Plancherel measure for GL(n,F) and GL(m,D): explicit formulas and Bernstein decomposition

Abstract

Let F be a nonarchimedean local field, let D be a division algebra over F, let GL(n) = GL(n,F). Let \nu denote Plancherel measure for GL(n). Each component \Omega in the Bernstein variety \Omega(GL(n)) has several numerical invariants attached to it. We provide explicit formulas for the Bernstein component \nu_{\Omega} of Plancherel measure in terms of these invariants. We also prove some new formal degree formulas, a transfer-of-measure formula for GL(n), and a transfer-of-measure formula from GL(n,F) to GL(m,D).