Unique Factorization for Tensor Products of Parabolic Verma Modules

Let \(\mathfrak g\) $\mathfrak{g}$ be a symmetrizable Kac-Moody Lie algebra with Cartan subalgebra \(\mathfrak h\) $\mathfrak{h}$ . 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 \(\mathfrak h\) $\mathfrak{h}$ . These include fixed point subalgebras of \(\mathfrak h\) $\mathfrak{h}$ under subgroups of diagram automorphisms of \(\mathfrak g\) $\mathfrak{g}$ and twisted graph automorphisms in the affine case.