doi:10.1090/S0002-9947-2011-05309-2

HTML articles powered by AMS MathViewer

by

Trans. Amer. Math. Soc. 363 (2011), 3799-3828

DOI: https://doi.org/10.1090/S0002-9947-2011-05309-2

Published electronically: February 15, 2011

PDF | Request permission

References

Alldridge, A.: “Toeplitz Operators on Semi-Simple Lie Groups”, Dissertation, Philipps-Universität Marburg, 2004.
Jonathan Arazy and Harald Upmeier, Boundary measures for symmetric domains and integral formulas for the discrete Wallach points, Integral Equations Operator Theory 47 (2003), no. 4, 375–434. MR 2021967, DOI 10.1007/s00020-003-1168-5
Wolfgang Bertram and Joachim Hilgert, Geometric Hardy and Bergman spaces, Michigan Math. J. 47 (2000), no. 2, 235–263. MR 1793623, DOI 10.1307/mmj/1030132532
Armand Borel and Lizhen Ji, Compactifications of symmetric and locally symmetric spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006. MR 2189882
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
Abdelhamid Boussejra and Khalid Koufany, Characterization of the Poisson integrals for the non-tube bounded symmetric domains, J. Math. Pures Appl. (9) 87 (2007), no. 4, 438–451 (English, with English and French summaries). MR 2317342, DOI 10.1016/j.matpur.2007.01.004
Hel Braun and Max Koecher, Jordan-Algebren, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 128, Springer-Verlag, Berlin-New York, 1966 (German). MR 0204470
Ralph Bremigan and John Lorch, Orbit duality for flag manifolds, Manuscripta Math. 109 (2002), no. 2, 233–261. MR 1935032, DOI 10.1007/s00229-002-0312-x
Mohammed Chadli, Noyau de Cauchy-Szegő d’un espace symétrique de type Cayley, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 1, 97–132 (French, with English and French summaries). MR 1614902
David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060
Mark G. Davidson and Ronald J. Stanke, Ladder representation norms for Hermitian symmetric groups, J. Lie Theory 10 (2000), no. 1, 157–170. MR 1748083
Jacques Faraut and Adam Korányi, Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1446489
I. M. Gel′fand and S. G. Gindikin, Complex manifolds whose spanning trees are real semisimple Lie groups, and analytic discrete series of representations, Funkcional. Anal. i Priložen. 11 (1977), no. 4, 19–27, 96 (Russian). MR 0492076
Glöckner, H.: “Infinite-Dimensional Complex Groups and Semigroups: Representations of Cones, Tubes, and Conelike Semigroups”, Dissertation, Technische Universität Darmstadt, 2000.
Harish-Chandra, Representations of semisimple Lie groups. VI. Integrable and square-integrable representations, Amer. J. Math. 78 (1956), 564–628. MR 82056, DOI 10.2307/2372674
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
Joachim Hilgert, Karl Heinrich Hofmann, and Jimmie D. Lawson, Lie groups, convex cones, and semigroups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. Oxford Science Publications. MR 1032761
Joachim Hilgert and Karl-Hermann Neeb, Lie semigroups and their applications, Lecture Notes in Mathematics, vol. 1552, Springer-Verlag, Berlin, 1993. MR 1317811, DOI 10.1007/BFb0084640
Joachim Hilgert, Karl-Hermann Neeb, and Bent Ørsted, The geometry of nilpotent coadjoint orbits of convex type in Hermitian Lie algebras, J. Lie Theory 4 (1994), no. 2, 185–235. MR 1337191
Joachim Hilgert, Karl-Hermann Neeb, and Bent Ørsted, Conal Heisenberg algebras and associated Hilbert spaces, J. Reine Angew. Math. 474 (1996), 67–112. MR 1390692
Joachim Hilgert, Karl-Hermann Neeb, and Werner Plank, Symplectic convexity theorems and coadjoint orbits, Compositio Math. 94 (1994), no. 2, 129–180. MR 1302314
Jaehyun Hong and Alan Huckleberry, On closures of cycle spaces of flag domains, Manuscripta Math. 121 (2006), no. 3, 317–327. MR 2271422, DOI 10.1007/s00229-006-0039-1
Toru Inoue, Unitary representations and kernel functions associated with boundaries of a bounded symmetric domain, Hiroshima Math. J. 10 (1980), no. 1, 75–140. MR 558849
Kenneth D. Johnson and Adam Korányi, The Hua operators on bounded symmetric domains of tube type, Ann. of Math. (2) 111 (1980), no. 3, 589–608. MR 577139, DOI 10.2307/1971111
Soji Kaneyuki, On orbit structure of compactifications of para-Hermitian symmetric spaces, Japan. J. Math. (N.S.) 13 (1987), no. 2, 333–370. MR 921587, DOI 10.4099/math1924.13.333
Soji Kaneyuki, Compactification of parahermitian symmetric spaces and its applications. II. Stratifications and automorphism groups, J. Lie Theory 13 (2003), no. 2, 535–563. MR 2003159
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239, DOI 10.1515/9781400883974
Anthony W. Knapp, Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389
Max Koecher, An elementary approach to bounded symmetric domains, Rice University, Houston, Tex., 1969. MR 0261032
Adam Korányi, The Poisson integral for generalized half-planes and bounded symmetric domains, Ann. of Math. (2) 82 (1965), 332–350. MR 200478, DOI 10.2307/1970645
Adam Korányi and Joseph A. Wolf, Realization of hermitian symmetric spaces as generalized half-planes, Ann. of Math. (2) 81 (1965), 265–288. MR 174787, DOI 10.2307/1970616
K. Koufany and B. Ørsted, Function spaces on the Ol′shanskiĭ semigroup and the Gel′fand-Gindikin program, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 3, 689–722 (English, with English and French summaries). MR 1411725
Khalid Koufany and Genkai Zhang, Hua operators and Poisson transform for bounded symmetric domains, J. Funct. Anal. 236 (2006), no. 2, 546–580. MR 2240174, DOI 10.1016/j.jfa.2006.02.014
Bernhard Krötz, On Hardy and Bergman spaces on complex Ol′shanskiĭ semigroups, Math. Ann. 312 (1998), no. 1, 13–52. MR 1645949, DOI 10.1007/s002080050211
Bernhard Krötz, Equivariant embeddings of Stein domains sitting inside of complex semigroups, Pacific J. Math. 189 (1999), no. 1, 55–73. MR 1687735, DOI 10.2140/pjm.1999.189.55
Bernhard Krötz, On the dual of complex Ol′shanskiĭ semigroups, Math. Z. 237 (2001), no. 3, 505–529. MR 1845335, DOI 10.1007/PL00004877
Bernhard Krötz and Eric Opdam, Analysis on the crown domain, Geom. Funct. Anal. 18 (2008), no. 4, 1326–1421. MR 2465692, DOI 10.1007/s00039-008-0684-5
Bernhard Krötz and Robert J. Stanton, Holomorphic extensions of representations. I. Automorphic functions, Ann. of Math. (2) 159 (2004), no. 2, 641–724. MR 2081437, DOI 10.4007/annals.2004.159.641
B. Krötz and R. J. Stanton, Holomorphic extensions of representations. II. Geometry and harmonic analysis, Geom. Funct. Anal. 15 (2005), no. 1, 190–245. MR 2140631, DOI 10.1007/s00039-005-0504-0
Ottmar Loos, Symmetric spaces. I: General theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0239005
Loos, O.: “Bounded Symmetric Domains and Jordan Pairs”, Lecture Notes, University of California, Irvine (1975)
Karl-Hermann Neeb, Invariant subsemigroups of Lie groups, Mem. Amer. Math. Soc. 104 (1993), no. 499, viii+193. MR 1152952, DOI 10.1090/memo/0499
Karl-Hermann Neeb, The classification of Lie algebras with invariant cones, J. Lie Theory 4 (1994), no. 2, 139–183. MR 1337190
Karl-Hermann Neeb, Invariant convex sets and functions in Lie algebras, Semigroup Forum 53 (1996), no. 2, 230–261. MR 1400650, DOI 10.1007/BF02574139
Karl-Hermann Neeb, Convexity properties of the coadjoint action of non-compact Lie groups, Math. Ann. 309 (1997), no. 4, 625–661. MR 1483827, DOI 10.1007/s002080050131
Karl-Hermann Neeb, On the complex and convex geometry of Ol′shanskiĭ semigroups, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 1, 149–203 (English, with English and French summaries). MR 1614894
Karl-Hermann Neeb, Holomorphy and convexity in Lie theory, De Gruyter Expositions in Mathematics, vol. 28, Walter de Gruyter & Co., Berlin, 2000. MR 1740617, DOI 10.1515/9783110808148
Toshihiko Matsuki, Equivalence of domains arising from duality of orbits on flag manifolds. III, Trans. Amer. Math. Soc. 359 (2007), no. 10, 4773–4786. MR 2320651, DOI 10.1090/S0002-9947-07-04076-7
Calvin C. Moore, Compactifications of symmetric spaces. II. The Cartan domains, Amer. J. Math. 86 (1964), 358–378. MR 161943, DOI 10.2307/2373170
G. Ólafsson and B. Ørsted, Causal compactification and Hardy spaces, Trans. Amer. Math. Soc. 351 (1999), no. 9, 3771–3792. MR 1458309, DOI 10.1090/S0002-9947-99-02101-7
G. I. Ol′shanskiĭ, Invariant cones in Lie algebras, Lie semigroups and the holomorphic discrete series, Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 53–66, 96 (Russian). MR 639200
G. I. Ol′shanskiĭ, Complex Lie semigroups, Hardy spaces and the Gel′fand-Gindikin program, Differential Geom. Appl. 1 (1991), no. 3, 235–246. MR 1244445, DOI 10.1016/0926-2245(91)90002-Q
Stephen M. Paneitz, Determination of invariant convex cones in simple Lie algebras, Ark. Mat. 21 (1983), no. 2, 217–228. MR 727345, DOI 10.1007/BF02384311
R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
Ichirô Satake, Algebraic structures of symmetric domains, Kanô Memorial Lectures, vol. 4, Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J., 1980. MR 591460
Robert J. Stanton, Analytic extension of the holomorphic discrete series, Amer. J. Math. 108 (1986), no. 6, 1411–1424. MR 868896, DOI 10.2307/2374530
Upmeier, H.: “Harmonische Analysis und Toeplitz-Operatoren auf beschränkten symmetrischen Gebieten”, Habilitationsschrift, Eberhard-Karls-Universität Tübingen, 1982.
Harald Upmeier, Toeplitz $C^{\ast }$-algebras on bounded symmetric domains, Ann. of Math. (2) 119 (1984), no. 3, 549–576. MR 744863, DOI 10.2307/2007085
Harald Upmeier, Symmetric Banach manifolds and Jordan $C^\ast$-algebras, North-Holland Mathematics Studies, vol. 104, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 96. MR 776786
Harald Upmeier, An index theorem for multivariable Toeplitz operators, Integral Equations Operator Theory 9 (1986), no. 3, 355–386. MR 846535, DOI 10.1007/BF01199351
Harald Upmeier, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math. 108 (1986), no. 1, 1–25 (1986). MR 821311, DOI 10.2307/2374466
È. B. Vinberg, Invariant convex cones and orderings in Lie groups, Funktsional. Anal. i Prilozhen. 14 (1980), no. 1, 1–13, 96 (Russian). MR 565090
Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Band 188, Springer-Verlag, New York-Heidelberg, 1972. MR 0498999
Joseph A. Wolf and Adam Korányi, Generalized Cayley transformations of bounded symmetric domains, Amer. J. Math. 87 (1965), 899–939. MR 192002, DOI 10.2307/2373253
Joseph A. Wolf, The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121–1237. MR 251246, DOI 10.1090/S0002-9904-1969-12359-1
Joseph A. Wolf, Flag manifolds and representation theory, Geometry and representation theory of real and $p$-adic groups (Córdoba, 1995) Progr. Math., vol. 158, Birkhäuser Boston, Boston, MA, 1998, pp. 273–323. MR 1486145
Joseph A. Wolf, Hermitian symmetric spaces, cycle spaces, and the Barlet-Koziarz intersection method for construction of holomorphic functions, Math. Res. Lett. 7 (2000), no. 5-6, 551–564. MR 1809282, DOI 10.4310/MRL.2000.v7.n5.a2
J. A. Wolf and R. Zierau, Cayley transforms and orbit structure in complex flag manifolds, Transform. Groups 2 (1997), no. 4, 391–405. MR 1486038, DOI 10.1007/BF01234542
Joseph A. Wolf and Roger Zierau, Linear cycle spaces in flag domains, Math. Ann. 316 (2000), no. 3, 529–545. MR 1752783, DOI 10.1007/s002080050342

Bibliographic Information

Published electronically: February 15, 2011
Journal: Trans. Amer. Math. Soc. 363 (2011), 3799-3828
DOI: https://doi.org/10.1090/S0002-9947-2011-05309-2
MathSciNet review: 2775828