ON THE FIRST ORDER COHOMOLOGY OF INFINITE-DIMENSIONAL UNITARY GROUPS

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.