The central support of the Plancherel measure of an affine Hecke algebra

We give conceptual proofs of certain basic properties of the arrangement of shifted root hyperplanes associated to a root system $R_0$ and a $W_0=W(R_0)$-invariant real valued parameter function on $R_0$. The method is based on the role of this shifted root hyperplane arrangement for the harmonic analysis of affine Hecke algebras. In addition this yields a conceptual proof of the description of the central support of the Plancherel measure of an affine Hecke algebra given by the author earlier.