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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. ==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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Translation functor」の詳細全文を読む スポンサード リンク
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