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References
Bernstein, I. N., Gelfand, I. M., Gelfand, S. I.: Structure of representations generated by highest weight vectors (in Russian). Funkt. Anal. i Ego Prilozheniya5, 1–9 (1971). English translation: Functional Analysis and its Applications5, 1–8 (1971)
Bernstein, I. M., Gelfand, I. N., Gelfand, S. I. Differential operators on the base affine space and a study of g-modules, Lie groups and their representations. Summer School of the Bolyai János Math. Soc., Gelfand, I. M. (ed.), pp. 21–64 New York: Division of Wiley and Sons, Halsted Press 1975
Bott, R.: Homogeneous vector bundles. Ann. of Math.66, 203–248 (1957)
Cartan, H., Eilenberg, S.: Homological algebra. Princeton University Press 1956
Cartier, P.: Remarks on “Lie algebra cohomology and the generalized Borel-Weil theorem”, by B.Kostant. Annals of Math.74, 388–390 (1961)
Curtis, C. W., Reiner, I.: Representation theory of finite groups and associative algebras. New York: Wiley-Interscience 1962
Dixmier, J.:Algèbres enveloppantes. Paris: Gauthier-Villars 1974
Garland, H.: Dedekind's ν-function and the cohomology of infinite dimensional Lie algebras. Proc. Nat. Acad. Sci. (U.S.A.)72, 2493–2495 (1975)
Garland, H., Raghunathan, M. S.: A Bruhat decomposition for the loop space of a compact group: a new approach to results of Bott. Proc. Nat. Acad. Sci. (U.S.A.)72, 4716–4717 (1975)
Humphreys, J. E.: Introduction to Lie algebras and representation theory. Berlin-Heidelberg-New York: Springer 1972
iwahori, N.: On the structure of the Hecke ring of a Chevalley group over a finite field. J. Fac. Sci. Univ. Tokyo Sect.I 10, 215–236 (1964)
Kac, V. G.: Simple irreducible graded Lie algebras of finite growth (in Russian). Izv. Akad. Nauk. SSSR32, 1323–1367 (1968), English translation: Math. USSR-Izvestija.2, 1271–1311 (1968)
Kac, V. G.: Inhnite-dimensional Lie algebras and Dedekind's ν-functions (in Russian). Funkt. Anal. i Ego Prilozheniya8, 77–78 (1974), English translation: Functional Analysis and its Applications8, 68–70 (1974)
Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Annals of Math.74, 329–387 (1961)
Lepowsky, J.: Conical vectors in induced modules. Trans. Amer. Math. Soc.208, 219–272 (1975)
Lepowsky, J.: Existence of conical vectors in induced modules Annals of Math.102, 17–40 (1975)
Lepowsky, J.: Generalized Verma modules and loop space cohomology. In preparation
Macdonald, I. G.: Affine root systems and Dedekind's ν-function. Inventiones math.15, 91–143 (1972)
Moody, R. V.: A new class of Lie algebras. J. Algebra10, 211–230 (1968)
Moody, R. V.: Euclidean Lie algebras. Can. J. Math.21, 1432–1454 (1969)
Moody, R. V.: Macdonald identities and Euclidean Lie algebras. Proc. Amer. Math. Soc.48, 43–52 (1975)
Moody, R. V., Teo, K. L.: Tits' systems with crystallographic Weyl groups. J. Algebra21, 178–190 (1972)
Serre, J.-P.: Algèbres de Lie semi-simples complexes. New York: Benjamin 1966
Sweedler, M.: Hopf algebras. New York: Benjamin 1969
Verma, D.-N.: Review of 1. G. Macdonald's paper “Affine root systems and Dedekind's ν-function”. Math. Reviews50, 1371–1374 (MR # 9996) (1975)
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Partially supported by NSF MPS 71-03469
Partially supported by NSF grants MPS 71-03469 and MPS 72-05055 A 03, and a Yale University Junior Faculty Fellowship
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Garland, H., Lepowsky, J. Lie algebra homology and the Macdonald-Kac formulas. Invent Math 34, 37–76 (1976). https://doi.org/10.1007/BF01418970
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DOI: https://doi.org/10.1007/BF01418970