Affine braids , Markov traces and the category

This paper provides a unified approach to results on representations of affine Hecke algebras, cyclotomic Hecke algebras, affine BMW algebras, cyclotomic BMW algebras, Markov traces, Jacobi-Trudi type identities, dual pairs [Ze], and link invariants [Tu2]. The key observation in the genesis of this paper was that the technical tools used to obtain the results in Orellana [Or] and Suzuki [Su], two a priori unrelated papers, are really the same. Here we develop this method and explain how to apply it to obtain results similar to those in [Or] and [Su] in more general settings. Some specific new results which are obtained are the following: (a) A generalization of the results on Markov traces obtained by Orellana [Or] to centralizer algebras coming from quantum groups of all Lie types. (b) A generalization of the results of Suzuki [Su] to show that Kazhdan-Lusztig polynomials of all finite Weyl groups occur as decomposition numbers in the representation theory of affine braid groups of type A, (c) A generalization of the functors used by Zelevinsky [Ze] to representations of affine braid groups of type A, (d) We define the affine BMW algebra (Birman-Murakami-Wenzl) and show that it has a representation theory analogous to that of affine Hecke algebras. In particular there are “standard modules” for these algebras which have composition series where multiplicities of the factors are given by Kazhdan-Lusztig polynomials for Weyl groups of types A,B,and C. (e) We generalize the results of Leduc and Ram [LR] to affine centralizer algebras.