D-modules on the basic affine space and large g-modules

In this paper, we treat 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 D-modules are also holonomic using the formula. As a consequence, we give a large algebra action on the u-cohomologies Hⁱ (u; V) of a g-module V when V is realized as a holonomic D-module. We consider affinity of the supports of the t-modules Hⁱ (u; V).