Say you have a semi-simple (in my case even simple) Lie algebra $\mathfrak{g}$ with two simple modules $V$ and $W$, which may be infinite dimensional. Is $V \otimes W$ always semi-simple?
My guess would be no, since I know about the existence of the direct integral, and for $\mathfrak{sl}_2(\mathbb{R})$, the tensor product of positive and negative discrete series can be decomposed as a direct integral of principal unitary representations. If the answer is indeed no, are there known conditions when it is semi-simple?
And a somewhat more general question: if this isn't the case, can we at least always decompose $V \otimes W$ as a direct integral?