Thank you very much, David! I am trying to rephrase what you said: Since $\mu: T^*\mathcal{B}\rightarrow \mathcal{N}$ is the moment map, by considering vector fields on $\mathcal{B}$ as functions on $T^*\mathcal{B}$, we can identify the comoment map $$ \mathfrak{g}\rightarrow C^{\infty}(T^*\mathcal{B}) $$ with the infinitesimal action $$ \mathfrak{g}\rightarrow T\mathcal{B}. $$ Then we quantize the later one and get the map from $U(\mathfrak{g})$ to global differential operators on the flag variety. Now we can construct the Beilinson-Bernstein equivalence. Is that (roughly) what you mean?

What do you mean by "each function $f_n(x)$ has a minimizer"?