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On the homomorphisms between scalar generalized Verma modules

Part of: Lie algebras and Lie superalgebras Lie groups

Published online by Cambridge University Press:  26 March 2014

Hisayosi Matumoto
Hisayosi Matumoto*
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo, 153-8914, Japan email hisayosi@ms.u-tokyo.ac.jp
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Abstract

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In this article, we study the homomorphisms between scalar generalized Verma modules. We conjecture that any homomorphism between scalar generalized Verma modules is a composition of elementary homomorphisms. The purpose of this article is to confirm the conjecture for some parabolic subalgebras under the assumption that the infinitesimal characters are regular.

Keywords

generalized Verma modulesemisimple Lie algebradifferential invariant

MSC classification

Secondary: 17B10: Representations, algebraic theory (weights) 22E47: Representations of Lie and real algebraic groups

Information

Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 5 , May 2014 , pp. 877 - 892
Copyright
© The Author 2014 

References

Bernstein, J. and Gelfand, S. I., Tensor product of finite and infinite dimensional representations of semisimple Lie algebras, Compositio Math. 41 (1980), 245–285.Google Scholar
Bernstein, J., Gelfand, I. M. and Gelfand, S. I., Structure of representations generated by vectors of highest weight, Funct. Anal. Appl. 5 (1971), 1–8.Google Scholar
Boe, B., Homomorphism between generalized Verma modules, Trans. Amer. Math. Soc. 288 (1985), 791–799.Google Scholar
Boe, B. and Collingwood, D. H., Multiplicity free categories of highest weight representations. I, II, Comm. Algebra 18 (1990), 947–1032; 1033–1070.Google Scholar
Borho, W. and Jantzen, J. C., Über primitive Ideale in der Einhüllenden einer halbeinfachen Lie-algebra, Invent. Math. 39 (1977), 1–53.Google Scholar
Borho, W. and Kraft, H., Über die Gelfand–Krillov-Dimension, Math. Ann. 220 (1976), 1–24.Google Scholar
Brink, B. and Howlett, R. B., Normalizers of parabolic subgroups in Coxeter groups, Invent. Math. 136 (1999), 323–351.Google Scholar
Carter, R. W., Finite groups of Lie type: conjugacy classes and complex characters, Pure and Applied Mathematics (Wiley, 1985).Google Scholar
Collingwood, D. H. and Shelton, B., A duality theorem for extensions of induced highest weight modules, Pacific J. Math. 146 (1990), 227–237.CrossRefGoogle Scholar
Deodhar, V. V., Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function, Invent. Math. 39 (1977), 187–198.Google Scholar
Dixmier, J., Enveloping algebras (North-Holland, 1977).Google Scholar
Dobrev, V. K., Canonical construction of intertwining differential operators associated with representations of real semisimple Lie groups, Rep. Math. Phys. 25 (1988), 159–181.Google Scholar
Duflo, M., Sur la classifications des idéaux primitifs dans l’algèbre de Lie semi-simple, Ann. of Math. (2) 105 (1977), 107–120.Google Scholar
Howlett, R. B., Normalizers of parabolic subgroups of reflection groups, J. Lond. Math. Soc. (2) 21 (1980), 62–80.Google Scholar
Huang, J.-S., Intertwining differential operators and reducibility of generalized Verma modules, Math. Ann. 297 (1993), 309–324.Google Scholar
Jakobsen, H. P., Basic covariant differential operators on Hermitian symmetric spaces, Ann. Sci. École Norm. Sup. (4) 18 (1985), 421–436.CrossRefGoogle Scholar
Jantzen, J. C., Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren, Math. Ann. 226 (1977), 53–65.Google Scholar
Joseph, A., On the classification of primitive ideals in the enveloping algebra of a semisimple Lie algebra, Lecture Notes in Mathematics, vol. 1024 (Springer, 1983), 30–76.Google Scholar
Knapp, A. W., Weyl group of a cuspidal parabolic, Ann. Sci. École Norm. Sup. (4) 8 (1975), 275–294.Google Scholar
Knapp, A. W., Lie groups beyond an introduction, Progress in Mathematics, vol. 140, second edition (Birkhäuser, 2002).Google Scholar
Kostant, B., Verma modules and the existence of quasi-invariant differential operators, in Non-commutative harmonic analysis, Marseille-Luminy, 1974, Lecture Notes in Mathematics, vol. 466 (Springer, 1975), 101–128.Google Scholar
Lepowsky, J., Conical vectors in induced modules, Trans. Amer. Math. Soc. 208 (1975), 219–272.CrossRefGoogle Scholar
Lepowsky, J., Existence of conical vectors in induced modules, Ann. of Math. (2) 102 (1975), 17–40.Google Scholar
Lepowsky, J., Uniqueness of embeddings of certain induced modules, Proc. Amer. Math. Soc. 56 (1976), 55–58.Google Scholar
Lepowsky, J., Generalized Verma modules, the Cartan–Helgason theorem, and the Harish-Chandra homomorphism, J. Algebra 49 (1977), 470–495.Google Scholar
Lepowsky, J., A generalization of the Bernstein–Gelfand–Gelfand resolution, J. Algebra 49 (1977), 496–511.Google Scholar
Lusztig, G., Coxeter orbits and eigenspaces of Frobenius, Invent. Math. 38 (1976/77), 101–159.Google Scholar
Matumoto, H., On the existence of homomorphisms between scalar generalized Verma modules, Contemporary Mathematics, vol. 145 (American Mathematical Society, Providence, RI, 1993), 259–274.Google Scholar
Matumoto, H., The homomorphisms between scalar generalized Verma modules associated to maximal parabolic subalgebras, Duke Math. J. 131 (2006), 75–118.Google Scholar
Matumoto, H. and Trapa, P. E., Derived functor modules arising as large irreducible constituents of degenerate principal series, Compositio Math. 143 (2007), 222–256.Google Scholar
Soergel, W., Kategorie 𝓞, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421–445.Google Scholar
Steinberg, R., Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. 80 (1968).Google Scholar
Verma, D. N., Structure of certain induced representations of complex semisimple Lie algebras, Bull. Amer. Math. Soc. 74 (1968), 160–166.Google Scholar