Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Splitting criteria for $\mathfrak {g}$-modules induced from a parabolic and the Berňsteĭn-Gel’fand-Gel’fand resolution of a finite-dimensional, irreducible $\mathfrak {g}$-module
HTML articles powered by AMS MathViewer

by Alvany Rocha-Caridi PDF
Trans. Amer. Math. Soc. 262 (1980), 335-366 Request permission

Abstract:

Let $\mathcal {g}$ be a finite dimensional, complex, semisimple Lie algebra and let V be a finite dimensional, irreducible $\mathcal {g}$-module. By computing a certain Lie algebra cohomology we show that the generalized versions of the weak and the strong Bernstein-Gelfand-Gelfand resolutions of V obtained by H. Garland and J. Lepowsky are identical. Let G be a real, connected, semisimple Lie group with finite center. As an application of the equivalence of the generalized Bernstein-Gelfand-Gelfand resolutions we obtain a complex in terms of the degenerate principal series of G, which has the same cohomology as the de Rham complex.
References
  • I. N. BernÅ¡teÄ­n, I. M. Gel′fand, and S. I. Gel′fand, A certain category of ${\mathfrak {g}}$-modules, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 1–8 (Russian). MR 0407097
  • —, Differential operators on the base affine space and a study of $\mathcal {g}$-modules, Lie groups and their representations, Proc. Summer School on Group Representations (I. M. Gelfand, ed.), Bolyai János Math. Soc., Wiley, New York, 1975, pp. 39-64.
  • N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163
  • Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0144979
  • Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
  • Thomas J. Enright, On the fundamental series of a real semisimple Lie algebra: their irreducibility, resolutions and multiplicity formulae, Ann. of Math. (2) 110 (1979), no. 1, 1–82. MR 541329, DOI 10.2307/1971244
  • Thomas J. Enright and Nolan R. Wallach, The fundamental series of representations of a real semisimple Lie algebra, Acta Math. 140 (1978), no. 1-2, 1–32. MR 476814, DOI 10.1007/BF02392301
  • Howard Garland and James Lepowsky, Lie algebra homology and the Macdonald-Kac formulas, Invent. Math. 34 (1976), no. 1, 37–76. MR 414645, DOI 10.1007/BF01418970
  • Peter John Hilton and Urs Stammbach, A course in homological algebra, Graduate Texts in Mathematics, Vol. 4, Springer-Verlag, New York-Berlin, 1971. MR 0346025, DOI 10.1007/978-1-4684-9936-0
  • James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842, DOI 10.1007/978-1-4612-6398-2
  • —, Finite and infinite dimensional modules for semi-simple Lie algebras, Queen’s Papers in Pure and Applied Math. 48 (1978), 1-64.
  • Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. MR 142696, DOI 10.2307/1970237
  • J. Lepowsky, A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra 49 (1977), no. 2, 496–511. MR 476813, DOI 10.1016/0021-8693(77)90254-X
  • J. Lepowsky, Generalized Verma modules, the Cartan-Helgason theorem, and the Harish-Chandra homomorphism, J. Algebra 49 (1977), no. 2, 470–495. MR 463360, DOI 10.1016/0021-8693(77)90253-8
  • Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Band 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879, DOI 10.1007/978-3-642-62029-4
  • D. N. Verma, Structure of certain induced representations of complex semi-simple Lie algebras, Dissertation, Yale University, 1966.
  • Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Applied Mathematics, No. 19, Marcel Dekker, Inc., New York, 1973. MR 0498996
  • —, Unpublished manuscript notes on the Borel-Weil theorem.
  • Nolan R. Wallach, On the Enright-Varadarajan modules: a construction of the discrete series, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 1, 81–101. MR 422518, DOI 10.24033/asens.1304
  • Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman & Co., Glenview, Ill.-London, 1971. MR 0295244
  • Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Band 188, Springer-Verlag, New York-Heidelberg, 1972. MR 0498999
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 17B10, 22E47
  • Retrieve articles in all journals with MSC: 17B10, 22E47
Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 262 (1980), 335-366
  • MSC: Primary 17B10; Secondary 22E47
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0586721-0
  • MathSciNet review: 586721