# A question about tensor product of $M,N\in\mathcal{O}$.

In the book "Representations of Semisimple Lie Algebras in the BGG Category O" by Humphrey.

Exercise 1.5: When $$\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$$, show that $$M(\lambda)\otimes M(\mu)$$ cannot lie in $$\mathcal{O}$$.

If $$M,N\in\mathcal{O}$$, why not $$M\otimes N\in\mathcal{O}$$ in general?

Here is my "argument" (should be wrong) to show $$M,N\in\mathcal{O}\implies M\otimes N\in\mathcal{O}$$:

1. Since $$M,N$$ are both finitely generated as $$U(\mathfrak{g})$$-module, then $$M\otimes N$$ is finitely generated as $$U(\mathfrak{g})$$-module.

2. Clearly, $$\bigoplus_{\lambda\in\mathfrak{h}^*}(M\otimes N)_\lambda\subseteq M\otimes N$$. Conversely, suppose $$v\otimes w\in M\otimes N$$, then $$v=v_1+\cdots+v_k$$, where $$v_i\in M_{\lambda_i}$$ and $$w=w_1+\cdots+w_l$$, where $$w_i\in N_{\mu_i}$$.

Then $$v\otimes w=\sum_{i=1}^k\sum_{j=1}^l v_i\otimes w_j\in \bigoplus_{\lambda\in\mathfrak{h}^*}(M\otimes N)_\lambda$$ since $$h.(v_i\otimes w_j)=(h.v_i)\otimes w_j+v_i\otimes (h.w_j)=(\lambda_i v_i)\otimes w_j+v_i\otimes (\mu_jw_j)=(\lambda_i+\mu_j)v_i\otimes w_j$$.

Hence $$M\otimes N=\bigoplus_{\lambda\in\mathfrak{h}^*}(M\otimes N)_\lambda$$.

1. For each $$v\otimes w\in M\otimes N$$, $$U(\mathfrak{n}).(v\otimes w)=(U(\mathfrak{n}).v)\otimes w+v\otimes (U(\mathfrak{n}).w)$$ is a finite dimensional since $$U(\mathfrak{n}).v$$ and $$U(\mathfrak{n}).w$$ are finite dimensional.

Where do I make a mistake? And how to do the exericse 1.5?

• Certainly the notation used in your question can be found in Humphrey's book. But the reader who doesn't have access to the book would like to know what $M(\lambda)$, $\mathcal{O}$ and $U(\mathfrak{g})$ are. Oct 2, 2018 at 13:41
• Point 1 is wrong. The tensor products of two finitely generated $\mathfrak{g}$ modules may not be finitely generated. Oct 2, 2018 at 13:53