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Let $G$ be a split connected semi-simple (or reductive) algebraic group over a (non-archimedean) field $k$ of characteristic zero. Denote by $\mathfrak{g}=\mathrm{Lie}(G)$ the semi-simple (or reductiv …

Let $\mathfrak{g}=\mathfrak{gl}_3$ over $\mathbb{C}$ with positive roots \begin{equation*} \Phi_+=\{\alpha_1=(1,-1,0),\alpha_2=(1,0,-1),\alpha_3=(0,1,-1)\}. \end{equation*} Consider the morphism \begi …

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ with Cartan subalgebra $\mathfrak{h}$, root system $\Phi \subset \mathfrak{h}^*$ and Weyl group $W$. Fix a set of positive roots $\Phi^ …

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ with root system $\Phi$, Weyl group $W$ and Cartan decomposition $\mathfrak{g}=\mathfrak{h}\oplus \bigoplus_{\alpha \in \Phi} \mathfrak …

As in the title I looking for technics to compute weights of kernels, images or cokernels in category $\mathcal{O}$ besides checking everything directly by hand. To be more concrete, consider the foll …

$\DeclareMathOperator\Hom{Hom}$I'm trying to understand morphism of Verma modules and consider the following example. PART 1: Consider $\mathfrak{g}=\mathfrak{gl}_3$ over $\mathbb{C}$ with positive ro …

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $G$ be a split connected reductive algebraic group over $K$ with Borel $B$. We have the associated Lie algebras $\mathfrak{g}=$Lie$(G)$ and $\mathfr …

What is the reason for the BBG category $\mathcal{O}$ failing to be closed under extensions i.e which of the 3 axioms of $\mathcal{O}$ does not hold under taking extensions? Is there a prototype of …