On the First Order Cohomology of Infinite--Dimensional Unitary Groups

The irreducible unitary highest weight representations $(\pi_\lambda,\mathcal{H}_\lambda)$ of the group $U(\infty)$, which is the countable direct limit of the compact unitary groups $U(n)$, are classified by the orbits of the weights $\lambda \in \mathbb{Z}^{\mathbb{N}}$ under the Weyl group $S_{(\mathbb{N})}$ of finite permutations. Here, we determine those weights $\lambda$ for which the first cohomology space $H^1(U(\infty),\pi_\lambda,\mathcal{H}_\lambda)$ vanishes. For finitely supported $\lambda \neq 0$, we find that the first cohomology space $H^1(U(\infty),\pi_\lambda,\mathcal{H}_\lambda)$ never vanishes. For these $\lambda$, the highest weight representations extend to norm-continuous irreducible representations of the full unitary group $U(\mathcal{H})$ (for $\mathcal{H}:= \ell^2(\mathbb{N},\mathbb{C})$) endowed with the strong operator topology and to norm-continuous representations of the unitary groups $U_p(\mathcal{H})$ ($p\in [1,\infty]$) consisting of those unitary operators $g\in U(\mathcal{H})$ for which $g-\mathbb{1}$ is of $p$th Schatten class. However, not every 1-cocycle on $U(\infty)$ automatically extends to one on these unitary groups, so we may not conclude that the first cohomology spaces of the extended representations are non-vanishing. On the contrary, for the groups $U(\mathcal{H})$ and $U_\infty(\mathcal{H})$, all first cohomology spaces vanish. This is different for the groups $U_p(\mathcal{H})$ with $1\leq p <\infty$, where only the natural representation on $\mathcal{H}$ and on its dual have vanishing first cohomology spaces.