Abstract
This is a survey paper about affine Hecke algebras. We start from scratch and discuss some algebraic aspects of their representation theory, referring to the literature for proofs. We aim in particular at the classification of irreducible representations.Only at the end we establish a new result: a natural bijection between the set of irreducible representations of an affine Hecke algebra with parameters in R≥1, and the set of irreducible representations of the affine Weyl group underlying the algebra. This can be regarded as a generalized Springer correspondence with affine Hecke algebras.
Highlights
Affine Hecke algebras typically arise in two ways
In the final paragraph we use Theorem A to derive some properties of affine Hecke algebras H of type belong to different W (Bn)/Cn with three independent positive q-parameters
We survey some of results on affine Hecke algebras obtained with methods from complex algebraic geometry
Summary
Affine Hecke algebras typically arise in two ways. Firstly, they are deformations of the group algebra of a Coxeter system (W, S) of affine type. Becomes an instance of the generalized Springer correspondence For such geometric affine Hecke algebras the whole of Irr(H) admits a natural parametrization in terms of data that are variations on Kazhdan–Lusztig parameters, see Paragraph 5.3. Via this parametrization, ζH becomes a generalized Springer correspondence for the (extended) affine Weyl group X W. In the final paragraph we use (the proof of) Theorem A to derive some properties of affine Hecke algebras H of type Bn/Cn with three independent positive q-parameters When these parameters are generic, we provide an explicit, effective classification of Irr(H). We apologize for these and other omissions and refer the reader to the literature
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