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Questions tagged [verma-modules]

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Simple formula for the dimension of weight spaces of Verma module?

Let $\mathfrak{g}$ be a simple Lie algebra (e.g. $\mathfrak{sl}_n$), and let $M_\lambda$ be the Verma module with highest weight $\lambda$. Is there a simple formula for $\dim (M_\lambda)_\mu$, where ...
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1answer
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Extension group between Verma modules for $\mathfrak{sl}_2$

Let $\mathcal{g}=\mathfrak{sl}_2(\mathbb C)$ be the simple lie algebra of $\text{SL}_2$, for any $\lambda \in \mathbb C$ one can consider the corresponding Verma module $M_{\lambda}$, which is ...
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Verma module analogue for non-semisimple Lie algebras

If $\mathfrak{g}$ is semisimple and complex, then we can identify a Cartan subalgebra $\mathfrak{h}$ and with choice of positive roots, a Borel subalgebra containing this Cartan. We can then consider ...
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Weight Vectors for $\mathfrak{sl}_3(\mathbb{C})$ Highest Weight Modules

I am studying Humphrey's book on BGG Category $\mathcal{O}$ and have been trying to understand the importance of Shapovalov's Theorem by means of Exercise 4.12. The question states, for $\mathfrak{sl}...
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Algebraic (g,B) modules

Let $\mathfrak{g}$ be a semisimple lie algebra and G its adjoint group, with Borel group B. I am trying to understand the theory of algebraic (g,B) modules as defined by Borho and Brylinski in https://...
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1answer
114 views

Extension of Dual Verma Module

In Humphrey's BGG Category $\mathcal{O}$ Exercise 3.3 he asks: Does the short exact sequence below always split? $$ 0 \rightarrow M(\lambda) \rightarrow E \rightarrow M(\mu)^\check{} \ \rightarrow 0.$...
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Verma module associated to a Lie algebra

Let $L$ be a Lie algebra with Cartan decomposition $L=H \oplus(\sum_{\alpha \in \Phi}L_{\alpha})$. Let $B=\sum_{\alpha \in \Phi^+}L_{\alpha}$ and $N=\sum_{\alpha \in \Phi^-}L_{\alpha}$ so that $L=B \...
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1answer
253 views

Verma flag (or standard filtration) on a direct summand.

Suppose $\mathfrak{g}$ semisimple Lie algebra and $M$ is a $\mathfrak{g}$-module, a Verma flag (or standard filtration) is a filtration $M=M_0>M_1> \ldots >M_n=0$ such that $M_i/M_{i+1} \...
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1answer
380 views

Composition series for Verma modules.

Let $L$ a Lie Algebra. I need prove that that every Verma module $\Delta(\lambda)$ admits a composition series, i.e a series of submodules with simple factors. I found a proof that is quite short in ...
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1answer
185 views

How to find the multiplicity of weight in a Verma module?

In particular, let $\mathfrak g$ be the semisimple Lie algebra of type $A_{2}$ et let $\alpha,\beta$ be its simple roots. How can the multiplicity of weight $-2\alpha -3\beta$ be calculated in the ...
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Definition of Verma modules

I have a question regarding different (but equivalent!?) definitions of Verma modules of semisimple Lie algebras: Let F be a field and denote the following: $ \mathfrak{g}$ , a semisimple Lie ...
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2answers
168 views

Equivalent definitions of Verma modules

This is a rather basic question. I was reading some notes on geometric representation theory by Gaitsgory and his defition of Verma module is the following: Let $ \lambda $ be a weight of $ \mathfrak{...
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1answer
182 views

Weight spaces of Verma modules

Let $\mathfrak g$ be a semisimple Lie algebra generated by $x_i^+,x_i^-$, $1\leq i\leq n$, via the Chevalley-Serre relations and let $V(\mu)$ be a Verma module with highest-weight $\mu$. I gather that ...