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.
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 χλ for the corresponding character of Z.
A representation of the Lie algebra is said to have central character χλ if every vector v is a generalized eigenvector of the center Z with eigenvalue χλ; in other words if z∈Z and v∈V then (z − χλ(z))n(v)=0 for some n.
The translation functor ψμ
λ takes representations V with central character χλ to representations with central character χμ. 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 χμ.
- 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, MR 0552943
- Knapp, Anthony W.; Vogan, David A. (1995), Cohomological induction and unitary representations, Princeton Mathematical Series, 45, Princeton University Press, ISBN 978-0-691-03756-1, MR 1330919
- Zuckerman, Gregg (1977), "Tensor products of finite and infinite dimensional representations of semisimple Lie groups", Ann. Math., 2, 106 (2): 295–308, JSTOR 1971097, MR 0457636