Etingof conjecture for quantized quiver varieties

Abstract

We compute the number of finite dimensional irreducible modules for the algebras quantizing Nakajima quiver varieties. We get a lower bound for all quivers and vectors of framing and provide an exact count in the case when the quiver is of finite type or is of affine type and the framing is the coordinate vector at the extending vertex. The latter case precisely covers Etingof's conjecture on the number of finite dimensional irreducible representations for Symplectic reflection algebras associated to wreath-product groups. We use several different techniques, the two principal ones are categorical Kac-Moody actions and wall-crossing functors. We finish the paper outlining some future directions of research.

v4, 95 pages, major changes, accepted version

References in corpus (6)

• 2-Kac-Moody algebras
• On the structure of the category O for W-algebras
• Proof of Varagnolo-Vasserot conjecture on cyclotomic categories O
• Rigidity in higher representation theory
• Poisson deformations and birational geometry
• Etingof conjecture for quantized quiver varieties II: affine quivers

Cited by in corpus (18)

• On categories O for quantized symplectic resolutions
• Koszul duality between Higgs and Coulomb categories $\mathcal{O}$
• Supports of simple modules in cyclotomic Cherednik categories O
• Etingof conjecture for quantized quiver varieties II: affine quivers
• Harish-Chandra bimodules over rational Cherednik algebras
• On modular categories O for quantized symplectic resolutions
• Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties
• On Localization for Quantum Hamiltonian Reductions in Arbitrary Characteristic
• Affine Springer Fibers, Procesi bundles, and Cherednik algebras
• Resolutions with conical slices and descent for the Brauer group classes of certain central reductions of differential operators in characteristic $p$
• Associated graded of Hodge modules and categorical sl_2 actions
• Real variations of stability conditions for noncommutative symplectic resolutions
• Abelian localization for cyclotomic Cherednik algebras
• Harish-Chandra bimodules for type A rational Cherednik algebras
• Namikawa-Weyl groups of affinizations of smooth Nakajima quiver varieties
• Localization theorems for quantized symplectic resolutions
• On categories $\mathcal{O}$ of quiver varieties overlying the bouquet graphs
• The image of the KZ functor for Cherednik algebras of varieties with finite group actions