Specialisations and Decomposition Maps

Abstract

In this chapter, we study non semisimple Iwahori-Hecke algebras in the spirit of Brauer’s classical modular representation theory. Using Lusztig’s a-function, we define the key concept of “canonical basic set”. This concept gives a theoritical way to classify the simple modules of Iwahori-Hecke algebras at roots of unity. It is in particular independent of the notion of cellular structure. We develop a general strategy to determine explicitly the canonical basic sets for Iwahori-Hecke algebras of classical types. A model case is given by the symmetric group. In another direction, we present a factorisation result for decomposition matrices and present a general formulation of James’ conjecture.