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

Anne Marie Aubert, Roger Plymen

    Research output: Contribution to journalArticlepeer-review


    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). © 2005 Elsevier Inc. All rights reserved.
    Original languageEnglish
    Pages (from-to)26-66
    Number of pages40
    JournalJournal of Number Theory
    Issue number1
    Publication statusPublished - May 2005


    • Bernstein decomposition
    • Division algebra
    • Local harmonic analysis
    • Plancherel measure


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