nLab translation functor

Definition

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

$$T^\mu_\lambda : M\to pr_\mu (L_{\bar{nu}}\otimes pr_\lambda M),$$

$T^\mu_\lambda : \mathcal{O}\to\mathcal{O}_{\chi_\mu}$ and its restriction to $\mathcal{O}_{\chi_\lambda}$ 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…

Literature

Translation

• 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 $\mathcal{O}$, 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