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Let $H_{n}$ be the affine Hecke algebra with parameter q, where q is not root of unity.
The classification of irreducible finite dimensional representations has been given by Kazhdan-Lusztig in terms of geometric data.. I wonder in the case of type A, is there any classification in terms of concrete combinatoric?

The reason I ask this question is that, for degenerate affine Hecke algebra in type A, all finite dimensional represetnations can be parametrized by two sequences of complex numbers, $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_n) $, $\mu=(\mu_1,\mu_2,\cdots,\mu_n) $, such that , $\lambda$ abnd $\mu$ satifisy some simple combinatorics conditions.

It is well known that affine hecke algebra is closely related to degenerate affine Hecke algebra.

I hope and expect there should be similar even the same classifciation in the case type A.

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  • $\begingroup$ Yes! The meta-principle is that the difference conditions in the case of degenerate affine Hecke algebra are replaced by the quotient conditions, e.g. $\lambda_i-\lambda_{i+1}=1$ becomes $\lambda_i/\lambda_{i+1}=q.$ Extensive work has been performed on this classification since Kazhdan and Lusztig's original papers. $\endgroup$
    – Victor Protsak
    Commented Jun 23, 2014 at 16:25
  • $\begingroup$ @VictorProtsak: Thank you. Could you give some related reference? $\endgroup$
    – JJH
    Commented Jun 23, 2014 at 16:55

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This is done in Orellana-Ram, `Affine braids, Markov Traces and the category O'. The answer is essentially the same as for the degenerate affine Hecke algebra.

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