Translation functor

In mathematical representation theory, a 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_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 ψ 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