Let $H$ be a subgroup of $G$ with Lie algebras $\mathfrak{h}$ and $\mathfrak{g}$ respectively. If I have 2 representations $V, W$ of $\mathfrak{h}$ equipped with a $\mathfrak{h}$ invariant inner product. I can use a generalized Verma module to induce up to representations on $\mathfrak{g}$ (this is from Kac's book "Infinite dimentional Lie Algebras").
$$ M_\mathfrak{g} (V) = \mathfrak{U}({\mathfrak{g}})\otimes_{\mathfrak{U}({\mathfrak{b}}_- + \mathfrak{h})\ }V$$
where $\mathfrak{b}_-$ is the negative Borel sub-algebra with $\mathfrak{b}_-/ \mathfrak{h}$ acting trivially on $V$. This has a maximal subalgebra $m_V$ so that $ M_\mathfrak{g} (V)/m_V$ is an $\mathfrak{g}$ representation. My question is there an induced inner product on $ M_\mathfrak{g} (V)/m_V$ that is $\mathfrak{g}$ invariant and if so then is this construction natural with respect to isometries?