# 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 the ($\mathfrak g$-module) isomorphism $V(\mu)\cong U(\mathfrak n^-)\otimes_\mathbb K \mathbb Kw$ (where $w$ is the highest-weight vector of weight $\mu$) shows that the weight spaces $V_\lambda$ are finite-dimensional, because every $x_i^-$ lowers the weight in some way.

When $n=1$, the highest weight is, say, $\mu$ and every other weight can be written as $\mu-2k$. In particular, $V_{\mu-2k}$ is spanned by $(x^-)^kw$, where $w$ is again the highest weight vector.

I don't quite understand the higher-dimensional analogue. Is it correct to write, for $n\geq 1$ the highest weight as $\mu=(k_1,\dotsc,k_n)$ ($k_i\in\mathbb N$) and the weight of the weight space spanned by the collection of $(x_1^-)^{a_1}\dotsb (x_n^-)^{a_n}w$ ($\sum a_i=a$ some integer) as $\mu-\sum a_i\alpha_i$, where $\alpha_i$ are the simple roots?

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