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Irreducible affine isometric actions on Hilbert spaces. (English) Zbl 1359.22007

For locally compact groups in place of a theory of (unitary) linear representations, one considers a theory of affine isometric actions on Hilbert spaces. The theory is quite parallel with the theory of unitary representations, but because of the specific definition of an affine action, there are some specific aspects. An affine isometric action is irreducible iff there is no nonempty, closed proper group invariant affine subspace.
An affine isometric action \(\alpha: g \times \mathcal H \to \mathcal H\), \(g\mapsto \pi(g)v - v + b(g), g\in G, v\in \mathcal H\), of a group \(G\) on a Hilbert space \(\mathcal H\) is defined by a unitary representation \(\pi\), a 1-cocycle \(b(g), g\in G\). This action is irreducible iff the cocycle \(g\mapsto \pi(g)v - v + b(g), g\in G, v\in \mathcal H\), has a total image in \(\mathcal H\) (Proposition 2.3). For any unitary or orthogonal representation \(\pi\) the existence of an affine isometric action with linear part \(\pi\) with enveloping orbits (i.e. the closed convex hull of every orbit is equal to \(\mathcal H\)) implies the existence of an irreducible affine isometric action with linear part \(\pi\), which implies also that \(\pi\) is strongly cohomological (i.e. \(H^1(G,\sigma) \neq 0\) for every nonzero subrepresentation \(\sigma\) of \(\pi\)) (Definition 8.1 and Proposition 8.2). For a compactly generated group \(G\), the existence of an irreducible affine isometric action is equivalent to the fact that \(G\) does not have the property (T) of Kazhdan, Haagerup (Corollary 4.21).
The affine Schur’s lemma looks a little different: the action \(\alpha\) is irreducible iff the commutant \(\alpha(G)'\) in the monoid of continuous affine maps on \(\mathcal H\) is the set of translations along \(\mathcal H^{\pi(G)}\) (Proposition 3.6). But there is also some strange fact that the direct sum of two irreducible actions could nevertheless still be irreducible (Theorem 5.2). For abelian groups, every irreducible action is given by some homomorphism \(b: G \to \mathcal H\) with dense linear span of \(\mathrm{span}(b(G))\). The fact is true also for nilpotent groups (Corollary 4.21).
The property of super-rigidity of Shalom: if \(\Gamma\) is an irreducible co-compact lattice in a product of two or more locally compact groups, then any irreducible affine action of \(\Gamma\) that has linear part not weakly contained in the trivial representations, extends to an affine action of the ambient groups (Theorem 6.6).

MSC:

22D05 General properties and structure of locally compact groups
22F05 General theory of group and pseudogroup actions
22D10 Unitary representations of locally compact groups
20F24 FC-groups and their generalizations
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