ON THE REGULARITY OF D-MODULES GENERATED BY RELATIVE CHARACTERS

Following the ideas of Ginzburg, for a subgroup K of a connected reductive ℝ-group G we introduce the notion of K-admissible D-modules on a homogeneous G-variety Z. We show that K-admissible D-modules are regular holonomic when K and Z are absolutely spherical. This framework includes: (i) the relative characters attached to two spherical subgroups H1 and H2, provided that the twisting character χi factors through the maximal reductive quotient of Hi, for i = 1; 2; (ii) localization on Z of Harish-Chandra modules; (iii) the generalized matrix coeficients when K(ℝ) is maximal compact. This complements the holonomicity proven by Aizenbud–Gourevitch–Minchenko. The use of regularity is illustrated by a crude estimate on the growth of K-admissible distributions based on tools from subanalytic geometry.