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 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.

Definition

By 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^μ
_λ 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

Jantzen, Jens Carsten (1979), Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Berlin, New York:
MR 0552943

Knapp, Anthony W.; Vogan, David A. (1995), Cohomological induction and unitary representations, Princeton Mathematical Series, vol. 45,
MR 1330919

Zuckerman, Gregg (1977), "Tensor products of finite and infinite dimensional representations of semisimple Lie groups", Ann. Math., 2, 106 (2): 295–308,
MR 0457636