Continuity conditions for finite-dimensional representations of connected locally compact groups
The discontinuity subgroup of a relatively compact homomorphism \(\pi\) between a connected locally compact group \(G\) and a topological group \(H\) is the intersection \(\bigcap\limits_U \overline{\pi(U)}\), where \(U\) runs over all the neighbourhoods of the identity of \(G\). It is a closed connected normal subgroup of the closure \(\overline{\pi(G)}\) of the image of \(\pi\), as proved in the author's paper [Izv. Math. 72, No. 1, 169--205 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 1, 183--224 (2008; Zbl 1158.22002)], and measures in some sense how discontinuous \(\pi\) is. The first result of the paper under review addresses the case of a locally bounded finite-dimensional homomorphism \(\pi\) between connected Lie groups. Using structural facts established by \textit{K. Iwasawa} in [Ann. Math. (2) 50, 507--558 (1949; Zbl 0034.01803)] and results of his own [Izv. Math. 72, No. 1, 169--205 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 1, 183--224 (2008; Zbl 1158.22002)], the author proves that the discontinuity subgroup of such a morphism is a finite-dimensional torus in the center of \(\overline{\pi(G)}\). The rest of the paper is devoted to the case where \(\pi\) is a locally bounded finite-dimensional representation of a connected locally compact group \(G\). Let \(K\) be a compact normal subgroup such that \(G/K\) is a Lie group, and denote by \(H\) the commutant of \(K\) so that \(G=HK\). It is proved that, if \(\pi\) is continuous on \(K\), then it is automatically continuous on \(H'K\), where \(H'\) is the commutator subgroup of \(H\). Building on this and results he obtained in [Trans. Mosc. Math. Soc. 2011 (2011); translation from Tr. Mosk. Mat. O.-va 2011, No. 1, 79--95 (2011; Zbl 1242.22010)] regarding the structure of irreducible finite-dimensional representations, the author finally states a continuity criterion: a locally bounded finite-dimensional representation of a connected locally compact group \(G\) is continuous if and only if its restriction to some compact normal subgroup and the restriction of its semisimple part to the radical of \(G\) are both continuous.
scientific article; zbMATH DE number 6151584