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My first encounter with Category $\mathcal{O}$ was (perhaps unusually) learning about Kac-Moody algebras from Kac's book. Kac takes the following definition: The Category $\mathcal{O}$ has objects $\ …

Let $\mathfrak{g}$ be a symmetrizable Kac–Moody algebra, $w \in W$ an element of the Weyl group, and $\lambda$ an integral dominant weight with $V(\lambda)$ the associated irreducible highest weight r …

You should be a bit careful, as this isn't precisely the action of the Casimir on $v \otimes v$, but instead follows from it. For each positive root $\alpha$, let $e_\alpha^{(1)}, \dots, e_\alpha^{(n_ …

To fix (albeit standard) notation, let $G$ be a complex semisimple algebraic group, and $T \subset B \subset G$ choices of maximal torus and Borel subgroup, respectively. Let $X^\ast(T)$ be the charac …

In his paper "Paths and Root Operators in Representation Theory," Littelmann gives an action of the Weyl group on the set of integral paths via $$ \tilde{s}_\alpha(\pi):= \begin{cases} f^n_\alpha(\pi …

Let $\mathfrak{g}$ be a symmetrizable Kac--Moody algebra, and let $\lambda$ be an associated dominant integral weight. Then two different objects we can relate to this data is $V(\lambda)$, the irredu …

If you are trying to recognize the irreducible representations in some larger ambient representation, then the form of the highest weight vector(s) will depend heavily on that scenario. So your questi …

In my experience, the Laurent polynomial construction is more suited to the "algebraic" aspects of the theory--in particular, if the power of $t$ corresponds to the coefficient of $\delta$ in the root …