We give a complete description of the category of smooth complex representations of the multiplicative group of a central simple algebra over a locally compact nonarchimedean local eld. More precisely, for each inertial class in the Bernstein spectrum, we construct a type and compute its Hecke algebra. The Hecke algebras that arise are all naturally isomorphic to products of a ne Hecke algebras...

We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q = −1. This allows us to define graded modules over the Hecke algebra at q = −1 that are ‘odd’ analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient...

We introduce and study the Koszul complex for a Hecke R-matrix. Its cohomologies, called the Berezinian, are used to de ne quantum superdeterminant for a Hecke R-matrix. Their behaviour with respect to Hecke sum of R-matrices is studied. Given a Hecke R-matrix in n-dimensional vector space, we construct a Hecke R-matrix in 2n-dimensional vector space commuting with a di erential. The notion of ...

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V ). This paper classifies all the algebras obtained by applying Drinfeld’s construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we...

Let K be a local non-archimedian field, F = K((t)) and let G be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical and Iwahori Hecke algebras for representations of the group G = G(F) and its central extension Ĝ . For instance our spherical Hecke algebra corresponds to the subgroup G(A) ⊂ G(F) where A ⊂ F is the subring OK((t)) where OK ⊂ K is the ring...

We apply the crystal basis theory for Fock spaces over quantum affine algebras to the modular representations of the cyclotomic Hecke algebras of type G(p, p, n). This yields a classification of simple modules over these cyclotomic Hecke algebras in the non-separated case, generalizing our previous work [J. Hu, J. Algebra 267 (2003), 7–20]. The separated case was completed in [J. Hu, J. Algebra...

A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for Kronecker coefficients, which are multiplicities of the decomposition of the tensor product of two Sr-irreducibles into irreducibles. Mulmuley and Sohoni attempt to solve this problem using canonical basis theory, by first constructing a nonstandard Hecke algebra Br, which, though not a Hopf al...

We examine blocks of the Ariki–Koike algebra, in an attempt to generalise the combinatorial representation theory of the Iwahori–Hecke algebra of type A. We identify a particular type of combinatorial block, which we call a core block, which may be viewed as an analogue of a simple block of the Iwahori–Hecke algebra. We give equivalent characterisations of core blocks and examine their basic co...

In studying residual automorphic representations, we need to parametrize the image of normalized local intertwining operators. This has been done by Moeglin in the case of the residual spectrum attached to the trivial character of the torus for split classical groups. In this paper, we extend her result to non-trivial characters of the torus. To do this, we use Roche’s Hecke algebra isomorphism...

We apply the crystal bases theory of Fock spaces over quantum affine algebras to the modular representations of the cyclotomic Hecke algebras of type G(p, p, n). This yields classification of simple modules over these cyclotomic Hecke algebras in the non-separated case, generalizing our previous work [J. Hu, J. Algebra 267 (2003) 7-20]. The separated case was completed in [J. Hu, J. Algebra 274...