A Bott-Borel-Weil theory for direct limits of algebraic groups

American Journal of Mathematics

Volumn 124, Issue 5, 2002, Pages 955-998

  Dimitrov, Ivan ^a   Penkov, Ivan ^a   Wolf, Joseph A ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036791503     PISSN: 00029327     EISSN: None     Source Type: Journal    
DOI: 10.1353/ajm.2002.0025     Document Type: Article

References (26)

1. Yu. Bahturin and G. Benkart, Highest weight modules for locally finite Lie algebras, Stud. Adv. Math. (S.-T. Yau, ed.), vol. 4, International Press, FL, 1997, pp. 1-31.
2. A. Baranov, Diagonal locally finite Lie algebras and a version of Ado's Theorem, J. Algebra 199 (1998), 1-39.
3. _, Complex finitary simple Lie algebras, Arch. Math. (Basel) 72 (1999), 101-106.
4. A. Baranov and A. Zhilinski, Diagonal direct limits of simple Lie algebras, Comm. Algebra 27 (1999), 2749-2766.
5. R. Bott, Homogeneous vector bundles, Ann. of Math. 66 (1957), 203-248.
6. M. Demazure, Une démonstration algébrique d'un théorème de Bott, Invent. Math. 5 (1968), 349-356.
7. _, A very simple proof of Bott's theorem, Invent. Math. 33 (1976), 271-272.
8. I. Dimitrov and I. Penkov, Weight modules of direct limit Lie algebras, Internat. Math. Res. Notices 5 (1999), 223-249.
9. R. Goodman and N. Wallach, Representations and Invariants of the Classical Groups, Cambridge University Press, New York, 1999.
10. A. Grothendieck, Éléments de Géometrie Algébrique, Inst. Hautes Études Sci. Publ. Math. 11 (1961).
11. R. Hartshorne, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1976), 5-99.
12. _, Algebraic Geometry, Springer-Verlag, New York, 1977.
13. G. Hochschild, Basic Theory of Algebraic Groups and Lie Algebras, Springer-Verlag, New York, 1981.
14. J.C. Jantzen, Representations of Algebraic Groups, Academic Press, London, 1987.
15. V. Kac, Infinite Dimensional Lie Algebras, 3rd edition, Cambridge University Press, 1990.
16. B. Kostant, Lie algebra cohomology and the generalized Borel-Weil Theorem, Ann. of Math. 74 (1961), 329-387.
17. S. Kumar, Infinite Grassmannians and moduli spaces of G-bundles, Vector Bundles on Curves-New Directions (M.S. Narasimhan, ed.), Lecture Notes in Math., vol. 1647, Springer-Verlag, New York, 1997, pp. 1-49.
18. O. Mathieu, Formules de charactères pour les algèbres de Kac-Moody générales, Astérisque 159-160 (1988).
19. G.D. Mostow, Fully reducible subgroups of algebraic groups, Amer. J. Math. 78 (1956), 200-221.
20. L. Natarajan, E. Rodríguez-Carrington and J.A. Wolf, The Bott-Borel-Weil Theorem for direct limit Lie groups, Trans. Amer. Math. Soc. 353 (2001), 4583-4622.
21. K.-H. Neeb, Holomorphic highest weight representations of infinite dimensional complex classical groups, J. Reine Angew. Math. 497 (1998), 171-222.
22. K.-H. Neeb and N. Stumme, The classification of locally finite split simple Lie algebras, J. Reine Angew. Math. 533 (2001), 25-53.
23. I. Penkov, Borel-Weil-Bott theory for classical Lie supergroups, Curt. Probl. in Math. 32, Moscow: VINITI 1988 (in Russian), 71-124.
24. A.N. Pressley and G.B. Segal, Loop Groups and their Representations, Oxford University Press, New York, 1985.
25. I.R. Shafarevich, On some infinite dimensional groups II, Izv. Akad. Nauk USSR Ser. Mat. 45 (1981), 214-226, 240 (Russian).
26. A. Tjurin, Finite dimensional bundles on infinite varieties, Izv. Akad. Nauk USSR Ser. Mat. 40 (1976), 1248-1268.