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 {{harvs|txt|last=Zuckerman|authorlink=Gregg Zuckerman|year=1977}} and {{harvs|txt|last=Jantzen|authorlink=Jens Carsten Jantzen|year=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 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{{su|p=μ|b=λ}} 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
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- {{Citation | last1=Jantzen | first1=Jens Carsten | title=Moduln mit einem höchsten Gewicht | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-09558-3 | doi=10.1007/BFb0069521 | mr=552943 | year=1979 | volume=750}}
- {{Citation | last1=Knapp | first1=Anthony W. | first2=David A. |last2=Vogan| title=Cohomological induction and unitary representations | publisher=Princeton University Press | series=Princeton Mathematical Series | isbn=978-0-691-03756-1 | mr=1330919 | year=1995 | volume=45 | doi=10.1515/9781400883936}}
- {{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}}
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