Translation Functor

In mathematical representation theory, a (Zuckerman) translation functor is a functor taking representations of a Lie algebra to representations with a possibly different central character. Translation functors were introduced independently by Zuckerman (1977) and Jantzen (1979). Roughly speaking, the functor is given by taking a tensor product with a finite-dimensional representation, and then taking a subspace with some central character.

Definition

By the Harish-Chandra isomorphism, the characters of the center Z of the universal enveloping algebra of a complex reductive Lie algebra can be identified with the points of L⊗C/W, where L is the weight lattice and W is the Weyl group. If λ is a point of L⊗C/W then write χ_λ for the corresponding character of Z.

A representation of the Lie algebra is said to have central character χ_λ if every vector v is a generalized eigenvector of the center Z with eigenvalue χ_λ; in other words if z∈Z and v∈V then (z − χ_λ(z))ⁿ(v)=0 for some n.

The translation functor ψμλ takes representations V with central character χ_λ to representations with central character χ_μ. It is constructed in two steps:

First take the tensor product of V with an irreducible finite dimensional representation with extremal weight λ−μ (if one exists).
Then take the generalized eigenspace of this with eigenvalue χ_μ.

References

Jantzen, Jens Carsten (1979), Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, 750, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0069521, ISBN 978-3-540-09558-3
Knapp, Anthony W.; Vogan, David A. (1995), Cohomological induction and unitary representations, Princeton Mathematical Series, 45, Princeton University Press, doi:10.1515/9781400883936, ISBN 978-0-691-03756-1
Zuckerman, Gregg (1977), "Tensor products of finite and infinite dimensional representations of semisimple Lie groups", Ann. Math., 2 106 (2): 295–308, doi:10.2307/1971097

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