[Paper Review] On categories of equivariant D-modules

András C. Lőrincz, Uli Walther|arXiv (Cornell University)|Jun 6, 2018

Algebraic structures and combinatorial models49 references3 citations

TL;DR

This paper studies G-equivariant D-modules on algebraic varieties X with finitely many G-orbits, showing that their category is equivalent to the category of finite-dimensional representations of a finite quiver with relations. For irreducible spherical G-varieties such as spaces of matrices, the quivers are typically representation-finite, and the authors explicitly describe characteristic cycles and the Pyasetskii pairing via twisted Fourier transform and moment map techniques.

ABSTRACT

Let $X$ be a variety with an action by an algebraic group $G$. In this paper we discuss various properties of $G$-equivariant $D$-modules on $X$, such as the decompositions of their global sections as representations of $G$ (when $G$ is reductive), and descriptions of the categories that they form. When $G$ acts on $X$ with finitely many orbits, the category of equivariant $D$-modules is isomorphic to the category of finite-dimensional representations of a finite quiver with relations. We describe explicitly these categories for irreducible $G$-modules $X$ that are spherical varieties, and show that in such cases the quivers are almost always representation-finite (i.e. with finitely many indecomposable representations).

Motivation & Objective

• To understand the structure of G-equivariant D-modules on varieties with finitely many G-orbits, particularly when G is reductive.
• To describe the category of strongly equivariant D-modules as equivalent to representations of a finite quiver with relations in the case of finitely many orbits.
• To analyze the global sections of simple equivariant D-modules in terms of weight multiplicities via the moment map.
• To compute characteristic cycles and the Pyasetskii pairing for simple equivariant D-modules on irreducible spherical G-varieties.
• To provide explicit quiver descriptions and representation-finiteness results for key examples such as spaces of symmetric, skew-symmetric, and m×n matrices under group actions.

Proposed method

• Use the orbit decomposition of X under G to classify simple strongly equivariant D-modules via irreducible equivariant local systems on orbits.
• Construct a finite quiver with relations whose representations classify the category of G-equivariant D-modules when G acts with finitely many orbits.
• Apply the twisted Fourier transform to relate D-modules on dual spaces and compute the Pyasetskii pairing via Fourier duality.
• Employ moment map techniques to estimate weight multiplicities in global sections of simple D-modules for linear reductive G.
• Use Bernstein–Sato polynomials and reduction methods to analyze D-module structure and characteristic varieties.
• Leverage the Riemann–Hilbert correspondence and the fact that strong equivariance is preserved under extensions in the semi-simple, simply connected case to relate characteristic varieties to conormal bundles of orbits.

Experimental results

Research questions

• RQ1 How can the category of G-equivariant D-modules be described when G acts on X with finitely many orbits?
• RQ2 What is the structure of the quiver with relations that classifies G-equivariant D-modules in the case of finitely many orbits?
• RQ3 For irreducible spherical G-varieties such as spaces of matrices, are the corresponding quivers representation-finite?
• RQ4 How do the characteristic cycles of simple equivariant D-modules relate to the conormal bundles of G-orbits?
• RQ5 What is the role of the twisted Fourier transform in computing the Pyasetskii pairing for equivariant D-modules?

Key findings

• When G acts on X with finitely many orbits, the category of G-equivariant D-modules is equivalent to the category of finite-dimensional representations of a finite quiver with relations.
• For irreducible spherical G-varieties such as the space of m×n matrices, skew-symmetric matrices, symmetric matrices, and certain tensor products, the corresponding quivers are almost always representation-finite.
• In the case of Sp_{2n} ⊗ GL_3 with n ≥ 3, the quiver has vertices (r,s) corresponding to orbits, with relations including zero 2-cycles and isolated vertices at (3,2) and (0,0).
• The twisted Fourier transform maps simple D-modules as follows: S_{3,2} ↦ S_{0,0}, S_{2,2} ↦ S_{1,0}, S_{3,0} ↦ S_{2,0}, and this determines the Pyasetskii pairing.
• The characteristic variety of each simple equivariant D-module S_{r,s} is irreducible and equals the closure of the conormal bundle to the orbit O_{r,s}, i.e., charC(S_{r,s}) = [T^*_{O_{r,s}}X].
• For linear reductive, topologically connected G and smooth X, the multiplicities of weights in global sections of simple D-modules are bounded by the moment map image, providing estimates in terms of geometric invariants.