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Let $\mathfrak{g}$ be a finite-dimensional (complex) semisimple Lie algebra. Then we denote the BGG category for $\mathfrak{g}$ by $\mathcal{O}$ as usual. It is well-known that $\mathcal{O}$ as well as its (indecomposable) blocks are highest weight categories in the sense of the paper "Finite-dimensional algebras and highest weight categories" by Cline, Parshall and Scott.

$\text{My question:}$ Is there any example giving an equivalence $F$ between blocks which does not preserve highest weight structure? That is, $F$ does not send standard objects to standard objects. Thanks!

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  • $\begingroup$ The most elementary example is the duality functor (see Theorem 3.2 in my 2008 AMS book for the effect on blocks). I've used the notation $M \mapsto M^\vee$ for duality, but there are other choices in the literature. $\endgroup$ Dec 19, 2016 at 15:18
  • $\begingroup$ @Jim Humphreys: Thank you so much for the duality functor. I am sorry for my neglect. I mean a covariant functor which is an equivalence. I would be appreciate it if you could share more ideas. $\endgroup$
    – Steven
    Dec 19, 2016 at 16:15
  • $\begingroup$ Please clarify the statement of your question. What exactly do you mean by $F$, for instance? I assumed you were looking for an equivalence of categories, such as an auto-equivalence of $\mathcal{O}$. But now you've added the further requirement of covariance. $\endgroup$ Dec 19, 2016 at 22:16

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