Question about definition of Verma modules

Question

I am studying Verma modules (reading Dixmier´s Enveloping Algebras) and have a question regarding the definition as a quotieng of the enveloping algebra.

Let $g$ be a Lie algebra and $h$ its Cartan subalgebra.

Verma module is constructed for a given highest weight $\lambda$, living in the dual space $h^\star$. It is defined as the quotient vector space $W_\lambda = U(g)/I_\lambda$ where $U(g)$is the universal enveloping algebra of $g$. I understand what $U(g)$ is.

My trouble is with the ideal $I_\lambda$, which is said to be generated by the elements $X_\alpha$, $\alpha \in R^+$. That is, $X_\alpha$ should be elements corresponding to positive roots. However, I am struggling to find the exact definition, in Dixmier or even in other sources. (Maybe I just overlooked them or got something wrong.)

What are these $X_\alpha$? And why does it make sense to use them for generating the ideal that is then used for defining the $W_\lambda$? I would like to understand more why are Verma modules defined this way and how does the quotient space look, so thats why I want to interpret the ideal correctly.

Thank you very much.

UPDATE: The first mention of $X_\alpha$ I have found in Dixmier is in this theorem, so I assume the definition is solved. However, I am still grateful for any insights about the ideal generated by such elements and the rest of my question.

Answer 1

I don't have my copy of Dixmier's textbook at hand, and therefore I can't tell exactly where the $X_\alpha$'s are defined (but I have no doubt that the definition is done in the first chapter). Anyway, what matters here is that $X_\alpha\in\mathfrak g_\alpha\setminus\{0\}$. Since $\dim\mathfrak g_\alpha=1$, the exact choice of $X_\alpha$ doesn't matter as far as the definition of $I_\lambda$ is concerned.

By the way, $I_\lambda$ is the ideal of $\mathcal U(\mathfrak g)$ generated by the $X_\alpha$'s and also by the elements of the form $H-\lambda(H)1$, with $H\in\mathfrak h$.