Last update: 26 February 2013
We show that the Young tableaux theory and constructions of the irreducible representations of the Weyl groups of type A, B and D, Iwahori-Hecke algebras of types A, B, and D, the complex reflection groups and the corresponding cyclotomic Hecke algebras can be obtained, in all cases, from the affine Hecke algebra of type A. The Young tableaux theory was extended to affine Hecke algebras (of general Lie type) in recent work of A. Ram. We also show how (in general Lie type) the representations of general affine Hecke algebras can be constructed from the representations of simply connected affine Hecke algebras by using an extended form of Clifford theory. This extension of Clifford theory is given in the Appendix.
Recent work of A. Ram [Ram2003,Ram0401326] gives a straightforward combinatorial construction of the simple calibrated modules of affine Hecke algebras (of general Lie type as well as type A). The first aim of this paper is to show that Young’s seminormal construction and all of its previously known generalizations are special cases of the construction in [Ram0401326]. In particular, the representation theory of
can be derived entirely from the representation theory of affine Hecke algebras of type A. Furthermore, the relationship between the affine Hecke algebra and the objects in (a)-(d) always produces a natural set of Jucys-Murphy type elements and can be used to prove the standard Jucys-Murphy type theorems. In particular, we are able to use Bernstein’s results about the center of the affine Hecke algebra to show that, in the semisimple case, the center of the cyclotomic Hecke algebra is the set of symmetric polynomials in the Jucys-Murphy elements.
A. Young’s seminormal construction of the irreducible representations of the symmetric group dates from 1931 [You1929]. Young himself generalized his tableaux to treat the representation theory of Weyl groups of types B and D [You1931]. In 1974 P.N. Hoefsmit [Hoe1974] generalized the seminormal construction to Iwahori-Hecke algebras of types A, B, and D. Hoefsmit’s work has never been published and, in 1985, Dipper and James [DJa1987], Theorem 4.9 and H. Wenzl [Wen1988] independently treated the seminormal construction for irreducible representations for Iwahori-Hecke algebras of type A. In 1994 Ariki and Koike [AKo1994] introduced (some of) the cyclotomic Hecke algebras and generalized Hoefsmit’s construction to these algebras. The construction was generalized to a larger class of cyclotomic Hecke algebras in [Ari1995]. For a summary of this work see [Ram1997] and [HRa1998].
General affine Hecke algebras are naturally associated to a reductive algebraic group and the size of the commutative part of the affine Hecke algebra depends on the structure of the corresponding algebraic group (simply connected, adjoint, etc.). The second aim of this paper is to show that it is sufficient to understand the representation theory of the affine Hecke algebra in the simply connected case. We describe explicitly how the representation theory of the other cases is derived from the simply connected case.
The machine which allows us to accomplish this reduction is a form of Clifford theory. Precisely, if is an algebra and is a finite group acting on by automorphisms then the representation theory of the ring of fixed points of the can be derived from the representation theory of and subgroups of This is an extension of the approach to Clifford theory given by Macdonald [Mac1980].
Let us state precisely what is new in this paper. The main result has three parts:
We use this method to work out the representation theory of the algebras in detail. Additional results include,
In a recent paper Reeder [Ree2002] has used our results to prove the Langlands classification of irreducible representations for general affine Hecke algebras. (Previously, this was known only in the simply connected case, see Kazhdan and Lusztig [KLu0862716].) This also provides a classification of the irreducible constituents of unramified principal series representations of general split reductive groups. (The Kazhdan-Lusztig result provides this classification for groups with connected center.)
A. Ram thanks P. Deligne for his encouragement, stimulating questions and helpful comments on the results in this paper. We thank D. Passman for providing some useful references about Clifford theory. We are grateful for the generous support of this research by the National Science Foundation, the National Security Agency and the Australian Research Council.
This is an excerpt of the paper entitled Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory authored by Arun Ram and Jacqui Ramagge. It was dedicated to Professor C.S. Seshadri on the occasion of his 70th birthday.
Research partially supported by the Natioanl Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).