Representation Theory of a Hecke Algebra of G( r, p, n)

Abstract

In our previous paper [1], we introduced a notion of a Hecke algebra for a series of complex reflection groups ( Z / r Z ) ≀ K n , which is the group of n by n permutation matrices with entries running through rth roots of unity. The ordinary representation of the algebra is also studied in it. The results are direct generalizations of Hoefsmit′s [4] for Hecke algebras of classical Weyl groups. In this paper, we establish a similar theory for the remaining series of complex reflection groups G( r, p, n) ( p\\ r). p = 1 is the case we previously considered in [1]. The construction of the Hecke algebra was soon generalized to the case p = 2 by Broué and Malle [2], motivated by a much deeper background. In [2], they conjectured that these Hecke algebras over a ring of l-adic integers are candidates for endomorphism rings of certain Deligne-Lusztig modules over the ring, which arise in modular representation theory related to algebraic groups. They also defined Hecke algebras of complex reflection groups for (not all but various kinds of) exceptional and two-dimensional cases in the same context. The first section starts with the definition of the Hecke algebra and then we prove that they actually have desirable properties. The second section is for explicit construction of irreducible representations. We call them the representations of semi-normal form. The author deeply thanks Professor Broué and Professor Malle, since he was able to learn much from them, and they hinted him to the problem.