In [1] (section 15.3), they say that the tensor product of finite dimensional highest weight semisimple Lie algebra modules $V_\lambda$ and $V_\mu$ decomposes into a direct sum of irreducible finite dimensional highest weight modules, and in particular one of the modules in the decomposition is $V_{\lambda + \mu}$.
I want to know if this is also true for infinite dimensional highest weight $Vir$-modules, with $Vir$ the Virasoro algebra. I expected this to be well explored in the literature, but found nothing so far.
Note: On page 268 of [1] the author says that such a decomposition of the weights still holds for any Lie algebra which admits a triangular decomposition (like $Vir$), but does not say if it is still true that the corresponding weight spaces are still irreducible. In particular: is $V_{\lambda+\mu}$ still irreducible?
[1] Fuchs, J. and Schweigert, C. (2003). Symmetries, Lie algebras and representations: A graduate course for physicists. Cambridge University Press.