translation functor


Let 𝔤\mathfrak{g} be a semisimple complex Lie algebra and 𝔥𝔤\mathfrak{h}\subset\mathfrak{g} a Cartan subalgebra. Given λ,μ𝔥 *\lambda,\mu\in\mathfrak{h}^* such that the difference is an integral weight μλΛ\mu-\lambda\in\Lambda. The orbit of the difference μλ\mu-\lambda under the Weyl group contains a unique positive integral weight nu¯\bar{nu}. Given any weight τ\tau denote by χ τ\chi_\tau the central character corresponding to the weight τ\tau. The Bernstein-Gelfand-Gelfand category 𝒪\mathcal{O} is the internal direct sum of the subcategories 𝒪 χ\mathcal{O}_{\chi} where χ\chi runs through central characters of the form χ τ\chi_\tau and where the full subcategory 𝒪 χ𝒪\mathcal{O}_\chi\subset\mathcal{O} for a central character χ\chi by the definition consists of all modules MM in 𝒪\mathcal{O} such that for each zz, the action (zχ(z)) nv=0(z-\chi(z))^n \cdot v = 0 for some n=n(z)>0n = n(z)\gt 0. There are canonical functors of projection pr λ:𝒪𝒪 χ λpr_\lambda:\mathcal{O}\to\mathcal{O}_{\chi_\lambda}. The functors

T λ μ:Mpr μ(L nu¯pr λM), T^\mu_\lambda : M\to pr_\mu (L_{\bar{nu}}\otimes pr_\lambda M),

T λ μ:𝒪𝒪 χ μT^\mu_\lambda : \mathcal{O}\to\mathcal{O}_{\chi_\mu} and its restriction to 𝒪 χ λ\mathcal{O}_{\chi_\lambda} is called the translation functor (because it changes the subcategories corresponding to different central characters).

Properties and applications

Translation functors are exact and preserve projective objects.

Used also in categorification in Lie theory…



  • J. C. Jantzen, Moduln mit einem höchsten Gewicht, (Modules with a highest weight) Lecture Notes in Math. 750, Springer 1979. ii+195 pp.

The translation functors are explained in detail in chapter 7 from

  • James E. Humphreys, Representations of semisimple Lie algebras in the BGG categry 𝒪\mathcal{O}, Grad. Studied in Math. 94, AMS 2008

Zuckerman came independently from Jantzen to the idea of using the translation functors

  • Greg Zuckerman, Tensor products of finite and infinite dimensional representations of semisimple Lie groups, Ann. of Math. 106 (1977) 295-308

Last revised on March 6, 2013 at 02:29:45. See the history of this page for a list of all contributions to it.