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 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 ψμ
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

  • Jantzen, Jens Carsten (1979), Moduln mit einem höchsten Gewicht, Lecture Notes in Mathematics, vol. 750, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0069521, ISBN 978-3-540-09558-3, MR 0552943
  • Knapp, Anthony W.; Vogan, David A. (1995), Cohomological induction and unitary representations, Princeton Mathematical Series, vol. 45, Princeton University Press, doi:10.1515/9781400883936, ISBN 978-0-691-03756-1, MR 1330919
  • Zuckerman, Gregg (1977), "Tensor products of finite and infinite dimensional representations of semisimple Lie groups", Ann. Math., 2, 106 (2): 295–308, doi:10.2307/1971097, JSTOR 1971097, MR 0457636

Content Disclaimer

Informasi ini disarikan dari Wikipedia dan disajikan kembali untuk tujuan edukasi. Konten tersedia di bawah lisensi CC BY-SA 3.0. Kami tidak bertanggung jawab atas ketidakakuratan data yang bersumber dari kontribusi publik tersebut.

  1. The information displayed on this website is sourced in part or in whole from Wikipedia and has been adapted for the purpose of restating it. We strive to provide accurate and relevant information, however:
  2. There is no guarantee of absolute accuracy. Wikipedia is an open, collaborative project that can be edited by anyone, so information is subject to change.
  3. It is not intended to constitute professional advice. The content displayed is for informational and educational purposes only. For important decisions (e.g., medical, legal, or financial), please consult a professional.
  4. Content copyright. Wikipedia is licensed under the Creative Commons Attribution-ShareAlike License (CC BY-SA). This means that content may be reused with appropriate attribution and shared under a similar license.
  5. Responsible use. Any risk arising from the use of information from this website is entirely the responsibility of the user.