Plancherel measure for [formula omitted] and [formula omitted]: 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 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 m 1 , … , m t , exponents e 1 , … , e t , torsion numbers r 1 , … , r t , formal degrees d 1 , … , d t and conductors f 11 , … , f tt . We provide explicit formulas for the Bernstein component ν Ω 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 ) .