Let $\mathfrak g$ be a Kac-Moody algebra.
Then $$ \mathfrak{n}_{-}=\oplus_{\alpha\in\varPhi_{+}}\mathfrak{g}_{-\alpha} $$
and for $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$ the Poincaré–Birkhoff–Witt basis is the family of monomials $$ \prod_{\alpha\in\Phi_{+}}\prod_{k=1}^{m_{\alpha}}\epsilon\left(e_{-\alpha,k}\right)$$ where $m_\alpha$ is the multiplicity of root $\alpha$ and $\epsilon:\mathfrak g \rightarrow \mathfrak{Ug}$ is the universal algebra of $\mathfrak g$ [Serre, pp.11-14; Kac, p. 152].
We now wish to show that $$ \mathrm{ch}\left(\mathfrak{U}\left(\mathfrak{n}_{-}\right)\right)=\prod_{\alpha\in\Phi_{+}}\left(1+\exp\left(-\alpha\right)+\exp^{2}\left(-\alpha\right)+\ldots\right)^{m_{\alpha}} $$
Questions:
- How can it be done?
- In particular, what would be the definition of the character of $\mathfrak{U}\left(\mathfrak{n}_{-}\right)$?
References:
[Kac] V. Kac. Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge, 1990
[Serre] J-P. Serre. Lie algebras and Lie groups. 1964 lectures given at Harvard University, Lecture Notes in Mathematics, 1500, (2006)
This is an immediate consequence of the PBW theorem; since taking associated graded does not change characters, the character is $U(\mathfrak{n}_-)$ is the same as $\mathrm{Sym}(\mathfrak{n}_-)$, which is exactly the product you wrote down.