  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 (Zelevinsky), and link invariants (Turaev). The key observation in the
genesis of this paper was that the technical tools used to obtain the results
in Orellana and Suzuki, 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 Orellana and Suzuki 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 to centralizer algebras coming
from quantum groups of all Lie types, (b) a generalization of the results of
Suzuki 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 to
representations of affine braid groups of type A, (d) a definition of 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 multiplicites 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
on constructing representations of centralizer algebras to affine centralizer
algebras.
