On the spectral decomposition of affine Hecke algebras. (English) Zbl 1102.22009
Summary: An affine Hecke algebra \(\mathcal{H}\) contains a large abelian subalgebra \(\mathcal{A}\) spanned by the Bernstein-Zelevinski-Lusztig basis elements \(\theta_x\), where \(x\) runs over (an extension of) the root lattice. The centre \(\mathcal{Z}\) of \(\mathcal{H}\) is the subalgebra of Weyl group invariant elements in \(\mathcal{A}\). 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 \(\mathcal{H}\)) over a cycle in the algebraic torus \(T=\text{Spec}(\mathcal{A})\). 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 \(W_0\setminus T\) of \(\mathcal{Z}\). From this result we derive the Plancherel formula of the affine Hecke algebra.
MSC:
22E35 | Analysis on \(p\)-adic Lie groups |
20C08 | Hecke algebras and their representations |
22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
43A85 | Harmonic analysis on homogeneous spaces |
43A32 | Other transforms and operators of Fourier type |
33D80 | Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics |