A version of van der Waerden's theorem and a proof of Mishchenko's conjecture on homomorphisms of locally compact groups

van der Waerden proved in 1933 that every finite-dimensional locally bounded representation of a semisimple compact Lie group is automatically continuous. This theorem evoked an extensive literature, which related the assertion of the theorem (and its converse) to properties of Bohr compactifications of topological groups and led to the introduction and study of classes of so-called van der Waerden groups and algebras. In the present paper we study properties of (not necessarily continuous) locally relatively compact homomorphisms of topological groups (in particular, connected locally compact groups) from the point of view of this theorem and obtain a classification of homomorphisms of this kind from the point of view of their continuity or discontinuity properties (this classification is especially simple in the case of Lie groups because it turns out that every locally bounded finite-dimensional representation of a connected Lie group is continuous on the commutator subgroup). Our main results are obtained by studying new objects, namely, the discontinuity group and the final discontinuity group of a locally bounded homomorphism, and the new notion of a finally continuous homomorphism from one locally compact group into another.
The notion of local relative compactness of a homomorphism is naturally related to the notion of point oscillation (at the identity element of the group) introduced by the author in 2002. According to a conjecture of A. S. Mishchenko, the (reasonably defined) oscillation at a point of any finite-dimensional representation of a ‘good’ topological group can take one of only three values: $0$, $2$ and $\infty$. We shall prove this for all connected locally compact groups.