## Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory

Last update: 26 February 2013

## Abstract

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 $G\left(r,p,n\right)$ and the corresponding cyclotomic Hecke algebras ${H}_{r,p,n},$ 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.

## Introduction

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

1. Weyl groups of types A, B, and D,
2. Iwahori-Hecke algebras of types A, B, and D,
3. the complex reflection groups $G\left(r,p,n\right),$ and
4. the cyclotomic Hecke algebras ${H}_{r,p,n},$

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 ${H}_{r,1,n}$ 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 $R$ is an algebra and $G$ is a finite group acting on $R$ by automorphisms then the representation theory of the ring ${R}^{G}$ of fixed points of the $G\text{-action}$ can be derived from the representation theory of $R$ and subgroups of $G\text{.}$ 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:

1. The Hecke algebras ${H}_{r,p,n}$ of the complex reflection groups $G\left(r,p,n\right)$ can be obtained as fixed point subalgebras of the Hecke algebra ${H}_{r,1,n}$ associated to the complex reflection group ${G}_{r,1,n}$ via $Hr,p,n= (Hr,1,n) ℤ/pℤ .$
2. The Hecke algebras ${\stackrel{\sim }{H}}_{L}$ of nonadjoint $p\text{-adic}$ groups can be obtained as fixed point subalgebras of the Hecke algebra ${\stackrel{\sim }{H}}_{P}$ associated to the corresponding adjoint $p\text{-adic}$ group, via $H∼L= (H∼P) P/L .$
3. There is a form of Clifford theory (to our knowledge new) that allows one to completely determine the representation theory of a fixed point subalgebra ${R}^{G}$ in terms of the representation theory of the algebra $R$ and the group $G\text{.}$

We use this method to work out the representation theory of the algebras ${H}_{r,p,n}$ in detail. Additional results include,

1. The discovery of the “right” affine braid groups ${ℬ}_{\infty ,p,n}$ and affine Hecke algebras ${H}_{\infty ,p,n}$ to associate to the cyclotomic Hecke algebras ${H}_{r,p,n}\text{.}$ The representation theory of these new groups and algebras is completely determined from the representation theory of the classical affine braid groups ${B}_{\infty ,1,n}$ and the classical affine Hecke algebras ${H}_{\infty ,1,n}$ of type $A$ as a consequence of the results in (3).

Finally,

1. We show how the classical trick (due to Cherednik) for determining the representation theory of the algebras ${H}_{r,1,n}$ from that of ${H}_{\infty ,1,n}$ arises from a map from the affine Hecke algebra of type $C$ to the finite Hecke algebra of type $C$ which corresponds to a folding of the Dynkin diagram. This explanation is new. We show that such maps from the affine Hecke algebra to the finite Hecke algebra cannot exist in general type, and we work out the details of the cases where such homomorphisms do arise from foldings.

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 $p\text{-adic}$ groups. (The Kazhdan-Lusztig result provides this classification for groups with connected center.)

## Acknowledgements

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.

## Notes and References

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).