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We study holonomic D -modules on SL n ( C ) × C n , called mirabolic modules , analogous to Lusztig's character sheaves. We describe the supports of simple mirabolic modules. We show that a mirabolic module is killed by the functor of Hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric Cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus. We introduce an analogue of Verdier's specialization functor for representations of Cherednik algebras which agrees, on category O , with the restriction functor of Bezrukavnikov and Etingof. In type A , we also consider a Verdier specialization functor on mirabolic D -modules. We show that Hamiltonian reduction intertwines specialization functors on mirabolic D -modules with the corresponding functors on representations of the Cherednik algebra. This allows us to apply known Hodge-theoretic purity results for nearby cycles in the setting considered by Bezrukavnikov and Etingof.

The theory of congruences for the partition function p ( n ) depends heavily on the properties of half-integral weight Hecke operators. The subject has been complicated by the absence of closed formulas for the Hecke images P ( z ) | T ( l 2 ) , where P ( z ) is the relevant modular generating function. We obtain such formulas using Eulerʼs Pentagonal Number Theorem and the denominator formula for the Monster Lie algebra. As a corollary, we obtain congruences for certain powers of Ramanujanʼs Delta-function.

In recent joint work with Wang, we have constructed graded Specht modules for cyclotomic Hecke algebras. In this article, we prove a graded version of the Lascoux-Leclerc-Thibon conjecture, describing the decomposition numbers of graded Specht modules over a field of characteristic zero., Comment: 57 pages; final version

We present a geometric description for tempered modules of the affine Hecke algebra of type C n with arbitrary (non-root of unity) unequal parameters, using the exotic Deligne–Langlands correspondence (Kato (2009) [18] ). Our description has several applications to the structure of the tempered modules. In particular, we provide a geometric and a combinatorial classification of discrete series which contain the sgn representation of the Weyl group, equivalently, via the Iwahori–Matsumoto involution, of spherical cuspidal modules. This latter combinatorial classification was expected from Heckman and Opdam (1997) [15] , and determines the L 2 -solutions for the Lieb–McGuire system.