Translation functor
Encyclopedia
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.
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:
Representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...
, 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 algebraUniversal enveloping algebra
In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie 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
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
. 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.