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 χ for the corresponding character of "Z". A representation of the Lie algebra is said to have central character χ if every vector "v" is a generalized eigenvector of the center "Z" with eigenvalue χ; in other words if "z"∈"Z" and "v"∈"V" then ("z" − χ("z"))("v")=0 for some "n". The translation functor ψ takes representations "V" with central character χ to representations with central character χ. It is constructed in two steps: