Geometrization of the Schr\"odinger Model for the Minimal Representation of an Even Orthogonal Group: The de Rham Setting

TL;DR

This paper develops and compares three geometric D-module models for the minimal representation of an even orthogonal group's conformal group, revealing new equivalences and constructions in the context of algebraic geometry and representation theory.

Contribution

It introduces three new D-module models for the minimal representation and establishes their equivalence, including a novel quadric Fourier transform and geometric insights into the algebra's structure.

Findings

Established equivalence between different D-module categories

Constructed a quadric Fourier transform on D_C

Proved D_C is finitely generated despite singularities

Abstract

We construct and compare three D-module models for the minimal representation of the conformal group of an even-dimensional quadratic space. Let $V$ be a quadratic space over a field $\kappa$ of characteristic $0$, let $C$ be the isotropic cone in $V^*$, and let $G$ be the conformal group of $V$. We prove an equivalence between the category of modules over the Grothendieck differential operator algebra $D_C$, a Kazhdan--Laumon glued category attached to the smooth locus of the cone, and a category of "harmonic" twisted D-modules on a flag variety $G/P$. Along the way, we construct a quadric Fourier transform on $D_C$, provide a geometric proof that the algebra $D_C$ is finitely generated despite the singularity of $C$, and explain the quasi-classical analogue of this minimal representation.