The Jantzen deformation and filtration of a Verma module

Put $R=\mathbb{C}\mathrm{⟦}t\mathrm{⟧}$ and ${\mathfrak{b}}_R=\mathfrak{b}{\otimes }_{\mathbb{C}}R$. Let $R_{\lambda +t\rho }$ be the free rank-one ${\mathfrak{b}}_R$-module on which ${\mathfrak{n}}_R^{+}$ acts by zero and $h\in \mathfrak{h}$ acts by $\lambda (h)+t\rho (h)$. The Jantzen deformation is $M_t(\lambda ):=U({\mathfrak{g}}_R){\otimes }_{U({\mathfrak{b}}_R)}{R}_{\lambda +t\rho }\mathrm{.}$ PBW makes each of its weight blocks finite free over $R$. Applying the same PBW-projection construction as for the Shapovalov form, now over $R$, gives a contravariant $R$-bilinear form and hence a deformed Shapovalov map $S_t:M_t(\lambda )\to M_t(\lambda {)}^{\vee }$, where the dual is taken weight-spacewise. Reduction modulo $t$ recovers the usual Shapovalov map on $M(\lambda )$. Define $M^i(\lambda )=\{\bar{v}\in M(\lambda ):\text{some lift }v\text{ has }{S}_t(v)\in t^i{M}_t(\lambda {)}^{\vee }\}\mathrm{.}$ Thus $M^0(\lambda )=M(\lambda )\supseteq M^1(\lambda )\supseteq \cdots$; the definition is made weight-spacewise, where the PBW blocks are finite free $\mathbb{C}\mathrm{⟦}t\mathrm{⟧}$-modules.