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

JOURNAL OF NUMBER THEORY

卷:112

Article

Aubert, AM ; Plymen, R

关键词: Plancherel measure;    Bernstein decomposition;    local harmonic analysis;    division algebra;   

DOI  :  10.1016/j.jnt.2005.01.005

来源: Elsevier

【 摘 要 】

Let F be a nonarchimedean local field, let D be a division algebra over F, let GL(n) = GL(n, F). Let v denote Plancherel measure for GL(n). Let Q be a component in the Bernstein variety Q(GL(n)). Then Q 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 v(Omega) 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). (c) 2005 Elsevier Inc. All rights reserved.

【 授权许可】

Free