Generalized Verma modules, loop space cohomology and MacDonald-type identities
Lepowsky, J.
Annales scientifiques de l'École Normale Supérieure, Tome 12 (1979), p. 169-234 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@article{ASENS_1979_4_12_2_169_0,
     author = {Lepowsky, J.},
     title = {Generalized Verma modules, loop space cohomology and MacDonald-type identities},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {12},
     year = {1979},
     pages = {169-234},
     doi = {10.24033/asens.1365},
     mrnumber = {81a:17004},
     zbl = {0414.17007},
     mrnumber = {543216},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_1979_4_12_2_169_0}
}

Lepowsky, J. Generalized Verma modules, loop space cohomology and MacDonald-type identities. Annales scientifiques de l'École Normale Supérieure, Tome 12 (1979) pp. 169-234. doi : 10.24033/asens.1365. http://gdmltest.u-ga.fr/item/ASENS_1979_4_12_2_169_0/

[1] G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, G.-C. ROTA, Ed., Addison-Wesley, Reading, Mass., 1976. | Zbl 0371.10001

[2] I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Differential Operators on the Base Affine Space and a Study of g-Modules in Lie Groups and Their Representations, Summer School of the Bolyai Janós Math. Soc., I. M. Gelfand, Ed., Division of Wiley and Sons, Halsted Press, New York, 1975, pp. 21-64. | MR 58 #28285 | Zbl 0338.58019

[3] R. Bott, (a) An Application of the Morse Theory to the Topology of Lie Groups (Bull. Soc. math. Fr., Vol. 84, 1956, pp. 251-281) ; (b) Homogeneous Vector Bundles (Ann. of Math., 66, 1957, pp. 203-248). | Numdam | MR 19,291a | Zbl 0073.40001

[4] N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5, 6, Hermann, Paris, 1969.

[5] H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, Princeton, 1956. | MR 17,1040e | Zbl 0075.24305

[6] C. Chevalley and S. Eilenberg, Cohomology Theory of Lie Groups and Lie Algebras (Trans. Amer. Math. Soc., Vol. 63, 1948, pp. 85-124). | MR 9,567a | Zbl 0031.24803

[7] M. Demazure, Identités de MacDonald (Séminaire Bourbaki, 28e année, 1975/1976, No. 483). | Numdam | Zbl 0345.17003

[8] J. Dixmier, Algèbres enveloppantes, Gauthier-Villars, Paris, 1974. | MR 58 #16803a | Zbl 0308.17007

[9] A. Feingold and J. Lepowsky, The Weyl-Kac Character Formula and Power Series Identities (Adv. in Math., Vol. 29, 1978, pp. 271-309). | MR 83a:17015 | Zbl 0391.17009

[10] H. Garland, Dedekind's ƞ-Function and the Cohomology of Infinite Dimensional Lie Algebras [Proc. Nat. Acad. Sc. (U.S.A.), Vol. 72, 1975, pp. 2493-2495]. | MR 52 #8204 | Zbl 0322.18010

[11] H. Garland and J. Lepowsky, (a) Lie Algebra Homology and the MacDonald-Kac Formulas (Invent. math., Vol. 34, 1976, pp. 37-76) ; (b) The MacDonald-Kac Formulas as a Consequence of the Euler-Poincaré Principle in Contributions to Algebra : A Collection of Papers Dedicated to Ellis Kolchin, BASS, CASSIDY and KOVACIC, Eds., Academic Press, New York, 1977, pp. 165-173. | MR 54 #2744 | Zbl 0366.17005

[12] H. Garland and M. S. Raghunathan, A Bruhat Decomposition for the Loop Space of a Compact Group : A New Approach to Results of Bott [Proc. Nat. Acad. Sc. (U.S.A.), Vol. 72, 1975, pp. 4716-4717]. | MR 54 #5389 | Zbl 0344.55009

[13] G. H. Hardy, Ramanujan, Cambridge Univ. Press, London, 1940, Reprinted by Chelsea, New York. | JFM 67.0002.09

[14] G. Hochschild, Relative Homological Algebra (Trans. Amer. Math. Soc., Vol. 82, 1956, pp. 246-269). | MR 18,278a | Zbl 0070.26903

[15] V. G. Kac, (a) Simple Irreducible Graded Lie Algebras of Finite Growth (in Russian) (Izv. Akad. Nauk S.S.S.R., Vol. 32, 1968, pp. 1323-1367), English translation : Math. U.S.S.R.-Izvestija, Vol. 2, 1968, pp. 1271-1311) ; (b) Automorphisms of Finite Order of Semisimple Lie algebras (in Russian) (Funkt. Anal. i Ego Prilozheniya, Vol. 3, 1969, pp. 94-96), English translation : Funct. Anal. and Appl., Vol. 3, 1969, pp. 252-254) ; (c) Infinite-Dimensional Lie algebras and Dedekind's ƞ-function (in Russian) (Funkt. Anal. i Ego Prilozheniya, Vol. 8, 1974, pp. 77-78), English translation : Funct. Anal. and Appl., Vol. 8, 1974, pp. 68-70) ; (d) Infinite-dimensional Algebras, Dedekind ƞ-Function, Classical Möbius Function and the very Strange Formula (Adv. in Math., Vol. 30, 1978, pp. 85-136). | MR 41 #4590

[16] B. Kostant, (a) The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group (Amer. J. Math., Vol. 81, 1959, pp. 973-1032) ; (b) Lie Algebra Cohomology and the Generalized Borel-Weil Theorem (Ann. of Math., Vol. 74, 1961, pp. 329-387) ; (c) Lie Algebra Cohomology and Generalized Schubert Cells (Ann. of Math., Vol. 77, 1963, pp. 72-144) ; (d) On MacDonald's ƞ-Function Formula, the Laplacian and Generalized Exponents (Adv. in Math., Vol. 20, 1976, pp. 179-212). | MR 22 #5693

[17] J. Lepowsky, (a) Generalized Verma Modules, the Cartan-Helgason Theorem and the Harish-Chandra Homomorphism (J. Algebra, Vol. 49, 1977, pp. 470-495) ; (b) Minimal K-Types for Certain Representations of Real Semisimple Groups (J. Algebra, Vol. 51, 1978, pp. 173-210) ; (c) MacDonald-type Identities (Adv. in Math., Vol. 27, 1978, pp. 230-234) ; (d) Application of the Numerator Formula to k-Rowed Plane Partitions (Adv. in Math., to appear) ; (e) Lie Algebras and Combinatorics (Proc. International Congress of Mathematicians, Helsinki, 1978, to appear). | MR 57 #3312

[18] J. Lepowsky and G. W. Mccollum, On the Determination of Irreducible Modules by Restriction to a Subalgebra (Trans. Amer. Math. Soc., Vol. 176, 1973, pp. 45-57). | MR 48 #2201 | Zbl 0266.17014

[19] J. Lepowsky and S. Milne, Lie Algebraic Approaches to Classical Partition Identities (Adv. in Math., Vol. 29, 1978, pp. 15-59). | MR 82f:17005 | Zbl 0384.10008

[20] J. Lepowsky and R. L. Wilson, Construction of the Affine Lie Algebra A1(1) (Comm. Math. Phys., Vol. 62, 1978, pp. 43-53). | MR 58 #28089 | Zbl 0388.17006

[21] I. G. Macdonald, Affine Root Systems and Dedekind's ƞ-Function, (Invent. math., Vol. 15, 1972, pp. 91-143). | MR 50 #9996 | Zbl 0244.17005

[22] R. V. Moody, (a) A New Class of Lie Algebras (J. Algebra, Vol. 10, 1968, pp. 211-230) ; (b) Euclidean Lie Algebras (Can. J. Math., Vol. 21, 1969, pp. 1432-1454) ; (c) MacDonald Identities and Euclidean Lie Algebras (Proc. Amer. Math. Soc., Vol. 48, 1975, pp. 43-52). | MR 37 #5261 | Zbl 0315.17003

[23] K. R. Parthasarathy, R. Ranga Rao and V. S. Varadarajan, Representations of Complex Semi-Simple Lie Groups and Lie Algebras (Ann. of Math., Vol. 85, 1967, pp. 383-429). | MR 37 #1526 | Zbl 0177.18004

[24] H. Rademacher, Topics in Analytic Number Theory, Die Grundlehren der Mathematischen Wissenschaften, Vol. 169, Springer-Verlag, New York, 1973. | MR 51 #358 | Zbl 0253.10002

[25] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. | MR 35 #1007 | Zbl 0145.43303

[26] L. Conlon, The Topology of Certain Spaces of Paths on a Compact Symmetric Space (Trans. Amer. Math. Soc., Vol. 112, 1964, pp. 228-248). | MR 29 #631 | Zbl 0146.19602

[27] F. J. Dyson, Missed Opportunities (Bull. Amer. Math. Soc., Vol. 78, 1972, pp. 635-653). | MR 58 #25442 | Zbl 0271.01005

[28] V. G. Kac, D. A. Kazhdan, J. Lepowsky and R. L. Wilson, Realization of the Basic Representations of the Euclidean Lie Algebras, to appear.