Centers and Simple Modules for Iwahori-hecke Algebras

نویسنده

  • MEINOLF GECK
چکیده

The work of Dipper and James on Iwahori-Hecke algebras associated with the finite Weyl groups of type An has shown that these algebras behave in many ways like group algebras of finite groups. Moreover, there are “generic” features in the modular representation theory of these algebras which, at present, can only be verified in examples by explicit computations. This paper arose from an attempt to provide a conceptual explanation of these phenomena, in the general framework of the representation theory of (symmetric) algebras. We will study relations between the center of such algebras and properties of decomposition maps, and we will use this to obtain a general result about the “genericity” of the number of simple modules of Iwahori-Hecke algebras. Usually, the formalism of decomposition maps is developed for algebras over a complete discrete valuation ring. However, in our applications to Iwahori-Hecke algebras, we have to make sure that this also works over the ring of Laurent polynomials in one indeterminate over the integers. Roughly speaking, this will be achieved by using the theory of Henselian rings (see [Ray]). In Section 2, we describe such a general setting for decomposition maps of algebras over integrally closed ground rings (see Proposition 2.11). Furthermore, we extend the standard results on the “BrauerCartan triangle” to the case of orders in non-semi-simple and non-split algebras, by using enlargements of the usual Grothendieck groups. As a formal consequence of the definition, we get a factorization property of decomposition maps (see Proposition 2.12). Previously, this factorization was only established using strong additional assumptions on the realizability of representations (cf. [Ge1], (2.4), (5.3)). Let H be an algebra over a local integrally closed domain O with residue field k. Then we have a canonical map from central functions on H to central functions on kH (induced by reduction modulo the maximal ideal of O). In Proposition 3.1, generalizing a theorem of Hattori, we show that the surjectivity of this map implies that the decomposition map has finite cokernel and that the Cartan matrix of kH

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تاریخ انتشار 1994