Translation functor

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:

• 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.

References