Generalised Springer correspondence for Z/m-graded Lie algebras

Let $G$ be a simple simply connected complex algebraic group and let ${\mathfrak{g}}_{\ast }$ be a $\mathbf{Z}\mathrm{/}m$-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 $i\in \mathbf{Z}\mathrm{/}m$, where $G_0$ is the exponentiation of the degree zero piece ${\mathfrak{g}}_0$. They proved a decomposition of the equivariant derived category of $\mathrm{\ell }$-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.