Lie Theory pp 213-328 | Cite as

Infinite-Dimensional Groups and Their Representations

  • Karl-Hermann Neeb
Part of the Progress in Mathematics book series (PM, volume 228)


This article provides an introduction to the representation theory of Banach-Lie groups of operators on Hilbert spaces, where our main focus lies on highest weight representations and their geometric realization as spaces of holomorphic sections of a complex line bundle. After discussing the finite-dimensional case in Section I, we describe the algebraic side of the theory in Sections II and III. Then we turn in Sections IV and V to Banach-Lie groups and holomorphic representations of complex classical groups. The geometry of the coadjoint action is discussed in Section VI, and in the concluding Section VII all threads lead to a full discussion of the theory for the group U 2(H) of unitary operators u on a Hilbert space H for which u1 is Hilbert-Schmidt.


Holomorphic Section Coadjoint Orbit Holomorphic Line Bundle High Weight Module High Weight Representation 
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© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Karl-Hermann Neeb
    • 1
  1. 1.Technische Universität DarmstadtDarmstadtGermany

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