Let G be a connected
unramified semi-simple group over a p-adic field F. In this note, we compute a
(Macdonald-)Plancherel-type formula: ∫G(F)×G(F)f(h)ϕ(g−1hg)dg dh =∫f∨(χ)I(χ,ϕ)dμ(χ).
Here f is a spherical function, f∨ is its Satake transform, and ϕ is a smooth function
supported on the elliptic set. For this, we use the Geometrical Lemma of Bernstein
and Zelevinsky, Macdonald’s Plancherel formula, Macdonald’s formula for the
spherical function, results of Casselman on intertwining operators of the unramified
series, and a combinatorial lemma of Arthur. This derivation follows a procedure of
Waldspurger rather closely, where the case of GL(n) was worked out in detail. We
may rewrite this formula as ∫G(F)f(g−1γg)dg =∫f∨(χ)I(χ,γ)dμ(χ), for γ
elliptic regular in G(F) and f spherical. Here I(χ,γ) is a distribution on the
support of the Plancherel measure (regarded as a compact complex analytic
variety).
