Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Boundary orbit strata and faces of invariant cones and complex Ol'shanskiĭ semigroups


Author: Alexander Alldridge
Journal: Trans. Amer. Math. Soc. 363 (2011), 3799-3828
MSC (2010): Primary 22E60, 32M15; Secondary 22A15, 52A05
DOI: https://doi.org/10.1090/S0002-9947-2011-05309-2
Published electronically: February 15, 2011
MathSciNet review: 2775828
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ D=G/K$ be an irreducible Hermitian symmetric domain. Then $ G$ is contained in a complexification $ G_{\mathbb{C}}$, and there exists a closed complex subsemigroup $ G\subset\Gamma\subset G_{\mathbb{C}}$, the so-called minimal Ol'shanskiĭ semigroup, characterised by the fact that all holomorphic discrete series representations of $ G$ extend holomorphically to $ \Gamma^\circ$.

Parallel to the classical theory of boundary strata for the symmetric domain $ D$, due to Wolf and Korányi, we give a detailed and complete description of the $ K$-orbit type strata of $ \Gamma$ as $ K$-equivariant fibre bundles. They are given by the conjugacy classes of faces of the minimal invariant cone in the Lie algebra.


References [Enhancements On Off] (What's this?)

  • 1. Alldridge, A.: ``Toeplitz Operators on Semi-Simple Lie Groups'',
    Dissertation, Philipps-Universität Marburg, 2004.
  • 2. Arazy, J., Upmeier, H.: 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 (2004m:32042)
  • 3. Bertram, W., Hilgert, J.: Geometric Hardy and Bergman spaces,
    Michigan Math. J. 47 (2000), no. 2, 235-263. MR 1793623 (2001k:32008)
  • 4. Borel, A., Ji, L.: ``Compactifications of Symmetric and Locally Symmetric Spaces'',
    Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, 2006. MR 2189882 (2007d:22030)
  • 5. Bourbaki, N.: ``Éléments de mathématique: Groupes et algèbres de Lie'',
    Act. Sci. Ind. 1337, Hermann, Paris, 1968. MR 0240238 (39:1590)
  • 6. Boussejra, A., Koufany, K.: Characterization of the Poisson integrals for the non-tube bounded symmetric domains,
    J. Math. Pures Appl. (9) 87 (2007), no. 4, 438-451. MR 2317342 (2008c:32009)
  • 7. Braun, H., Koecher, M.: ``Jordan-Algebren'',
    Grundlehren 128, Springer-Verlag, Berlin, 1966. MR 0204470 (34:4310)
  • 8. Bremigan, R., Lorch, J.: Orbit duality for flag manifolds,
    Manuscripta Math. 109 (2002), no. 2, 233-261. MR 1935032 (2004b:22020)
  • 9. Chadli, M.: Noyau de Cauchy-Szegö d'un espace symétrique de type Cayley,
    Ann. Inst. Fourier (Grenoble) 48 (1998), no. 1, 97-132. MR 1614902 (99b:22022)
  • 10. Collingwood, D., McGovern, W.: ``Nilpotent Orbits in Semisimple Lie Algebras'',
    Van Nostrand, New York, 1993. MR 1251060 (94j:17001)
  • 11. Davidson, M.G., Stanke, R.J.: Ladder representation norms for Hermitian symmetric groups,
    J. Lie Theory 10 (2000), no. 1, 157-170. MR 1748083 (2001h:22018)
  • 12. Faraut, J., Korányi, A.: ``Analysis on Symmetric Cones'',
    Oxford Mathematical Monographs, The Clarendon Press, New York, 1994. MR 1446489 (98g:17031)
  • 13. Gel'fand, I.M., Gindikin, S.G.: Complex manifolds whose skeletons are semi-simple Lie groups, and analytic discrete series of representations,
    Funct. Anal. Appl. 11 (1977), no. 4, 19-27. MR 0492076 (58:11230)
  • 14. Glöckner, H.: ``Infinite-Dimensional Complex Groups and Semigroups: Representations of Cones, Tubes, and Conelike Semigroups'',
    Dissertation, Technische Universität Darmstadt, 2000.
  • 15. Harish-Chandra: Representations of semisimple Lie groups. VI. Integrable and square-integrable representations,
    Amer. J. Math. 78 (1956), 564-628. MR 0082056 (18:490d)
  • 16. Helgason, S.: ``Differential Geometry, Lie Groups, and Symmetric Spaces'',
    Pure and Appl. Math. 80. Academic Press, Inc., New York, 1978. MR 514561 (80k:53081)
  • 17. Hilgert, J., Hofmann, K.H., Lawson, J.D.: ``Lie Groups, Convex Cones, and Semigroups'',
    Oxford Mathematical Monographs, The Clarendon Press, New York, 1989. MR 1032761 (91k:22020)
  • 18. Hilgert, J., Neeb, K.H.: ``Lie Semigroups and their Applications'',
    Lect. Notes Math. 1552, Springer-Verlag, Berlin, 1993. MR 1317811 (96j:22002)
  • 19. Hilgert, J., Neeb, K.H., Ørsted, B.: The geometry of nilpotent coadjoint orbits of convex type in Hermitian Lie algebras,
    J. Lie Theory 4 (1994), no. 2, 185-235. MR 1337191 (96e:17016)
  • 20. Hilgert, J., Neeb, K.H., Ørsted, B.: Conal Heisenberg algebras and associated Hilbert spaces,
    J. Reine Angew. Math. 474 (1996), 67-112. MR 1390692 (97d:22014)
  • 21. Hilgert, J., Neeb, K.H., Plank, W.: Symplectic convexity theorems and coadjoint orbits,
    Compositio Math. 94 (1994), no. 2, 129-180. MR 1302314 (96d:53053)
  • 22. Hong, J., Huckleberry, A.: On closures of cycle spaces of flag domains,
    Manuscripta Math. 121 (2006), no. 3, 317-327. MR 2271422 (2007h:32017)
  • 23. Inoue, T.: Unitary representations and kernel functions associated with boundaries of a bounded symmetric domain,
    Hiroshima Math. J. 10 (1980), no. 1, 75-140. MR 558849 (81e:22017)
  • 24. Johnson, K.D., Korányi, A.: The Hua operators on bounded symmetric domains of tube type,
    Ann. of Math. (2) 111 (1980), no. 3, 589-608. MR 577139 (81j:32032)
  • 25. Kaneyuki, S.: On orbit structure of compactifications of para-Hermitian symmetric spaces,
    Japan. J. Math. (N.S.) 13 (1987), no. 2, 333-370. MR 921587 (88m:53094)
  • 26. Kaneyuki, S.: Compactification of parahermitian symmetric spaces and its applications. II. Stratifications and automorphism groups,
    J. Lie Theory 13 (2003), no. 2, 535-563. MR 2003159 (2004k:32036)
  • 27. Knapp, A.W.: ``Representation Theory of Semisimple Groups--An Overview Based on Examples'',
    Princeton Mathematical Series 36, Princeton University Press, Princeton, 1986. MR 855239 (87j:22022)
  • 28. Knapp, A.W.: ``Lie Groups Beyond an Introduction. Second Edition'',
    Progress in Math. 140, Birkhäuser Boston, Inc., Boston, 2002. MR 1920389 (2003c:22001)
  • 29. Koecher, M.: ``An Elementary Approach to Bounded Symmetric Domains'',
    Rice University, Houston, 1969. MR 0261032 (41:5652)
  • 30. Korányi, A.: The Poisson integral for generalized half-planes and bounded symmetric domains,
    Ann. of Math. (2) 82 (1965), 332-350. MR 0200478 (34:371)
  • 31. Korányi, A., Wolf, J.A.: Realization of hermitian symmetric spaces as generalized half-planes,
    Ann. of Math. (2) 81 (1965), 265-288. MR 0174787 (30:4980)
  • 32. Koufany, K., Ørsted, B.: Function spaces on the Ol'shanskiĭ semigroup and the Gel'fand-Gindikin program,
    Ann. Inst. Fourier (Grenoble) 46 (1996), no. 3, 689-722. MR 1411725 (97k:22021)
  • 33. Koufany, K., Zhang, G.: Hua operators and Poisson transform for bounded symmetric domains,
    J. Funct. Anal. 236 (2006), no. 2, 546-580. MR 2240174 (2007h:32011)
  • 34. Krötz, B.: On Hardy and Bergman spaces on complex Ol'shanskiĭ semigroups,
    Math. Ann. 312 (1998), no. 1, 13-52. MR 1645949 (99h:22009)
  • 35. Krötz, B.: Equivariant embeddings of Stein domains sitting inside of complex semigroups,
    Pacific J. Math. 189 (1999), no. 1, 55-73. MR 1687735 (2000f:32006)
  • 36. Krötz, B.: On the dual of complex Ol'shanskiĭ semigroups,
    Math. Z. 237 (2001), no. 3, 505-529. MR 1845335 (2002j:22003)
  • 37. Krötz, B., Opdam, E.: Analysis on the crown domain,
    Geom. Funct. Anal. 18 (2008), no. 4, 1326-1421. MR 2465692 (2010a:22011)
  • 38. Krötz, B., Stanton, R.J.: Holomorphic extensions of representations. I. Automorphic functions,
    Ann. of Math. (2) 159 (2004), no. 2, 641-724. MR 2081437 (2005f:22018)
  • 39. Krötz, B., Stanton, R.J.: Holomorphic extensions of representations. II. Geometry and harmonic analysis,
    Geom. Funct. Anal. 15 (2005), no. 1, 190-245. MR 2140631 (2006d:43010)
  • 40. Loos, O.: ``Symmetric Spaces. I: General Theory'',
    W.A. Benjamin, New York, 1969. MR 0239005 (39:365a)
  • 41. Loos, O.: ``Bounded Symmetric Domains and Jordan Pairs'',
    Lecture Notes, University of California, Irvine (1975)
  • 42. Neeb, K.H.: Invariant subsemigroups of Lie groups,
    Mem. Amer. Math. Soc. 104 (1993), no. 899. MR 1152952 (94a:22001)
  • 43. Neeb, K.H.: The classification of Lie algebras with invariant cones,
    J. Lie Theory 4 (1994), no. 2, 139-183. MR 1337190 (96e:17013)
  • 44. Neeb, K.H.: Invariant convex sets and functions in Lie algebras,
    Semigroup Forum 53 (1996), no. 2, 230-261. MR 1400650 (97j:17033)
  • 45. Neeb, K.H.: Convexity properties of the coadjoint action of non-compact Lie groups,
    Math. Ann. 309 (1997), no. 4, 625-661. MR 1483827 (98j:22030)
  • 46. Neeb, K.-H.: On the complex and convex geometry of Ol'shanskiĭ semigroups,
    Ann. Inst. Fourier (Grenoble) 48 (1998), no. 1, 149-203. MR 1614894 (99e:22013)
  • 47. Neeb, K.H.: ``Holomorphy and Convexity in Lie Theory'',
    Expositions in Mathematics 28. Walter De Gruyter & Co., Berlin, 2000. MR 1740617 (2001j:32020)
  • 48. Matsuki, T.: Equivalence of domains arising from duality of orbits on flag manifolds. III,
    Trans. Amer. Math. Soc. 359 (2007), no. 10, 4773-4786 (electronic). MR 2320651 (2009e:14077)
  • 49. Moore, C.C.: Compactifications of symmetric spaces. II. The Cartan domains,
    Amer. J. Math. 86 (1964), 358-378. MR 0161943 (28:5147)
  • 50. Ólafsson, G., Ørsted, B.: Causal compactification and Hardy spaces,
    Trans. Amer. Math. Soc. 351 (1999), no. 9, 3771-3792. MR 1458309 (99m:22005)
  • 51. Ol'shanskiĭ, G.I.: Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series,
    Funct. Anal. Appl. 15 (1982), 275-285. MR 639200 (83e:32032)
  • 52. Ol'shanskiĭ, G.I.: Complex Lie semigroups, Hardy spaces, and the Gelfand-Gindikin program,
    Diff. Geom. Appl. 1 (1991), 297-308. MR 1244445 (94h:22006)
  • 53. Paneitz, S.M.: Determination of invariant convex cones in simple Lie algebras,
    Ark. Mat. 21 (1983), 217-228. MR 727345 (86h:22031)
  • 54. Rockafellar, R.T.: ``Convex Analysis'',
    Princeton Math. Series 28, Princeton University Press, Princeton, 1970. MR 0274683 (43:445)
  • 55. Satake, I.: ``Algebraic Structures of Symmetric Domains'',
    Kanō Memorial Lectures 4. Iwanami Shōten, Tōkyō, 1980. MR 591460 (82i:32003)
  • 56. Stanton, R.J.: Analytic extension of the holomorphic discrete series,
    Amer. J. Math. 108 (1986), 1411-1424. MR 868896 (88b:22013)
  • 57. Upmeier, H.: ``Harmonische Analysis und Toeplitz-Operatoren auf beschränkten symmetrischen Gebieten'',
    Habilitationsschrift, Eberhard-Karls-Universität Tübingen, 1982.
  • 58. Upmeier, H: Toeplitz $ C\sp{\ast} $-algebras on bounded symmetric domains,
    Ann. of Math. (2) 119 (1984), no. 3, 549-576. MR 744863 (86a:47022)
  • 59. Upmeier, H.: ``Symmetric Banach Manifolds and Jordan C$ ^*$-Algebras'',
    Math. Studies 104, North-Holland Publishing Co., Amsterdam, 1985. MR 776786 (87a:58022)
  • 60. Upmeier, H.: An index theorem for multivariable Toeplitz operators,
    Integral Equations Operator Theory 9 (1986), no. 3, 355-386. MR 846535 (88b:32072)
  • 61. Upmeier, H.: Jordan algebras and harmonic analysis on symmetric spaces,
    Amer. J. Math. 108 (1986), 1-25. MR 821311 (87e:32047)
  • 62. Vinberg, E.B.: Invariant convex cones and orderings in Lie groups,
    Funct. Anal. Appl. 14 (1980), 1-10. MR 565090 (82c:32034)
  • 63. Warner, G.: ``Harmonic Analysis on Semi-Simple Lie Groups. I-II'',
    Grundlehren 188-189, Springer-Verlag, Berlin, 1972. MR 0498999 (58:16979); MR 0499000 (58:16980)
  • 64. Wolf, J.A., Korányi, A.: Generalized Cayley transformations of bounded symmetric domains,
    Amer. J. Math. 87 (1965), 899-939. MR 0192002 (33:229)
  • 65. Wolf, J.A.: 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 0251246 (40:4477)
  • 66. Wolf, J.A.: Flag manifolds and representation theory. In: Tirao, J., Vogan, D.A., Wolf, J.A. (eds.): ``Geometry and Representation Theory of Real and $ p$-Adic Groups (Córdoba, 1995)'', pp. 273-323.
    Progr. Math. 158, Birkhäuser Boston, Inc., Boston, 1998. MR 1486145 (99a:22029)
  • 67. Wolf, J.A.: Hermitian symmetric spaces, cycle spaces, and the Barlet-Koziarz intersection method for construction of holomorphic functions,
    Math. Res. Lett. 7 (2000), nos. 5-6, 551-564. MR 1809282 (2003a:32036)
  • 68. Wolf, J.A., Zierau, R.: Cayley transforms and orbit structure in complex flag manifolds,
    Transform. Groups 2 (1997), no. 4, 391-405. MR 1486038 (99b:32049)
  • 69. Wolf, J.A., Zierau, R.: Linear cycle spaces in flag domains,
    Math. Ann. 316 (2000), no. 3, 529-545. MR 1752783 (2001g:32054)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 22E60, 32M15, 22A15, 52A05

Retrieve articles in all journals with MSC (2010): 22E60, 32M15, 22A15, 52A05


Additional Information

Alexander Alldridge
Affiliation: Institut für Mathematik, Universität Paderborn, Warburger Strasse 100, 33100 Paderborn, Germany
Address at time of publication: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Cologne, Germany
Email: alldridg@math.upb.de, alldridg@math.uni-koeln.de

DOI: https://doi.org/10.1090/S0002-9947-2011-05309-2
Keywords: Invariant cone, complex Lie semigroup, boundary stratum, convex face, Hermitian symmetric space of non-compact type
Received by editor(s): August 25, 2009
Received by editor(s) in revised form: February 5, 2010
Published electronically: February 15, 2011
Additional Notes: This research was partially supported by the IRTG “Geometry and Analysis of Symmetries”, funded by Deutsche Forschungsgemeinschaft (DFG), Ministère de l’Éducation Nationale (MENESR), and Deutsch-Französische Hochschule (DFH-UFA)
This paper is a completely rewritten and substantially expanded version of a part of the author’s doctoral thesis under the supervision of Harald Upmeier. The author thanks him for his support and guidance. Furthermore, the author extends his thanks to an anonymous referee whose constructive criticism helped to improve the paper.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society