct limit of the finite-dimensional unitary groups U(n), the corresponding 1-cohomology space H 1 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 U p ( 2 ) of the direct limit U() with respect to the pth Schatten norm for 1p. For p<, the corresponding 1-cohomology spaces H 1 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 p=, the first cohomology spaces all vanish because U ( 2 ) has property (FH) as a bounded topological group.
Parmi les représentations unitaires irréductibles du groupe U(), la limite directe dénombrable des groupes unitaires de dimension finie U(n), 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 U p ( 2 ) de la limite directe U() par rapport à la norme p de Schatten pour 1p. Si p<, 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 p=, la 1-cohomologie est triviale puisque le groupe topologique U ( 2 ) possède la propriété (FH).
Received : 2016-07-05
Revised : 2017-05-15
Accepted : 2017-11-14
Published online : 2018-11-23
DOI : https://doi.org/10.5802/aif.3205
Classification:  22E41
Keywords: First order group cohomology, unitary representation, (Banach–)Lie group, Lie algebra, direct limit group, Kazhdan’s property (T)
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {5},
     year = {2018},
     pages = {2149-2176},
     doi = {10.5802/aif.3205},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/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, Volume 68 (2018) no. 5, pp. 2149-2176. doi : 10.5802/aif.3205. https://aif.centre-mersenne.org/item/AIF_2018__68_5_2149_0/

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