Produced representations of Lie algebras and Harish-Chandra modules

Heumos, Michael J.

doi:10.1090/S0002-9947-1987-0891633-5

Produced representations of Lie algebras and Harish-Chandra modules
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by Michael J. Heumos

Trans. Amer. Math. Soc. 302 (1987), 523-534

DOI: https://doi.org/10.1090/S0002-9947-1987-0891633-5

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Abstract:

The comultiplication of the universal enveloping algebra of a Lie algebra is used to give modules produced from a subalgebra, an additional compatible structure of a module over an algebra of formal power series. When only the $\mathfrak {k}$-finite elements of this algebra act on a module, conditions are given that insure that it is the Harish-Chandra module of a produced module. The results are then applied to Zuckerman derived functor modules for reductive Lie algebras. The main application describes a setting where the Zuckerman functors and production from a subalgebra commute.

References

Robert J. Blattner, Induced and produced representations of Lie algebras, Trans. Amer. Math. Soc. 144 (1969), 457–474. MR 308223, DOI 10.1090/S0002-9947-1969-0308223-4
Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, No. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 554917
Jacques Dixmier, Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). MR 0498737
T. J. Enright and N. R. Wallach, Notes on homological algebra and representations of Lie algebras, Duke Math. J. 47 (1980), no. 1, 1–15. MR 563362
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
George W. Mackey, Imprimitivity for representations of locally compact groups. I, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 537–545. MR 31489, DOI 10.1073/pnas.35.9.537

Representations of real reductive groups

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