Geometric interplay between function subspaces and their rings of differential operators

2004/03/24 by Rikard Bøgvad, Rikard Bögvad, Bögvad, Rikard +2
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math.AG #math.RT

paper · pdf · doi:10.48550/arxiv.math/0403409

36 pages, LaTeX

arxiv created 2004/03/24 · openalex publication_date 2004/03/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study, in the setting of algebraic varieties, finite-dimensional spaces of functions V that are invariant under a ring D^V of differential operators, and give conditions under which D^V acts irreducibly. We show how this problem, originally formulated in physics (Kamran-Milson-Olver), is related to the study of principal parts bundles and Weierstrass points (Laksov-Thorup), including a detailed study of Taylor expansions. Under some conditions it is possible to obtain V and D^V as global sections of a line bundle and its ring of differential operators. We show that several of the published examples of D^V are of this type, and that there are many more -- in particular arising from toric varieties.

Related

• Représentations irréductibles de certaines algèbres d'opérateurs différentiels
• Differential operators on toric varieties and Fourier transform
• Finite dimensional representations of invariant differential operators
• Differential Isomorphism and Equivalence of Algebraic Varieties
• Differential operators on Hitchin variety