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)

 
 

 

The Bott-Borel-Weil Theorem for direct limit groups


Authors: Loki Natarajan, Enriqueta Rodríguez-Carrington and Joseph A. Wolf
Journal: Trans. Amer. Math. Soc. 353 (2001), 4583-4622
MSC (1991): Primary 222E30, 22E65; Secondary 22C05, 32C10, 46G20, 22E70
DOI: https://doi.org/10.1090/S0002-9947-01-02452-7
Published electronically: July 3, 2001
MathSciNet review: 1650034
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show how highest weight representations of certain infinite dimensional Lie groups can be realized on cohomology spaces of holomorphic vector bundles. This extends the classical Bott-Borel-Weil Theorem for finite-dimensional compact and complex Lie groups. Our approach is geometric in nature, in the spirit of Bott's original generalization of the Borel-Weil Theorem. The groups for which we prove this theorem are strict direct limits of compact Lie groups, or their complexifications. We previously proved that such groups have an analytic structure. Our result applies to most of the familiar examples of direct limits of classical groups. We also introduce new examples involving iterated embeddings of the classical groups and see exactly how our results hold in those cases. One of the technical problems here is that, in general, the limit Lie algebras will have root systems but need not have root spaces, so we need to develop machinery to handle this somewhat delicate situation.


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

  • [1] Bahturin, Yu. A. and Benkart, G., Highest weight modules for locally finite Lie algebras, in ``Modular Interfaces'' (Proceedings, Riverside, CA, 1995), AMS/IP Studies Adv. Math. 4, Amer. Math. Soc., 1997, 1-31. MR 98m:17009
  • [2] Bahturin, Yu. A. and Strade, H, Locally finite dimensional simple Lie algebras, Mat. Sbornik 81 (1995), 137-161. MR 95a:17025
  • [3] -, Some examples of locally finite simple Lie algebras, Archives Math. 65 (1995), 23-26. MR 96f:17010
  • [4] Baranov, A. A., Simple diagonal locally finite Lie algebras, Proc. London Math. Soc. 77 (1998), 362-386. MR 99k:17041
  • [5] Baranov, A. A. & Zhilinskii, A. G., Diagonal direct limits of simple Lie algebras, Comm. Algebra 27 (1999), 2749-2766. MR 2000f:17031
  • [6] Bott, R., Homogeneous vector bundles, Annals of Math. 66 (1957), 203-248. MR 19:681d
  • [7] Bourbaki, N., Groupes et Algèbres de Lie (Ch. 4, 5 et 6), Éléments de Mathématique, Fasc. XXXIV, Hermann, Paris, 1968. MR 39:1590
  • [8] Boyer, R. P., Representation theory of the Hilbert-Lie group $U_{2}(H)$, Duke J. Math. 47 (1980), 205-236. MR 81j:22024
  • [9] -, Characters and factor representations of the infinite classical groups, J. Operator Theory 28 (1993). MR 96e:22036
  • [10] -, Representation theory of infinite dimensional unitary groups, Contemporary Math. 145 (1993). MR 94g:22038
  • [11] Dimitrov, I, Private communications, May and June 1999.
  • [12] Dimitrov, I. & Penkov, I., Partially integrable highest weight modules, Transformation Groups 3 (1998), 241-253. MR 99j:17035
  • [13] -, Weight modules of direct limit Lie algebras, International Math. Research Notices, No. 5 (1999), 223-249. MR 2000a:17005
  • [14] Dimitrov, I., Penkov, I. & J. A. Wolf, A Bott-Borel-Weil theory for direct limits of algebraic groups, to appear.
  • [15] Eda, K., Kiyosawa, T., and Ohta, H., N-completeness and its applications, in ``Topics in General Topology'', North-Holland, Amsterdam, 1989, pp. 459-521. MR 91m:54018
  • [16] Gel'fand, I.M. and Graev, M.I., Principal representations of the group $U(\infty )$, in ``Representations of Lie Groups and Related Topics", Adv. Stud. Contemp. Math., vol. 7, Gordon & Breach, New York, 1990, pp. 119-153. MR 92k:22033
  • [17] Glöckner, H., Private communications.
  • [18] Habib, A., Direct limits of Zuckerman derived functor modules, Doctoral Dissertation, University of California, Berkeley, 1997. Journal publication to appear.
  • [19] Hartshorne, R., On the de Rham cohomology of algebraic varieties, Publ. Math. IHES 45 (1976), 5-99. MR 55:5633
  • [20] -, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, 1977. MR 57:3116
  • [21] Kakutani, S. & Klee, V., The finite topology of a linear space, Arch. Math. 14 (1963), 55-58. MR 27:1799
  • [22] Kirillov, A. A., Representations of the infinite unitary group, Soviet Math. Doklady 14 (1973), 288-290. MR 49:5239
  • [23] Natarajan, L., Unitary highest weight modules of inductive limit Lie algebras and groups, J. Algebra 167 (1994), 9-28. MR 95f:22022
  • [24] Natarajan, L., Rodríguez-Carrington, E. and Wolf, J. A., Differentiable structure for direct limit groups, Letters in Math. Physics 23 (1991), 101-123. MR 92k:22035
  • [25] -, Locally convex Lie groups, Nova J. Algebra and Geometry 2 (1993), 59-88. MR 94j:22022
  • [26] -, New classes of infinite dimensional groups, ``Algebraic Groups and their Generalizations;'' Proceedings of Symposia in Pure Mathematics 56, Part II (1994), 377-392. MR 95c:22024
  • [27] Neretin, Yu. A., Categories of Symmetries and Infinite-Dimensional Groups, Clarendon Press, Oxford, 1996. MR 98b:22003
  • [28] Ol'shanskii, G. I., Unitary representations of the group $SO_{0}(\infty ,\infty )$ as limits of unitary representations of the groups $SO_{0}(n,\infty )$ as $n\to \infty $, Functional Analysis & Applications 20 (1988), 292-301. MR 88b:22025
  • [29] -, Representations of infinite dimensional classical groups, limits of enveloping algebras, and Yangians, Advances Soviet Math. 2 (1991), 1-66. MR 92g:22039
  • [30] Pickrell, D., The separable representations of $U(H)$, Proceedings Amer. Math. Soc. 102 (1988), 416-420. MR 89c:22036
  • [31] Rodríguez-Carrington, E. and Wolf, J. A., Infinite Weyl groups, to appear.
  • [32] Stratila, S. and Voiculescu, D., ``Representations of AF-algebras and of the group $U(\infty )$'', Lecture Notes in Math. 486 (1975). MR 56:16391
  • [33] Yanson, I. A. and Zhdanovich, D. V., The set of direct limits of Lie algebras of type A, Communications in Algebra 24 (1996), 1125-1156. MR 97c:17035
  • [34] Zhdanovich, D. V., Doctoral Dissertation, Moscow State University, 1996.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 222E30, 22E65, 22C05, 32C10, 46G20, 22E70

Retrieve articles in all journals with MSC (1991): 222E30, 22E65, 22C05, 32C10, 46G20, 22E70


Additional Information

Loki Natarajan
Affiliation: Department of Mathematics 0112, University of California at San Diego, La Jolla, California 92093
Email: loki@euclid.ucsd.edu

Enriqueta Rodríguez-Carrington
Affiliation: Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854
Email: carringt@math.rutgers.edu

Joseph A. Wolf
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720–3840
Email: jawolf@math.berkeley.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02452-7
Keywords: Direct limit, direct limit Lie group, diagonal Lie algebra, diagonal embedding, Borel--Weil Theorem, Bott--Borel--Weil Theorem, direct limit representation, inverse limit representation, direct limit cohomology, inverse limit cohomology, infinite dimensional Lie group
Received by editor(s): May 6, 1997
Received by editor(s) in revised form: July 6, 1998, and April 26, 2000
Published electronically: July 3, 2001
Additional Notes: LN: research partially supported by NSF Grant DMS 92 08303.
ERC: research partially supported by PSF–CUNY Grant 6–66386.
JAW: research partially supported by NSF Grants DMS 93 21285 and DMS 97 05709.
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society