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.
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:
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