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Let $\Delta$ be a irreducible root system and $\Delta^+$ be its positive roots. We say a subset $\Delta^{\prime}\subset \Delta^+$ can generate the Weyl group if reflections of roots in $\Delta^{\prim …

Let $(\mathfrak{g},\mathfrak{h},\Phi)$ be a root system of a complex simple Lie algebra, where $\Phi$ is the set of all roots. For each $\alpha\in \Phi$, let $\alpha^{\vee}=2\alpha/(\alpha,\alpha)$ be …

I think the answer is yes because $(\Phi_{[\lambda]})^{\vee}$ and $(\Phi_{[\mu]})^{\vee}$ are closed sub-root systems of the dual root system $\Phi^{\vee}$. Closed means if $\alpha$ and $\beta$ are ro …

Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{h}$ be a Cartan subalgebra. For any $\lambda\in \mathfrak{h}^{*}$ let $M(\lambda)$ and $L(\lambda)$ be the Verma module and the simple mo …

This question may be a simple problem for experts. Let $G$ be a connected compact Lie group and $T$ be its maximal torus. Let $\mathfrak{g}$ and $\mathfrak{t}$ be the corresponding Lie algebras. We kn …

The question is related to this MO question. Let $(\Phi, E)$ be a irreducible crystallographic root system where $\Phi$ is the set of all roots and $E$ is the $\mathbb{R}$-span of $\Phi$. As in the st …

$\def\abs#1{\lvert#1\rvert}\DeclareMathOperator\Span{Span}$I think I get a proof inspired by the comment of @LSpice. First we can prove that $\Phi_1\setminus \Phi_2$ is orthogonal to $\Phi_2\setminus …

It may be a easy question for experts. The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra $\mathfrak{h} …