Tensoring a Verma module by a finite-dimensional module shifts the type

Put $N_k:=U(\mathfrak{g})(v_{\psi }\otimes v_1,\dots ,v_{\psi }\otimes v_k)$ for $0\le k\le n$. These are $\mathfrak{g}$-submodules forming an increasing filtration $0=N_0\subseteq \cdots \subseteq N_n$ of $M(\psi )\otimes V$; and $N_n=M(\psi )\otimes V$. To see the latter, induct on the PBW degree of $u\in U({\mathfrak{n}}^{-})$: the diagonal action satisfies $u(v_{\psi }\otimes v_i)=(u{v}_{\psi })\otimes v_i$ plus terms of strictly smaller PBW degree in the first factor. Those terms lie in $N_n$ by induction, and the degree-zero tensors are its generators, so all $(u{v}_{\psi })\otimes v_i$ lie in $N_n$.

The vector $v_{\psi }\otimes v_k$ is a weight vector of weight $\psi +{\lambda }_k$, since $h(v_{\psi }\otimes v_k)=(\psi +{\lambda }_k)(h)(v_{\psi }\otimes v_k)$, and ${\mathfrak{n}}^{+}(v_{\psi }\otimes v_k)=v_{\psi }\otimes {\mathfrak{n}}^{+}{v}_k\in v_{\psi }\otimes \mathrm{span}(v_1,\dots ,v_{k-1})\subseteq N_{k-1}$ by [F1]. Hence the class of $v_{\psi }\otimes v_k$ in $N_k\mathrm{/}{N}_{k-1}$ is a highest weight vector of weight $\psi +{\lambda }_k$, and because all the other generators of $N_k$ lie in $N_{k-1}$ this class generates $N_k\mathrm{/}{N}_{k-1}$ as a $U(\mathfrak{g})$-module.

$N_k$ is free over $U({\mathfrak{n}}^{-})$ on the generators $v_{\psi }\otimes v_1,\dots ,v_{\psi }\otimes v_k$. Indeed, $N_k=U({\mathfrak{n}}^{-})(v_{\psi }\otimes v_1,\dots ,v_{\psi }\otimes v_k)$: by [F1] the action of $U(\mathfrak{b})U({\mathfrak{n}}^{+})$ on $v_{\psi }\otimes v_i$ stays in $v_{\psi }\otimes \mathrm{span}(v_1,\dots ,v_i)$, and $U({\mathfrak{n}}^{-})$ carries $v_{\psi }$ to $M(\psi )$. If ${\sum }_i{\xi }_i(v_{\psi }\otimes v_i)=0$ with ${\xi }_i\in U({\mathfrak{n}}^{-})$, choose the largest PBW degree $d$ occurring among the ${\xi }_i$ and take the degree-$d$ part of the relation in the associated graded $S({\mathfrak{n}}^{-})\otimes V$: it reads ${\sum }_i\sigma ({\xi }_i)\otimes v_i=0$ with $\sigma ({\xi }_i)=0$ whenever $\mathrm{deg}{\xi }_i<d$ and $\sigma ({\xi }_i)\mathrm{\ne }0$ for the maximal ones; since $S({\mathfrak{n}}^{-})$ is a domain and the $v_i$ are linearly independent over $\mathbb{C}$, all $\sigma ({\xi }_i)$ vanish, a contradiction.

Consequently $N_k\mathrm{/}{N}_{k-1}$ is free of rank one over $U({\mathfrak{n}}^{-})$, generated by the class $c_k$ of $v_{\psi }\otimes v_k$. By The universal property of Verma modules there is a nonzero (hence surjective) homomorphism $M(\psi +{\lambda }_k)\to N_k\mathrm{/}{N}_{k-1}$ carrying the highest weight vector to $c_k$; source and target are both free of rank one over $U({\mathfrak{n}}^{-})$ by [F2] and step 3.1, and the map carries a free generator to a free generator, so it is an isomorphism. Thus $N_k\mathrm{/}{N}_{k-1}\cong M(\psi +{\lambda }_k)$.

The filtration $0=N_0\subseteq \cdots \subseteq N_n=M(\psi )\otimes V$ therefore exhibits $M(\psi )\otimes V$ as Verma-filtered with type $\{\psi +{\lambda }_1,\dots ,\psi +{\lambda }_n\}=\psi +\mathrm{Wt}V$.