Invariant algebraic D-modules over affine algebraic groups

2025/05/19 by Wei, Yunsong
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2505.12755

openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the invariant algebraic D-modules on an affine variety under the action of an algebraic group.For linear algebraic groups with the multiplication action by themselves, such D-modules correspond to representations of their Lie algebra. For unipotent algebraic groups, we show that two invariant D-modules are isomorphic if and only if they lie in the same fiber of the GIT (Geometric Invariant Theory) quotient of the space of representations under the action of conjugation. Additionally, we classify invariant D-modules over the algebraic torus and the Borel subgroup of the general linear group.

Citations

• Components and singularities of Quot schemes and varieties of commuting matrices
• Invariant theory for the commuting scheme of symplectic Lie algebras
• Solution Formulas for Differential Sylvester and Lyapunov Equations
• Lie algebras of vector fields on smooth affine varieties
• Springer Isomorphisms In Characteristic p
• Nilpotent centralizers and Springer isomorphisms
• The invariant theory of n × n matrices

Related

• Module schemes in invariant theory
• Reductive and unipotent actions of affine groups
• Computing invariants of algebraic group actions in arbitrary characteristic
• Geometric aspects of representation theory for DG algebras: answering\n a question of Vasconcelos
• Filtrations of D-modules along semi-invariant functions