Mathematics > Representation Theory
arXiv:2604.11023 (math)
[Submitted on 13 Apr 2026 (v1), last revised 25 Jun 2026 (this version, v2)]
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 $0$, $C\subset V^*$ 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 $\kappa[\overline{O}_{min}]$.
| Comments: | Updated from the earlier version, which incorrectly identified the conformal twisting line bundle as O(k-1) rather than O(1-k). This rules out a naive use of Beilinson-Bernstein localization; the present version replaces it with direct arguments, modifies the harmonic category, and discusses the local harmonic quotient filtration. Comments welcome! |
| Subjects: | Representation Theory (math.RT); Number Theory (math.NT); Rings and Algebras (math.RA) |
| Cite as: | arXiv:2604.11023 [math.RT] |
| (or arXiv:2604.11023v2 [math.RT] for this version) | |
| https://doi.org/10.48550/arXiv.2604.11023 arXiv-issued DOI via DataCite |
Submission history
From: Aaron Slipper [view email]
[v1]
Mon, 13 Apr 2026 05:41:22 UTC (103 KB)
[v2]
Thu, 25 Jun 2026 07:35:35 UTC (140 KB)
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