Weights of universal enveloping algebra

Question

Let $L$ be a semi simple Lie algebra over an algebraically closed field $F$ with

1. Cartan decomposition $L = h \oplus n_+ \oplus n_- $,

2. Root system $\Phi$,

3. Set of positive roots $\Phi_+$,

4. Simple roots $\Delta$.

Consider the universal enveloping algebra $U(n_+)$ of $n_+$ which can be consider as a $n_+$ - module.

What is the dimension of the weight space $U(n_+)_\beta$ for a root $\beta$ ?

I am trying to prove that, this dimension is equal to $K(\beta,q)$ where $K(.;q)$ is the q-kostant partition function.

$K(\beta,q)$ = co-efficient of $e^{-\beta}$ in the product $\prod_{\alpha \in \Phi_+}(1-qe^{-\alpha})^{-mult \alpha}$

Thanks in Advance.

Answer 1

Denoting ${\mathscr S}(V)$ the symmetric algebra over a vector space $V$, realized as the subalgebra of symmetric tensors within the tensor algebra ${\mathscr T}(V)$ (as opposed to a quotient), the PBW-Theorem implies that $$\bigotimes\limits_{\alpha\in\Phi_+} {\mathscr S}({\mathfrak g}_{\alpha})\to\bigotimes\limits_{\alpha\in\Phi_+}{\mathscr U}({\mathfrak n}_+)\xrightarrow{\text{mult}}{\mathscr U}({\mathfrak n}_+)$$ is an isomorphism of vector spaces, which moreover is quickly seen to be ${\mathfrak h}$-linear. Hence, $$\chi\left({\mathscr U}({\mathfrak n}_+)\right) = \prod\limits_{\alpha\in\Phi_+}\chi\left({\mathscr S}({\mathfrak g}_{\alpha})\right).\quad\quad(\ddagger)$$ Similarly, denoting ${\mathbb C}_\alpha$ the simple ${\mathfrak h}$-module with weight $\alpha$, we have ${\mathscr S}({\mathfrak g}_\alpha)\cong {\mathscr S}({\mathbb C}_\alpha)^{\otimes \dim({\mathfrak g}_\alpha)}$, so $$\chi\left({\mathscr S}({\mathfrak g}_\alpha)\right)=\chi\left({\mathscr S}({\mathbb C}_\alpha)\right)^{\dim({\mathfrak g}_\alpha)}=(1+e^{\alpha} + e^{2\alpha} + ...)^{\dim({\mathfrak g}_\alpha)}=(1-e^{-\alpha})^{-\dim({\mathfrak g}_\alpha)},$$ which together with $(\ddagger)$ gives your claim, up to the $q$ of which I also don't know what it's supposed to be.