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

If G is a semisimple Lie group and (π, $\cal H$) an irreducible unitary representation of G with square integrable matrix coefficients, then there exists a number d(π) such that ($\forall$v, v′, w, w′ ∈ $\cal H$) 1/d(π) 〈v, v′〉 〈w′, w〉 = ∫G 〈π(g).v,w〉 $\overline$ 〈π(g).v′.w′〉 dμG(g). The constant d(π) is called the formal dimension of (π, $\cal H$) and was computed by Harish-Chandra in [HC56, 66]. If now H\G is a semisimple symmetric space and (π, $\cal H$) an irreducible H-spherical unitary (π, $\cal H$) belonging to the holomorphic discrete series of H\G, then one can define a formal dimension d(π) in an analogous manner. In this paper we compute d(π) for these classes of representations.