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Let $W$ be the Weyl group of a complex semisimple Lie algebra $\frak{g}$. Certainly $W$ acts on the root system $R$ of $\frak{g}$ but can it be made to act on $\frak{g}$ or on the universal enveloping algebra?

Lorenzo Del Vecchiopontopolos
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    If you write $W = N_G(T)/T$ you see that there is no reason that $W$ should acts on $U(g)$. $W$ will usually acts on objects related to $Lie(T)$, for example center of category $\mathcal O$ with fixed character. This is related to so-called "translation functor". – Nicolas Hemelsoet Jun 21 '23 at 12:28
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    There is a standard ``lift'' of $W $ to $ G $. You can make this lift act on $ \mathfrak g $ by the adjoint action. – Joel Kamnitzer Jun 21 '23 at 13:50
  • @JoelKamnitzer: Thanks for the comment. What is the standard lift of $W$ to $G$? – Lorenzo Del Vecchiopontopolos Jun 21 '23 at 20:51
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    @JoelKamnitzer Unfortunately, if by $G$ you mean the Lie group, the extension of $W$ by $T$ to give $N_G(T)$ is not always split, so there isn't a "standard lift" or necessarily any lift. You can find a full discussion of this in Curtis, Wiederhold and Williams, "Normalizers of maximal tori" Springer LNM 418 (1974). – Dave Benson Jun 21 '23 at 21:32
  • https://mathoverflow.net/questions/13139/can-we-realize-weyl-group-as-a-subgroup – Bugs Bunny Jun 23 '23 at 07:42
  • @Bugs Bunny: Yes, I think this does answer my question. Thanks a lot! – Lorenzo Del Vecchiopontopolos Jun 25 '23 at 15:51

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