  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 $H^i(\mathfrak{u}; V)$.
