Stability of the van der Waerden theorem on the continuity of homomorphisms of compact semisimple Lie groups

Abstract

The famous van der Waerden theorem [B.L. van der Waerden, Stetigkeitssätze für halbeinfache Liesche Gruppen, Math. Z. 36 (1933) 780–786] concerns the continuity of finite-dimensional representations of compact semisimple Lie groups. It turns out that this theorem is stable, i.e., if G is a compact semisimple Lie group and ρ : G → U( N) is a mapping for which the norm ∥ ρ( gg′) − ρ( g) ρ( g′)∥, g, g′ ∈ G, is uniformly small enough, then the mapping ρ is a small perturbation of a (necessarily continuous) ordinary unitary representation of G into U( N). This completely answers the question attributed to Milman by Kazhdan [D. Kazhdan, On ε-representations, Israel J. Math. 43 (4) (1982) 315–323] and gives a partial answer to Gromov’s question [M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, Functional Analysis on the Eve of the 21st Century, vol. II, Dordrecht, Boston, MA, 1996, pp. 1–213].