$\mathscr{D}$-modules on the basic affine space and large $\mathfrak{g}$-modules

In this paper, we treat $\mathscr{D}$-modules on the basic affine space $G/U$ and their global sections for a semisimple complex algebraic group $G$. Our aim is to prepare basic results about large non-irreducible modules for the branching problem and harmonic analysis of reductive Lie groups. A main tool is a formula given by Bezrukavnikov--Braverman--Positselskii. The formula is about a product of functions and their Fourier transforms on $G/U$ like Capelli's identity. Using the formula, we give a generalization of the Beilinson--Bernstein correspondence. We show that the global sections of holonomic $\mathscr{D}$-modules are also holonomic using the formula. As a consequence, we give a large algebra action on the $\mathfrak{u}$-cohomologies $H^i(\mathfrak{u}; V)$ of a $\mathfrak{g}$-module $V$ when $V$ is realized as a holonomic $\mathscr{D}$-module. We consider affinity of the supports of the $\mathfrak{t}$-modules $H^i(\mathfrak{u}; V)$.