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

Let F be a nonarchimedean local field, let D be a division algebra over F, let GL(n)=GL(n,F). Let ν denote Plancherel measure for GL(n). Let Ω be a component in the Bernstein variety Ω(GL(n)). Then Ω yields its fundamental invariants: the cardinality q of the residue field of F, the sizes m1,…,mt, exponents e1,…,et, torsion numbers r1,…,rt, formal degrees d1,…,dt and conductors f11,…,ftt. We provide explicit formulas for the Bernstein component νΩ of Plancherel measure in terms of the fundamental invariants. We prove a transfer-of-measure formula for GL(n) and establish some new formal degree formulas. We derive, via the Jacquet–Langlands correspondence, the explicit Plancherel formula for GL(m,D).