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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 …

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 …

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 …

$\def\abs#1{\lvert#1\rvert}\DeclareMathOperator\Span{Span}$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 …

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 …

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 …

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra (according to a comment of Victor Ostrik, we need to further require that $\mathfrak{g}$ is simple) and we can consider its BG …

Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra and we can consider its BGG category $\mathcal{O}$. It is well-known that $\mathcal{O}$ is not closed under tensor product, i. …

Let $G$ be a real semisimple Lie group, $K$ be a maximal compact subgroup. Let $\mathfrak{g}_0$ and $\mathfrak{k}_0$ be their real Lie algebras respectively. Let $\mathfrak{g}$ and $\mathfrak{k}$ be t …

Let $G$ be a real semisimple Lie group and $K$ a maximal compact subgroup. Let $\mathfrak{g}$ and $\mathfrak{k}$ be the complexified Lie algebra of $G$ and $K$, respectively. Then the Hecke algebra …

For a first introduction you can read Michel Brion's "http://arxiv.org/pdf/math/0410240v1.pdf". He gives a nice introduction (for G=GL(n)) in Section 1. I'm not sure whether your curve method works b …

In general cases Verma modules are corresponding to D-modules on flag variety and then via Riemann-Hilbert correspondence to constructible sheaves on flag variety. Hence the suitable generalization of …

For your question 2, the reason is in fact we can prove $$ K\times \mathfrak{p}\xrightarrow{\sim} G\\ (k,p)\mapsto k\exp(p) $$ is an diffeomorphism. Here $\mathfrak{p}$ is the $-1$ eigen space of the …

This question may be trivial for experts. Let $\mathfrak{g}$ be a Lie algebra over a field $k$ and consider the category $\mathfrak{g}$-mod of $\mathfrak{g}$ modules. We can add suitable conditions, …

If we have two Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ over a field $k$, and if we have a Lie algebra homomorphism $\mathfrak{g}\rightarrow \text{Der}_k(\mathfrak{h})$, then we can define the s …