Van der Waerden continuity theorem for semisimple Lie groups
The famous van der Waerden continuity theorem asserts that a finite-dimensional, locally bounded representation of a compact semisimple Lie group is automatically continuous. The present paper offers surprising generalizations of this result. The author begins with a beautiful exposition of the history of the original van der Waerden theorem, its impact and its ramifications. The restriction to a subclass of the class of all Lie groups is indispensable because of the discontinuous characters of the real line obtained from a Hamel basis. However, the author shows that the compactness condition in van der Waerden's theorem is superfluous. In the case of a compact Lie group, he gives several other conditions that can replace local boundedness, e.g., Haar measurability of the representation, or local boundedness of its restriction to some maximal torus. In the general case, it suffices to require local boundedness of the restriction to any of the three factors \(K\), \(A\), \(N\) of an Iwasawa decomposition of the group \(G\). The case of \(G = \text{ SL}(2,\mathbb R)\) is studied in detail. The author announces a further paper, where he applies his result in order to show that every locally bounded finite-dimensional representation of a Lie group \(G\) is continuous if and only if \(G\) coincides with its commutator subgroup.
scientific article; zbMATH DE number 5136623
scientific article