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 and . 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 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 ψ takes representations "V" with central character χλ to representations with central character χμ. It is constructed in two steps: