Irreducible Locally Bounded Finite-Dimensional Pseudorepresentations of Connected Locally Compact Groups Revisited
- PDF / 310,350 Bytes
- 3 Pages / 612 x 792 pts (letter) Page_size
- 94 Downloads / 250 Views
c Pleiades Publishing, Ltd., 2020.
Irreducible Locally Bounded Finite-Dimensional Pseudorepresentations of Connected Locally Compact Groups Revisited A. I. Shtern∗,∗∗,∗∗∗,1 ∗
Moscow Center for Fundamental and Applied Mathematics, Moscow, 119991 Russia ∗∗ Department of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia ∗∗∗ Scientific Research Institute for System Analysis of the Russian Academy of Sciences (FGU FNTs NIISI RAN), Moscow, 117312 Russia, E-mail: 1 [email protected] Received November 21, 2019; Revised December 4, 2019; Accepted December 15, 2019
Abstract. The irreducible locally bounded finite-dimensional pure pseudorepresentations of connected locally compact groups are described. DOI 10.1134/S1061920820030097
1. INTRODUCTION For the definitions and notation, see [1–4]. As was noted in [5], bounded quasirepresentations always admit small (in norm) perturbations that define irreducible bounded quasirepresentations. The pseudorepresentations corresponding to these quasirepresentations are defined more rigidly, and therefore it is reasonable to seek for irreducible pseudorepresentations, for example, in the finite-dimensional case. However, the problem becomes even more interesting for unbounded quasirepresentations. Indeed, even for the simplest class of bounded (or even small) perturbations of ordinary unbounded pseudorepresentations, the family of perturbations turns out to be subjected to strong conditions, see the introduction to [5]. This increases the interest to irreducible locally bounded finite-dimensional pseudorepresentations of topological groups. The paper [5] promised to give a description of irreducible locally bounded finite-dimensional pseudorepresentations of connected locally compact groups. Instead, as follows from the text of the proofs of the theorem in [5], the statement of the theorem is related to semisimple Lie groups only rather than to connected locally compact groups (everywhere in the text of [5], the words “connected locally compact group” should read “connected semisimple Lie group”). In [6], we clarified the matter for irreducible locally bounded finite-dimensional pure pseudorepresentations of connected Lie groups. In the present note, we describe the irreducible locally bounded finite-dimensional pure pseudorepresentations of all connected locally compact groups. 2. PRELIMINARIES Recall the general result concerning the structure of quasirepresentations of arbitrary groups. Theorem A [1–4] Let G be a group and let π be a quasirepresentation of G on a finite-dimensional vector space Eπ . Let Eπ∗ be the space dual to Eπ . Let L be the set of vectors ξ ∈ Eπ whose orbit {π(g)ξ | g ∈ G} is bounded in E; let M be the set of functionals f ∈ Eπ∗ whose orbit {π(g)∗ f | g ∈ G} is bounded in Eπ∗ ; then L and the annihilator M ⊥ are π-invariant vector subspaces in Eπ . Let us consider an increasing family of subspaces {0}, L ∩ M ⊥ , M ⊥ , L + M ⊥ , E = Eπ and write out the matrix t(g) of the operator π(g), g ∈ G, in a block form corresponding to the decomposition of
Data Loading...
