Radon transforms of twisted D-modules on partial flag varieties
Kohei Yahiro
TL;DR
The paper develops a geometric framework for Radon transforms (intertwining functors) on twisted D-modules over partial flag varieties and connects these transforms to representations of the complex semisimple Lie algebra $\mathfrak g$. It extends Beilinson–Bernstein localization to a broad class of orbits in $G/P_J \times G/P_I$, proving that the associated intertwining functors $R^{w,\mu}_{+}$ and $R^{w,\mu}_{!}$ are equivalences of derived categories, and establishes a localization-type compatibility with global sections in the dominant-to-antidominant direction. A key technical achievement is a set of isomorphism criteria for the natural maps between derived global sections and Radon transforms, including a maximal parabolic case governed by irreducibility of certain generalized Verma modules. The results generalize Marastoni's work and provide a unified approach to translating geometric Radon transforms into representation-theoretic data, with potential applications to category equivalences and Kazhdan–Lusztig type phenomena in both finite and affine settings.
Abstract
In this paper we study intertwining functors (Radon transforms) for twisted D-modules on partial flag varieties and their relation to the representations of semisimple Lie algebras. We show that certain intertwining functors give equivalences of derived categories of twisted D-modules. This is a generalization of a result by Marastoni. We also show that these intertwining functors from dominant to antidominant direction are compatible with taking global sections.