ON THE SPECTRAL DECOMPOSITION OF AFFINE HECKE ALGEBRAS

By E. Opdam. Journal of the Institute of Mathematics of Jussieu, 2001

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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 $T=\textrm{Spec}(\mathcal{A})$. 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. AMS 2000 Mathematics subject classification: Primary 20C08; 22D25; 22E35; 43A32

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

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In Discrete Mathematics and Combinatorics

  1. M. Solleveld
  2. E. Opdam
  3. I. Gordon

In Mathematical Physics

  1. M. Solleveld
  2. E. Opdam
  3. Jean-Philippe Anker
  4. I. Gordon

In Statistics and Probability

  1. M. Solleveld

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