Tensor product of Verma modules

Question

Let $M(\mu_1)$ and $M(\mu_2)$ be two Verma modules of lie algebra $\mathfrak{sl_2}$.

Why is the tensor product $M(\mu_1) \otimes_{\mathbb{C}} M(\mu_2)$ not finitely generated?

Answer 1

The tensor product is again locally ${\mathfrak U}({\mathfrak n}^+)$-finite, but it's no longer finitely generated. You can see this by working out the weight space dimensions - for a module in ${\mathscr O}({\mathfrak s}{\mathfrak l}_2)$, these are eventually constant.