In mathematical

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 studies modules over these abstract algebraic structures. In essen ...

, a (Zuckerman) translation functor is a functor taking representations of a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

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 In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...

with a finite-dimensional representation, and then taking a subspace with some central character.

Definition

By the

Harish-Chandra isomorphism In mathematics, the Harish-Chandra isomorphism, introduced by , is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center \mathcal(U(\mathfrak)) of the universal enveloping algebra U(\mathfr ...

, the characters of the center ''Z'' of the

universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representati ...

of a complex reductive Lie algebra can be identified with the points of ''L''⊗C/''W'', where ''L'' is the

weight lattice In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplic ...

and ''W'' is the

Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...

. 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

* * *{{Citation , last1=Zuckerman , first1=Gregg , title=Tensor products of finite and infinite dimensional representations of semisimple Lie groups , jstor=1971097 , mr=0457636 , year=1977 , journal=Ann. Math. , series=2 , volume=106 , issue=2 , pages=295–308 , doi=10.2307/1971097 Representation theory Functors