Geometrization of the Schrödinger 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 $0$, $C\subset V^{\ast }$ be the isotropic cone, $\Delta \in D_V$ be the associated Laplace--Beltrami operator, $G$ be the conformal group of $V$, and $D_C$ be the algebra of Grothendieck differential operators on $C$. We prove that the category of finitely generated $D_C$-modules is equivalent both to a Kazhdan--Laumon glued category attached to the smooth locus $C^o$ and to a category of ``harmonic'' twisted $D$-modules on the projective conformal compactification $G/P\supset V$. The gluing is governed by the quadric Fourier transform, while the harmonic model is built from a distinguished $G$-equivariant sheaf $H$ on $G/P$ extending the local quotient $D_V/D_V\Delta$. We prove a new geometric interpretation of higher symmetries of the Laplacian as global sections of $H$, and use this connection to give a geometric proof of the theorem of Levasseur, Smith, and Stafford that $D_C$ is Noetherian despite the singularity of $C$. We also study, via a descent procedure we call ``$F$-moment descent,'' the algebraic geometry of the closure of the minimal nilpotent orbit of $G$, which is the quasiclassical analogue of the minimal representation. Finally, we analyze the filtered structure of $D_V/D_V\Delta$ as a right $D_C$-module, identifying its associated graded layers through a flat degeneration of an affine flag multicone whose special fiber is the Rees space of a natural ideal in $\kappa [{\overline{O}}_{min}]$.

Corrected an editing splice in one proof and fixed minor typos; results unchanged. Comments welcome!