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We give an introduction to the $\mathbb{Z}$-graded representation theory ofthe BGG category $\mathcal{O}$ of a complex semisimple Lie algebras, with anemphasis on Soergel's combinatorial $\mathbb{V}$ functor, definitions of$\mathbb{Z}$-graded duality functors and definitions of $\mathbb{Z}$-gradedtranslation functors.

BGG category $\mathcal{O}$ and $\mathbb{Z}$-graded representation theory