Generalised Springer correspondence for Z/m-graded Lie algebras

2018/06/28 by Liu, Wille
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1806.10791

Abstract

Let G be a simple simply connected complex algebraic group and let \mathfrakg_* be a Z/m-grading on its Lie algebra \mathfrakg. In a recent series of articles, G. Lusztig and Z. Yun, studied the classification of simple G₀-equivariant perverse sheaves on the nilpotent cone of \mathfrakg_i for i∈ Z/m, where G₀ is the exponentiation of the degree zero piece \mathfrakg₀. They proved a decomposition of the equivariant derived category of ℓ-adic sheaves on the nilpotent cone of \mathfrakg_i into blocks, each generated by a certain cuspidal local system via \itshape spiral inductions. We prove a conjecture of them, which predicts the bijectivity of a map from 1) the set of simple perverse sheaves in a fixed block to 2) the set of simple modules of a block of a (trigonometric) degenerate double affine Hecke algebra (dDAHA). This is a dDAHA analogue of the Deligne--Langlands correspondence for affine Hecke algebras proven by Kazhdan--Lusztig. Our results generalise a previous work of E. Vasserot, where the perverse sheaves in the principal block were considered.

Related

• ℤ/mℤ-graded Lie algebras and perverse sheaves, III: graded double affine Hecke algebra
• Z/m-graded Lie algebras and perverse sheaves, I
• Formality and Lusztig's generalized Springer correspondence
• Z/m-graded Lie algebras and perverse sheaves, II
• Formality for the nilpotent cone and a derived Springer correspondence