  Let $\mathfrak{g}$ be a symmetrizable Kac-Moody Lie algebra with Cartan
subalgebra $\mathfrak{h}$. We prove a unique factorization property for tensor
products of parabolic Verma modules. More generally, we prove unique
factorization for products of characters of parabolic Verma modules when
restricted to certain subalgebras of $\mathfrak{h}$. These include fixed point
subalgebras of $\mathfrak{h}$ under subgroups of diagram automorphisms of
$\mathfrak{g}$ and twisted graph automorphisms in the affine case.
