We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-archimedian field. We prove a generalization of the Satake isomorphism for these algebras, relating it to integrable representations of the Langlands dual affine Kac-Moody group. In the next publication we shall use these results to define and study the notion of Hecke eigenfunction for the group Ga...

We prove a commutative algebra result which has consequences for congruences between automorphic forms modulo prime powers. If C denotes the congruence module for a fixed automorphic Hecke eigenform π0 we prove an exact relation between the p-adic valuation of the order of C and the sum of the exponents of p-power congruences between the Hecke eigenvalues of π0 and other automorphic forms. We a...

Abstract. A general method for computing irreducible representations of Weyl groups and Iwahori-Hecke algebras was introduced by the first author in [8]. In that paper the representations of the algebras of types An, Bn, Dn and G2 were computed and it is the purpose of this paper to extend these computations to F4. The main goal here is to compute irreducible representations of the Iwahori-Heck...

Using the formalism of noncommutative symmetric functions, we derive the basic theory of the peak algebra of symmetric groups and of its graded Hopf dual. Our main result is to provide a representation theoretical interpretation of the peak algebra and its graded dual as Grothendieck rings of the tower of Hecke–Clifford algebras at q 1⁄4 0: r 2004 Elsevier Inc. All rights reserved.

We classify the \fully tight" simply-laced Coxeter groups, that is, the ones whose iji-avoiding Kazhdan{Lusztig basis elements are monomials in the generators B s i. We then investigate the basis of the Temperley{Lieb algebra arising from the Kazhdan{Lusztig basis of the associated Hecke algebra, and prove that the basis coincides with the usual (monomial) basis.

We construct a model of the affine nil-Hecke algebra as a subalgebra of the Nichols-Woronowicz algebra associated to a Yetter-Drinfeld module over the affine Weyl group. We also discuss the Peterson isomorphism between the homology of the affine Grassmannian and the small quantum cohomology ring of the flag variety in terms of the braided differential calculus.

Let W be the wreath product of a symmetric group with a cyclic group of order l. The corresponding restricted rational Cherednik algebra is a finite dimensional algebra whose block structure has a combinatorial description in terms of J-hearts. We show that this description is equivalent to one given in terms of residues of multipartitions. This establishes links with Rouquier families for the ...

We introduce an algebra H consisting of difference-reflection operators and multiplication operators that can be considered as a q = 1 analogue of Sahi's double affine Hecke algebra related to the affine root system of type (C ∨ 1 , C1). We study eigenfunctions of a Dunkl-Cherednik-type operator in the algebra H, and the corresponding Fourier transforms. These eigenfunctions are non-symmetric v...

We prove that a certain genuine Hecke algebra $\mathcal{H}$ on the non-linear double cover of simple, simply-laced, simply-connected, Chevalley group $G$ over $\mathbb{Q}_{2}$ admits Bernstein presentation. This presentation has two consequences. First, component containing unramified principal series is equivalent to $\mathcal{H}$-mod. Second, isomorphic Iwahori-Hecke linear $G/Z_{2}$, where $...

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz’ external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinit...