Title:
The number of simple modules of the Hecke algebras of type $G(r,1,n)$

Authors: Susumu Ariki and Andrew Mathas

Keywords: Cyclotomic Hecke algebras, affine Hecke algebras,
          Kac-Moody algebras, crystal graphs, quantum groups.

Status: Math. Z. (to appear)

Abstract:
This paper is concerned with the problem of classifying the simple
modules of a Hecke algebra $H$ of type $G(r,1,n)$.
Using Kac-Moody algebra techniques we first show that the number of
simple $H$-modules is, in a certain sense, independent of the
choice of parameters for the Hecke algebra.  Next, by studying
Kashiwara's crystal graph, we show that the simple
$H$-modules are indexed by the set of {\it Kleshchev
multipartitions} and we give a generating function for this set.

As an application of these results we give a classification of
the number of simple modules of the affine Hecke algebras of type
{\bf A}.