Geometric representations of graded and rational Cherednik algebras

2014/07/21 by Alexei Oblomkov, Oblomkov, Alexei, Zhiwei Yun +1 · 5 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1407.5685

openalex publication_date 2014/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide geometric constructions of modules over the graded Cherednik algebra \mathfrakH^gr_ν and the rational Cherednik algebra \mathfrakH^rat_ν attached to a simple algebraic group \mathbbG together with a pinned automorphism θ. These modules are realized on the cohomology of affine Springer fibers (of finite type) that admit ℂ^*-actions. In the rational Cherednik algebra case, the standard grading on these modules is derived from the perverse filtration on the cohomology of affine Springer fibers coming from its global analog: Hitchin fibers. When θ is trivial, we show that our construction gives the irreducible finite-dimensional spherical modules \mathfrakL_ν(triv) of \mathfrakH^gr_ν and of \mathfrakH^rat_ν. We give a formula for the dimension of \mathfrakL_ν(triv) and give a geometric interpretation of its Frobenius algebra structure. The rank two cases are studied in further details.

Citations

• Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves
• Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0
• Supports of irreducible spherical representations of rational Cherednik algebras of finite Coxeter groups
• Finite dimensional representations of rational Cherednik algebras
• The spherical part of the local and global Springer actions
• Proof of Varagnolo-Vasserot conjecture on cyclotomic categories O

Cited by

• Cohomology of compactified Jacobians for locally planar integral curves
• Bigraded Polynomials for the Cohomology of Wild Hitchin Systems
• Character sheaves on loop Lie algebras: polar partition
• An explicit local geometric Langlands for supercuspidal representations: the toral case
• Intersections of blocks of cyclotomic Hecke algebras

Related

• Rational Cherednik algebras and Hilbert schemes II: representations and sheaves
• Rational Cherednik algebras and Hilbert schemes
• Finite dimensional representations of rational Cherednik algebras
• Representations of rational Cherednik algebras
• Parabolic degeneration of rational Cherednik algebras