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Representation theory and convexity. (English) Zbl 0964.22004

In Section I the essential results on Ol’shanskii semigroups \(S=G \operatorname {Exp} (iW)\), where \(G\) is a Lie group, and invariant convex functions are collected. These semigroups are complex domains generalizing complex reductive groups that show up for \(W=\mathfrak g\) where \(\mathfrak g\) is a compact Lie algebra. Let \(D\subset S\) be a \(G\)-biinvariant domain. Then \(G\times G\) acts on \(D\), and one has to do with harmonic analysis of Hilbert spaces of holomorphic functions \({\mathcal H}\subset Hol(D)\) on which \(G\times G\) acts unitarily. These spaces are called biinvariant Hilbert spaces. In Section II it is shown that irreducible biinvariant Hilbert spaces come from highest weight representations. For example, the unitary representation of \(G\times G\) on a biinvariant Hilbert space \({\mathcal H}\) extends to an action of the semigroup \(S_{\min}\times S_{\min}\) by contractions on \({\mathcal H}\). In Section III the author explains how to construct biinvariant Kähler structures on biinvariant Stein domains and demonstrates with the help of a certain Legendre transform that the studied symplectic manifolds are isomorphic to domains in the cotangent bundle \(T^*(G)\).
Reviewer: A.K.Guts (Omsk)

MSC:

22E10 General properties and structure of complex Lie groups
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
32M05 Complex Lie groups, group actions on complex spaces
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[1] [AL92] D. Arnal, J. Ludwig, La convexit? de l’application moment d’un groupe de Lie, J. Funct. Anal.105:2 (1992), 256-300. · Zbl 0763.22006 · doi:10.1016/0022-1236(92)90080-3
[2] [BtD85] T. Br?cker, T. tom Dieck,Representations of Compact Lie Groups, Springer Verlag, New York, Heidelberg, 1985.
[3] [FaTh99] J. Faraut, E. G. F. Thomas,Invariant Hilbert spaces of holomorphic functions, J. Lie Theory9:2 (1999), 383-402. · Zbl 1014.32005
[4] [Fe49] M. Fenchel,On conjugate convex functions, Canad. J. Math.1 (1949), 73-77. · Zbl 0038.20902 · doi:10.4153/CJM-1949-007-x
[5] [GR77] H. Grauert, R. Remmert,Theorie der Steinschen R?ume, Grundlehren der math. Wissenschaften227, Springer Verlag, Heidelberg, 1977. Russian translation: ?. ???????, ?. ???????,?????? ??????????? ??????, ?????, ?., 1989.
[6] [GS91] V. Guillemin, M. Stenzel,Grauert tubes and the homogeneous Monge-Amp?re equation, J. Diff. Geom.34 (1991), 561-570. · Zbl 0746.32005
[7] [GS92] V. Guillemin, M. Stenzel,Grauert tubes and the homogeneous Monge-Amp?re equation, J. Diff. Geom.35 (1992), 627-641. · Zbl 0789.32010
[8] [Ha97] B. C. Hall,Phase space bounds for quantum mechanics on a compact Lie group, Comm. Math. Phys.184 (1997), 233-250. · Zbl 0869.22013 · doi:10.1007/s002200050059
[9] [HHL89] J. Hilgert, K. H. Hofmann, J. D. Lawson,Lie Groups, Convex Cones, and Semigroups, Oxford University Press, 1989. · Zbl 0701.22001
[10] [Kr97] B. Kr?tz,The Plancherel theorem for biinvariant Hilbert spaces, R.I.M.S. Publ.35 (1999), 91-122. · Zbl 0999.22018 · doi:10.2977/prims/1195144190
[11] [Kr98] B. Kr?tz,On Hardy and Bergman spaces on complex Ol’shanski? semigroups, Math. Ann.312 (1998), 13-52. · Zbl 0922.22007 · doi:10.1007/s002080050211
[12] [Kr99] B. Kr?tz,Equivariant embeddings of Stein domains sitting inside of complex semigroups, Pac. J. Math.189:1 (1999), 55-73. · Zbl 0942.32013 · doi:10.2140/pjm.1999.189.55
[13] [KN?97] B. Kr?tz, K.-H. Neeb, G. ?lafsson,Spherical representations and mixed symmetric spaces, J. Representation Theory1 (1997), 424-461. · Zbl 0887.22022 · doi:10.1090/S1088-4165-97-00035-6
[14] [Ne95] K.-H. Neeb,On the convexity of the moment mapping for a unitary highest weight representation, J. Funct. Anal.127:2 (1995), 301-325. · Zbl 0829.22012 · doi:10.1006/jfan.1995.1014
[15] [Ne96a] K.-H. Neeb,Invariant convex sets and functions in Lie algebras, Semigroup Forum53 (1996), 230-261. · Zbl 0873.17009 · doi:10.1007/BF02574139
[16] [Ne96b] K.-H. Neeb,Coherent states, holomorphic extensions, and highest weight representations, Pac. J. Math.,174:2 (1996), 497-542. · Zbl 0894.22008
[17] [Ne97] K.-H. Neeb,On some classes of multiplicity free representations, Manuscripta Math.92 (1997), 389-407. · Zbl 0882.43002 · doi:10.1007/BF02678201
[18] [Ne98a] K.-H. Neeb,On the complex and convex geometry of Ol’shanski? semigroups, Annales de l’Institut Fourier48:2 (1996), 149-203.
[19] [Ne98b] K.-H. Neeb,Some open problems in representation theory related to complex geometry, in Positivity in Lie Theory: Open Problems (J. Hilgert et. al.); Expositions in Math., vol. 26, de Gruyter, 1998, pp. 192-200.
[20] [Ne99a] K.-H. Neeb,Holomorphy and Convexity in Lie Theory, Expositions in Mathematics, vol. 28, de Gruyter, Berlin, 1999.
[21] [Ne99b] K.-H. Neeb,On the complex geometry of invariant domains in complexified symmetric spaces, Annales de l’Institut Fourier49:1, 177-225.
[22] [Sz95] R. Sz?ke,Automorphisms of certain Stein manifolds, Math. Zeit.219:3 (1995), 357-386. · Zbl 0829.32009 · doi:10.1007/BF02572371
[23] [Th94] E. G. F. Thomas,Integral representations in conuclear cones, J. Convex Analysis1:2 (1994), 225-258. · Zbl 0837.46009
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