Generalised Springer correspondence for Z/m-graded Lie algebras

Abstract

Let $G$ be a simple simply connected complex algebraic group and let ${\mathfrak{g}}_{\ast }$ be a $\mathbf{Z}/m$-grading on its Lie algebra $\mathfrak{g}$. In a recent series of articles, G. Lusztig and Z. Yun, studied the classification of simple $G_0$-equivariant perverse sheaves on the nilpotent cone of ${\mathfrak{g}}_i$ for $i\in \mathbf{Z}/m$, where $G_0$ is the exponentiation of the degree zero piece ${\mathfrak{g}}_0$. They proved a decomposition of the equivariant derived category of $\ell$-adic sheaves on the nilpotent cone of ${\mathfrak{g}}_i$ 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.

arXiv comment60 pages, major modifications of organisation with respect to last version. To appear in Ann. Sci. Éc. Norm. Supér

References in corpus (1)

• Affine highest weight categories and affine quasihereditary algebras