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Geometry of biinvariant subsets of complex semisimple Lie groups. (English) Zbl 0922.32011
Let \(G\) be a complex semi-simple Lie group and \(G^\mathbb{R}\) a real form of \(G\). The group \(G^\mathbb{R}\times G^\mathbb{R}\) acts on \(G\) by left and right translations, i.e. \(x\to gxh^{-1}\).
The authors define the notion of the generic \(G^\mathbb{R}\times G^\mathbb{R}\)-orbits in \(G\) and then investigate the relationships between the representation theory and the CR-geometry of the generic orbits in \(G\). First, the authors recall the basic notions about CR-structures and Levi-forms as well as Levi-cones and some general facts about group actions, Cartan subalgebras and the Bremingan’s description of the generic \(G^\mathbb{R}\times G^\mathbb{R}\)-orbits in \(G\). After describing the Levi-cones of a generic \(G^\mathbb{R}\times G^\mathbb{R}\)-orbit, the authors prove, among other things, that if \(S\) is a generic orbit in \(G\) satisfying certain conditions, then \(S\) cannot be contained in a level set of a non-constant biinvariant plurisubharmonic function.

MSC:
32V05 CR structures, CR operators, and generalizations
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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References:
[1] A. Boggess , CR-Manifolds and Tangential Cauchy-Riemann Complex, Studies in Advanced Math ., C.R.C. Press , 1991 . MR 1211412 | Zbl 0760.32001 · Zbl 0760.32001
[2] R. Bremigan , Invariant analytic domains in complex semisimple groups , Transformation Groups 1 ( 1996 ), 279 - 305 . MR 1424446 | Zbl 0867.22004 · Zbl 0867.22004 · doi:10.1007/BF02549210
[3] G. Fels - A.T. Huckleberry , A Characterisation of K-Invariant Stein Domains in Symmetric Embeddings, Complex Analysis and Geometry , Plenum Press , New York , 1993 , 223 - 234 . MR 1211883 | Zbl 0790.32030 · Zbl 0790.32030
[4] S.J. Greenfield , Cauchy-Riemann equations in several complex variables , Ann. Scuola Norm. Sup. Pisa 22 ( 1968 ), 257 - 314 . Numdam | MR 237816 | Zbl 0159.37502 · Zbl 0159.37502 · numdam:ASNSP_1968_3_22_2_275_0 · eudml:83459
[5] I.M. Gelfand - S.G. Gindikin , Complex manifolds whose skeletons are semisimple real Lie groups, and analytic discrete series of representations , Functional Anal. Appl . 7 - 4 ( 1977 ), 19 - 27 . MR 492076 | Zbl 0444.22006 · Zbl 0444.22006 · doi:10.1007/BF01077140
[6] J. Hilgert - K.H. Neeb , Lie Semigroups and their Applications , Lecture Notes in Math . 1552 , Springer Verlag , 1993 . MR 1317811 | Zbl 0807.22001 · Zbl 0807.22001
[7] J.E. Humphreys , Conjugacy Classes in Semisimple Algebraic Groups, Mathematical Surveys and Monographs , Vol. 43 , AMS , Providence, Rhole Island , 1995 . MR 1343976 | Zbl 0834.20048 · Zbl 0834.20048
[8] M. Lassalle , Séries de Laurent des fonctions holomorphes dans la complexification d’une espace symétrique compact , Ann. Sci. École Norm. Sup. 4 ( 1973 ), 267 - 290 . Numdam | Zbl 0452.43011 · Zbl 0452.43011 · numdam:ASENS_1978_4_11_2_167_0 · eudml:82012
[9] J.J. Loeb , Plurisubharmonicité et convexité sur les groupes reductifs complexes , Pub. IRMA - Lille 2 VIII ( 1986 ), 1 - 12 .
[10] J.J. Loeb , Action d’une forme réelle d’un groupe de Lie complexe sur les fonctions plurisubharmoniques , Ann. Inst. Fourier 35 ( 1985 ), 59 - 97 . Numdam | MR 812319 | Zbl 0563.32013 · Zbl 0563.32013 · doi:10.5802/aif.1028 · numdam:AIF_1985__35_4_59_0 · eudml:74698
[11] T. Matsuki - T. Oshima , Orbits of affine symmetric spaces under the action of the isotropy groups , J. Math. Soc. Japan 32 ( 1980 ), 399 - 414 . Article | MR 567427 | Zbl 0451.53039 · Zbl 0451.53039 · doi:10.2969/jmsj/03220399 · minidml.mathdoc.fr
[12] K.H. Neeb , Invariant convex sets and functions in Lie Algebras , Semigroups Forum 53 ( 1996 ), 305 - 349 . Article | MR 1400650 | Zbl 0873.17009 · Zbl 0873.17009 · doi:10.1007/BF02574139 · eudml:135488
[13] K.H. Neeb , On the Complex and Convex Geometry of Ol’shanskiĭ semigroups, preprint 10 , Institut Mittag-Leffler , 1995 / 96 . MR 1614894
[14] G I. OL’SHANSKIĭ, Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series , Functional Anal. Appl. 15 ( 1982 ), 275 - 285 . MR 639200 | Zbl 0503.22011 · Zbl 0503.22011 · doi:10.1007/BF01106156
[15] G.I. Ol’shanski , Complex Lie semigroups, Hardy spaces and the program of Gelfand Gindikin , Differential Geom. Appl. 1 ( 1982 ), 235 - 246 . MR 1244445 | Zbl 0789.22011 · Zbl 0789.22011 · doi:10.1016/0926-2245(91)90002-Q
[16] P.K. Rashevskij , On the connectedness of the fixed point set of an automorphism of a Lie group , Funct. Anal. Appl. 6 ( 1973 ), 341 - 342 . Zbl 0285.54036 · Zbl 0285.54036 · doi:10.1007/BF01077664
[17] R.J. Stanton , Analytic extension of the holomorphic discrete series , Amer. J. Math. 108 ( 1986 ), 1411 - 1424 . MR 868896 | Zbl 0626.43008 · Zbl 0626.43008 · doi:10.2307/2374530
[18] M. Sugiura , Conjugate classes of Cartan subalgebras in real semisimple Lie Algebras , J. Math. Soc. Japan 11 ( 1959 ), 374 - 434 . Article | MR 146305 | Zbl 0204.04201 · Zbl 0204.04201 · doi:10.2969/jmsj/01140374 · minidml.mathdoc.fr
[19] A.E. Tumanov , The geometry of CR-Manifolds, Encyclopaedia of Mathematical Sciences , Vol. 9 , Springer Verlag , 1989 , 201 - 221 . Zbl 0658.32007 · Zbl 0658.32007
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