Wonderful asymptotics of matrix coefficient D-modules

TL;DR

This paper explores the asymptotic behavior of matrix coefficient D-modules in the context of Lie group representations, revealing how localization and degeneration techniques connect growth properties with geometric structures.

Contribution

It establishes a new algebraic framework linking Beilinson-Bernstein localization, asymptotics of D-modules, and the geometry of the wonderful compactification.

Findings

Asymptotics of matrix coefficient D-modules are described by parabolic restrictions.

Localization is compatible with degenerations to asymptotic cones.

Provides an algebraic derivation of the relation between matrix coefficient growth and $ $-homology.

Abstract

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 $\overline{G}$, or equivalently in the special fiber of the Vinberg semigroup of $G$. We show that Beilinson-Bernstein localization for $U\mathfrak g$-bimodules arises naturally as the specialization at infinity in $\overline{G}$ 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 $\overline{G}$ 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 $\mathfrak n$-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.