Wonderful Asymptotics of Matrix Coefficient D-modules

Beilinson–Bernstein localization realizes representations of complex reductive Lie algebras as monodromic D-modules on the “basic affine space” G/N, a torus bundle over the flag variety. A doubled version of the same space appears as the horocycle space describing the geometry of the reductive group G at infinity, near the closed stratum of the wonderful compactification G‾, or equivalently in the special fiber of the Vinberg semigroup of G. We show that Beilinson–Bernstein localization for Ug-bimodules arises naturally as the specialization at infinity in G‾ of the D-modules on G describing matrix coefficients of Lie algebra representations. More generally, the asymptotics of matrix coefficient D-modules along any stratum of G‾ are given by the matrix coefficient D-modules for parabolic restrictions. This provides a simple algebraic derivation of the relation between growth of matrix coefficients of admissible representations and n-homology. The result is an elementary consequence of the compatibility of localization with the degeneration of affine G-varieties to their asymptotic cones; analogous results hold for the asymptotics of the equations describing spherical functions on symmetric spaces.