Calibrated representations of affine Hecke algebras

This paper introduces the notion of calibrated representations for affine Hecke algebras and classifies and constructs all finite dimensional irreducible calibrated representations. The main results are that (1) irreducible calibrated representations are indexed by placed skew shapes, (2) the dimension of an irreducible calibrated representation is the number of standard Young tableaux corresponding to the placed skew shape and (3) each irreducible calibrated representation is constructed explicitly by formulas which describe the action of each generator of the affine Hecke algebra on a specific basis in the representation space. This construction 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. 0. Introduction The affine Hecke algebra was introduced by Iwahori and Matsumoto [IM] as a tool for studying the representations of a p-adic Lie group. In some sense, all irreducible principal series representations of the p-adic group can be determined by classifying the representations of the corresponding affine Hecke algebra. Unfortunately, it is not so easy to determine the irreducible representations of the affine Hecke algebra. Kazhdan and Lusztig [KL] (see also the important work of Ginzburg [CG]) gave a geometric classification of the irreducible representations of the affine Hecke algebra. This classification is a q-analogue of Springer’s construction of the irreducible representations of the Weyl group on the cohomology of unipotent varieties. In the q-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 special subvarieties of the flag variety. Although the classification of Kazhdan and Lusztig is an incredible tour-de-force it is difficult to obtain combinatorial information from this geometric construction. For example, it is difficult to determine the dimensions of the irreducible modules. In this paper I give a new construction of a large family of irreducible modules of the affine Hecke algebra. The basis vectors are labeled by generalized standard Young tableaux and the action of each generator on each basis element is given explicitly. This construction is a generalization of Young’s seminormal construction of the irreducible representations of the symmetric group. In order to obtain this generalization I have had to generalize the concept of standard Young tableaux to arbitrary Lie type. The modules which I construct I have termed “calibrated” modules. Specifically, a calibrated module is a module which has a basis of simultaneous eigenvectors for all the elements of a large ∗ Research supported in part by National Science Foundation grant DMS-9622985, and a Postdoctoral Fellowship at Mathematical Sciences Research Institute.