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

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.