Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $V$ be a highest weight module of a semisimple Lie algebra $\mathfrak{g}$ with highest weight $\lambda$. Then there is a highest weight vector $v$ such that $V=U(\mathfrak{g})^{-}\cdot v$. Let $x_i^{-}$ ($i\in I ={1, \ldots, n}$) be a set of generators of $U(\mathfrak{g})^{-}$. Let $W=\oplus_{\mu\in \mathfrak{h},v(\mu)\leq p}V_{\mu}$, where $\mathfrak{h}$ is generated by coroots $\alpha_1^{\vee},\ldots, \alpha_n^{\vee}$, and $v(\mu)=b_1+\ldots +b_n$ for $\mu=\sum_{i=1}^{n}b_i \alpha_i^{\vee}$. Does $W$ has the property: $\oplus_{\mu\in\mathfrak{h},v(\mu)\leq p+1}V_{\mu}\subset \sum_{i\in I} x_i^{-}\cdot W$? If $V$ is not a highest weight module, does $W$ have this property? Thank you very much.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.