## Affine Hecke algebras and generalized standard Young tableaux

Last update: 19 February 2013

## Abstract

This paper introduces calibrated representations for affine Hecke algebras and classifies and constructs all finite-dimensional irreducible calibrated representations. The primary technique is to provide indexing sets for controlling the weight space structure of finite-dimensional modules for the affine Hecke algebra. Using these indexing sets we show that (1) irreducible calibrated representations are indexed by skew local regions, (2) the dimension of an irreducible calibrated representation is the number of chambers in the local region, (3) each irreducible calibrated representation is constructed explicitly by formulas which describe the action of the generators of the affine Hecke algebra on a specific basis in the representation space. The indexing sets for weight spaces are generalizations of standard Young tableaux and the construction of the irreducible calibrated affine Hecke algebra modules is a generalization of A. Young’s seminormal construction of the irreducible representations of the symmetric group. In this sense Young’s construction has been generalized to arbitrary Lie type.

## Introduction

The classical representation theory of the symmetric group, as developed by G. Frobenius and A. Young [You1977,You1931], has the following features:

1. The irreducible representations ${S}^{\lambda }$ of the symmetric group ${S}_{n}$ are indexed by partitions $\lambda$ with $n$ boxes.
2. The dimension of ${S}^{\lambda }$ is the number of standard tableaux of shape $\lambda \text{.}$
3. The ${S}_{n}\text{-module}$ has an elegant explicit construction: ${S}^{\lambda }$ is the span of a basis $\left\{{v}_{T}\right\}$ parametrized by standard tableaux $T$ and the action of each generator of ${S}_{n}$ is given by a simple formula,
$sivT= 1 c(T(i))- c(T(i+1)) vT+ ( 1+ 1 c(T(i))-c (T(i+1)) ) vsiT.$

In this paper we prove analogous results for representations of affine Hecke algebras.

1. The irreducible calibrated representations ${\stackrel{\sim }{H}}^{\left(t,J\right)}$ of the affine Hecke algebra $\stackrel{\sim }{H}$ are indexed by skew local regions $\left(t,J\right)\text{.}$
2. The dimension of ${\stackrel{\sim }{H}}^{\left(t,J\right)}$ is the number of chambers in the local region $\left(t,J\right)\text{.}$
3. The $\stackrel{\sim }{H}\text{-module}$ ${\stackrel{\sim }{H}}^{\left(t,J\right)}$ has an elegant explicit construction: ${\stackrel{\sim }{H}}^{\left(t,J\right)}$ is the span of a basis $\left\{{v}_{w} \mid w\in {ℱ}^{\left(t,J\right)}\right\}$ parametrized by chambers in the local region and the action of each generator of $\stackrel{\sim }{H}$ is given by a simple formula,
$Xλvw= q⟨λ,wγ⟩ vw,Tivw= q-q-1 1-t(Xw-1αi) + ( q-1+ q-q-1 1-t(Xw-1αi) ) vsiw.$

In fact, the classical theory of standard Young tableaux and partitions is a special case of our theory of chambers and local regions; this is proved in Sections 5 and 6 of this paper. Section 1 serves to fix notations and fundamental data in the form which will need it. The bulk of this material can be found in [6, Chapitres IV–VI] and Steinberg’s Yale Lecture Notes [Ste1968]. Two known results which are included in Section 1 are:

1. the determination of the center of the affine Hecke algebra, and
2. the Pittie–Steinberg theorem, which provides a nice basis for the affine Hecke algebra over its center.

In each case we have given an elementary proof, which, hopefully, illustrates the beautiful simplicity of these powerful results. Section 2 treats the notion of weight spaces for affine Hecke algebra representations and shows how certain combinatorially defined indexing sets ${ℱ}^{\left(t,J\right)}$ give explicit information about the weight space structure of affine Hecke algebra modules. Section 3 classifies and constructs all irreducible calibrated affine Hecke algebra modules (for any $q$ such that ${q}^{2}\ne ±1,$ including roots of unity). Section 4 gives the main results about the structure of the labeling sets ${ℱ}^{\left(t,J\right)}$ and defines a conjugation involution on them. Sections 5 and 6 show that the classical theory of standard Young tableaux is very special case of the analysis of the combinatorial structure of the sets ${ℱ}^{\left(t,J\right)}\text{.}$ Section 7 works out the generalized standard Young tableaux in the type A, root of unity case. The resulting objects are $\ell \text{-periodic}$ standard Young tableaux. Section 8 describes how the generalized standard Young tableaux look in the type C, nonroot of unity case. In this case the objects are negative rotationally symmetric standard Young tableaux. It should not be difficult to work out similar explicit tableaux in terms of fillings of boxes in the other classical types.

Let us put these results into perspective.

1. p-adic groups and affine Hecke algebras.

The affine Hecke algebra was introduced by Iwahori and Matsumoto [IMa1972] as a tool for studying the representations of a $p\text{-adic}$ Lie group. In some sense, all irreducible principal series representations of the $p\text{-adic}$ group can be determined by classifying the representations of the corresponding affine Hecke algebra. Kazhdan and Lusztig [KLu0862716] (see also [Bou1968]) gave a geometric classification of all irreducible representations of the affine Hecke algebra. This classification is a $q\text{-analogue}$ of Springer’s construction of the irreducible representations of the Weyl group on the cohomology of unipotent varieties. In the $q\text{-case,}$ K-theory takes the place of cohomology and the irreducible representations of the affine Hecke algebra are constructed as quotients of the K-theory of the Steinberg varieties. It is difficult to obtain combinatorial information from this geometric construction. So the combinatorial approach in this paper gives new information.

1. The theory of Young tableaux.

The word “Young tableau” is commonly used for three very different objects in representation theory:

(1a) partitions with $n$ boxes, which index representations of the symmetric group ${S}_{n},$
(1b) partitions with $\le$ n rows, which index the polynomial representations of $G{L}_{n}\left(ℂ\right),$
(2) standard tableaux, which label the basis elements of an irreducible representation of ${S}_{n},$
(3) column strict tableaux, which label the basis elements of an irreducible polynomial representation of $G{L}_{n}\left(ℂ\right)\text{.}$

The partitions in (1b) were generalized to all Lie types by H. Weyl in 1926, who showed that finite-dimensional irreducible representations of compact Lie groups are indexed by the dominant integral weights. There was much important work generalizing the column strict tableaux in (3) to other Lie types, for a survey of this work see [Sun1990]. The problem of generalizing the column strict tableaux in (3) to all Lie types was finally solved by the path model of Littelmann [Lit1995,Lit1995-2]. This paper provides a generalization of the partitions of (1a) and the standard tableaux of (2) which are valid for all Lie types. For important earlier work in this direction see [Mac1995, I Appendix B], Hoefsmit [Hoe1974], and Ariki and Koike [AKo1994].

This paper is a revised, expanded, and updated version of the preprints [Ram1998,Ram1998-2]. The original preprints will not be published since the results there are contained in and expanded in this paper. Those preprints will remain available at http://www.math.wisc.edu/∼ram/ preprints.html.

## Acknowledgements

During this work I have benefited from conversations with many people, including, but not limited to, G. Benkart, H. Barcelo, P. Deligne, S. Fomin, T. Halverson, F. Knop, R. Macpherson, R. Simion, L. Solomon, J. Stembridge, M. Vazirani, D.-N. Verma, and N. Wallach. I sincerely thank everyone who has let me tell them my story. Every one of these sessions was helpful to me in solidifying my understanding. I thank A. Kleshchev for thrilling energetic conversations which pushed me to work the examples out carefully for type A root of unity case and I thank J. Olsson for his wonderful gift to me of A. Young’s collected papers [You1977].

## Notes and References

This is an excerpt of the paper entitled Affine Hecke algebras and generalized standard Young tableaux written by Arun Ram in 2002, published in the Academic Press Journal of Algebra 260 (2003) 367-415. The paper was dedicated to Robert Steinberg.

Research supported in part by the National Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).