We prove the Mirković–Vilonen conjecture: the integral local intersection cohomology groups of spherical Schubert varieties on the affine Grassmannian have no$p$-torsion, as long as$p$is outside a certain small and explicitly given set of prime numbers. (Juteau has exhibited counterexamples when$p$is a bad prime.) The main idea is to convert this topological question into an algebraic question about perverse-coherent sheaves on the dual nilpotent cone using the Juteau–Mautner–Williamson theory of parity sheaves.

Let $G$ be a simple simply connected algebraic group and let $\mathfrak{g}_*$ be a $\mathbf{Z}/m$-grading on its Lie algebra $\mathfrak{g}$. G. Lusztig and Z. Yun, in a recent series of articles, 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 the procedure of {\itshape spiral inductions}. We prove a conjecture of them, stating that there is a bijection between 1) the set of simple perverse sheaves in a fixed block and 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.

We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.

In this paper, we study the relation between the cocenter \overline {{ilde {\mathcal H}}} and the representations of an affine pro-\overline {{ilde {\mathcal H}}} Hecke algebra \overline {{ilde {\mathcal H}}} . As a consequence, we obtain a new criterion on supersingular representations: a (virtual) representation of \overline {{ilde {\mathcal H}}} is supersingular if and only if its character vanishes on the non-supersingular part of the cocenter \overline {{ilde {\mathcal H}}} .

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