[Paper Review] Hamiltonian reduction and nearby cycles for Mirabolic D-modules

Gwyn Bellamy, Victor Ginzburg|ENLIGHTEN (Jurnal Bimbingan dan Konseling Islam)|Jul 7, 2012

Algebraic structures and combinatorial models22 references3 citations

TL;DR

This paper establishes a deep connection between mirabolic D-modules on $SL_n(bC) \times \bbC^n$ and representations of the trigonometric Cherednik algebra via Hamiltonian reduction. It proves that a mirabolic D-module is killed by this reduction if and only if its characteristic variety lies in the unstable locus, and shows that Hamiltonian reduction intertwines specialization functors on D-modules with those on Cherednik algebra representations, enabling the application of Hodge-theoretic purity results.

ABSTRACT

We study holonomic D-modules on SL_n(C)xC^n, called mirabolic modules, analogous to Lusztig's character sheaves. We describe the supports of simple mirabolic modules. We show that a mirabolic module is killed by the functor of Hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric Cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus. We introduce an analogue of the Verdier specialization functor for representations of Cherednik algebras which agrees, on category O, with the restriction functor of Bezrukavnikov and Etingof. In type A, we also consider a Verdier specialization functor on mirabolic D-modules. We show that Hamiltonian reduction intertwines specialization functors on mirabolic D-modules with the corresponding functors on representations of the Cherednik algebra. This allows us to apply known purity results for nearby cycles in the setting considered by Bezrukavnikov and Etingof.

Motivation & Objective

• To establish an equivalence between two definitions of mirabolic D-modules: one geometric (via characteristic varieties), the other algebraic (via enveloping algebra actions).
• To characterize when a mirabolic D-module is killed by Hamiltonian reduction, relating this to the unstable locus in the characteristic variety.
• To define and study a Verdier specialization functor for Cherednik algebra representations and show its agreement with the Bezrukavnikov-Etingof restriction on category $\mathcal{O}$.
• To prove that Hamiltonian reduction intertwines specialization functors on mirabolic D-modules with those on Cherednik algebra representations.
• To apply Hodge-theoretic purity results for nearby cycles in the context of Cherednik algebra representations via this interplay.

Proposed method

• Uses the Riemann-Hilbert correspondence to relate mirabolic D-modules to perverse sheaves, analogous to Lusztig’s character sheaves.
• Defines mirabolic D-modules as regular holonomic D-modules with characteristic varieties contained in the Lagrangian subvariety $\mathbb{M}_{\text{nil}}(SL)$.
• Applies Hamiltonian reduction from the category of mirabolic D-modules to representations of the trigonometric Cherednik algebra via a $G = GL(V)$-action on $\mathfrak{X} = SL_n \times \bbC^n$.
• Introduces a specialization functor for Cherednik algebra representations that agrees with the restriction functor of Bezrukavnikov and Etingof on category $\mathcal{O}$.
• Constructs a corresponding specialization functor for mirabolic D-modules and proves its compatibility with Hamiltonian reduction.
• Employs radial parts maps and shift functors to relate differential operators on $SL_n$ to Cherednik algebra actions, using $\delta$-twisted differential operators and $\kappa$-dependent radial parts.

Experimental results

Research questions

• RQ1 When is a mirabolic D-module annihilated by the Hamiltonian reduction functor?
• RQ2 How do specialization functors on mirabolic D-modules relate to those on Cherednik algebra representations?
• RQ3 Can Hodge-theoretic purity results for nearby cycles be extended to the setting of Cherednik algebra representations via this framework?
• RQ4 What is the precise relationship between the geometric and algebraic definitions of mirabolic D-modules?
• RQ5 How does the radial parts map relate to the Harish-Chandra homomorphism in the context of $\kappa$-deformations?

Key findings

• A mirabolic D-module is killed by Hamiltonian reduction if and only if its characteristic variety is contained in the unstable locus of $\mathbb{M}_{\text{nil}}(SL)$.
• The specialization functor on Cherednik algebra representations agrees with the restriction functor of Bezrukavnikov and Etingof on category $\mathcal{O}$.
• Hamiltonian reduction intertwines the specialization functors on mirabolic D-modules and on Cherednik algebra representations.
• The radial parts map $\mathrm{Rad}_w(\Omega_{\mathfrak{sl}})$ is computed explicitly, yielding a formula involving $\Omega_\mathfrak{t}$, $\partial_\alpha$, and $w(w+2)(\rho,\rho)$.
• The radial parts map $\mathfrak{R}$ agrees with the Harish-Chandra homomorphism on $\mathfrak{Z}$ for all $\kappa$, not just positive integers, via polynomial continuation.
• The equivalence between the geometric and algebraic definitions of mirabolic D-modules is established, resolving a key foundational question.