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

TL;DR

This paper classifies when the first cohomology of infinite-dimensional unitary groups vanishes or not, revealing how certain representations extend and how cohomology behaves across different unitary groups.

Contribution

It determines the weights for which the first cohomology vanishes and analyzes the extension of representations and cohomology properties across various infinite-dimensional unitary groups.

Findings

First cohomology vanishes for certain weights and groups.

Finitely supported non-zero weights have non-vanishing cohomology.

Cohomology vanishes for full unitary groups but not for all Schatten class groups.

Abstract

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.