2010/11/04 by Karl‐Hermann Neeb, Karl-Hermann Neeb, Neeb, Karl-Hermann
Mathematics · Physics and Astronomy · #22E45 #22E65 #Advanced Algebra and Geometry #Advanced Differential Geometry Research #FOS: Mathematics #Mathematical Analysis and Transform Methods #Representation Theory (math.RT) #math.RT #msc:22E45 #msc:22E65
paper · pdf · doi:10.48550/arxiv.1011.1210
36 pages
arxiv created 2010/11/04 · openalex publication_date 2010/11/04 · arxiv updated 2010/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we explore the method of holomorphic induction for unitary representations of Banach--Lie groups. First we show that the classification of complex bundle structures on homogeneous Banach bundles over complex homogeneous spaces of real Banach--Lie groups formally looks as in the finite dimensional case. We then turn to a suitable concept of holomorphic unitary induction and show that this process preserves commutants. In particular, holomorphic induction from irreducible representations leads to irreducible ones. Finally we develop criteria to identify representations as holomorphically induced ones and apply these to the class of so-called positive energy representations. All this is based on extensions of Arveson's concept of spectral subspaces to representations on Fréchet spaces, in particular on spaces of smooth vectors.