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首页> 外文期刊>Annales de l'Institut Fourier >ON THE FIRST ORDER COHOMOLOGY OF INFINITE-DIMENSIONAL UNITARY GROUPS
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ON THE FIRST ORDER COHOMOLOGY OF INFINITE-DIMENSIONAL UNITARY GROUPS

机译：ON THE FIRST ORDER COHOMOLOGY OF INFINITE-DIMENSIONAL UNITARY GROUPS

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摘要

We determine precisely for which irreducible unitary highest weight representation of the group U(infinity), 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 N, 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(l(2)) of the direct limit U(infinity) with respect to the pth Schatten norm for 1 = p = infinity. For p infinity, 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 = infinity, the first cohomology spaces all vanish because U-infinity(l(2)) has property (FH) as a bounded topological group.

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