On the spectral decomposition of affine Hecke algebras

Abstract: An affine Hecke algebra H contains a large abelian subalgebra A spanned by the Bernstein-Zelevinski-Lusztig basis elements theta x, where x runs over (an extension of) the root lattice. The centre Z of H is the subalgebra of Weyl group invariant elements in A. The natural trace ('evaluation at the identity') of the affine Hecke algebra can be written as integral of a certain rational n-form (with values in the linear dual of H) over a cycle in the algebraic torus T = Spec(A). This cycle is homologous to a union of 'local cycles'. We show that this gives rise to a decomposition of the trace as an integral of positive local traces against an explicit probability measure on the spectrum W0 \ T of Z. From this result we derive the Plancherel formula of the affine Hecke algebra.
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AMS 2000 Mathematics subject classification: Primary 20C08; 22D25; 22E35; 43A32