Sp(n,1) admits a proper 1-cocycle for a uniformly bounded representation

TL;DR

This paper proves that the Lie group Sp(n,1) admits a proper affine isometric action on a Hilbert space with a uniformly bounded linear part, confirming Shalom's conjecture for all n.

Contribution

It provides two different proofs—one abstract and one explicit—demonstrating the existence of a proper 1-cocycle for Sp(n,1) with a uniformly bounded representation.

Findings

Sp(n,1) admits a proper affine isometric action on a Hilbert space.

Two distinct proofs are provided, one using automatic-properness results and the other based on Sobolev embeddings.

The work confirms Shalom's conjecture for all n in the context of Sp(n,1).

Abstract

We verify Shalom's conjecture for the simple real-rank-one Lie group Sp(n ,1) for any n: i.e. we show that it admits a metrically proper affine action on a Hilbert space whose linear part is a uniformly bounded representation. We provide two different proofs. Both approaches crucially use results on uniformly bounded representations by Michael Cowling. The first approach is quite abstract: it uses an automatic-properness result of Shalom and requires almost no computations. The second approach is explicit: we deduce the properness of cocycles from the non-continuity of a critical case of the Sobolev embedding. This work is inspired from Pierre Julg's work on the Baum--Connes conjecture for Sp(n,1).