This paper is concerned with some important properties of r-polynomials in Hecke algebras with unequal parameters . In chapter one, we briefly introduce some conceptions in Coxeter group W and some notations in Hecke algebra H. In chapter two, we give some important properties of some r-polynomials which satisfy ry,w = ∏l(w)−l(y) i=1 (vsi − v−1 si ) when y ≤ w in W and w = s1s2 · · · sq is any ...

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 (`), where P (z) is the relevant modular generating function. We obtain such formulas using Euler’s Pentagonal Number Theorem and the denominator formula for the Mons...

We study the local factor at p of the semi-simple zeta function of a Shimura variety of Drinfeld type for a level structure given at p by the pro-unipotent radical of an Iwahori subgroup. Our method is an adaptation to this case of the Langlands-Kottwitz counting method. We explicitly determine the corresponding test functions in suitable Hecke algebras, and show their centrality by determining...

In the representation theory of Iwahori–Hecke algebras of type A (and in particular for representations of symmetric groups) the notion of the weight of a block, introduced by James, plays a central rôle. Richards determined the decomposition numbers for blocks of weight 2, and here the same task is undertaken for weight two blocks of Iwahori–Hecke algebras of type B, using the author’s own def...

For a prime p we describe an algorithm for computing the Brandt matrices giving the action of the Hecke operators on the space V of modular forms of weight 2 and level p2. For p ≡ 3 mod 4 we define a special Hecke stable subspace V0 of V which contains the space of modular forms with CM by the ring of integers of Q( √−p) and we describe the calculation of the corresponding Brandt matrices.

The goal of these lectures is twofold: first I would like to introduce the basic concepts of the theory, like the Siegel modular group and its action on the Siegel upper half-space, reduction theory, examples of Siegel modular forms, Hecke operators and L-functions. Secondly, following up, I would like to discuss two rather recent research topics, namely the ”Ikeda lifting” and sign changes of ...

James’s Conjecture suggests that in certain cases, the decomposition numbers for the Iwahori– Hecke algebra of the symmetric group over a field of prime characteristic (and in particular, the decomposition numbers for the symmetric group itself) coincide with the decomposition numbers for a corresponding Iwahori–Hecke algebra defined over C, and hence can be computed using the LLT algorithm. We...

A general method for computing irreducible representations of Weyl groups and Iwahori–Hecke algebras was introduced by the first author in [10]. In that paper the representations of the algebras of typesAn,Bn,Dn andG2 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–Hecke algebra of ...

Let F be a real quadratic field with ring of integers Ø and with class number 1. Let Γ be a congruence subgroup of GL2(Ø). We describe a technique to compute the action of the Hecke operators on the cohomology H(Γ ;C). For F real quadratic this cohomology group contains the cuspidal cohomology corresponding to cuspidal Hilbert modular forms of parallel weight 2. Hence this technique gives a way...