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

Plancherel measure for GL(n,F) and GL(m,D): explicit formulas and Bernstein decomposition
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 ? denote Plancherel measure for GL(n). Let ? be a component in the Bernstein variety ?(GL(n)). Then ? 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 ?? 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).
0022-314X
26-66
Aubert, Anne-Marie
f1ab184a-28bd-49a7-bec5-d0abcec2fed0
Plymen, Roger
76de3dd0-ddcb-4a34-98e1-257dddb731f5
Aubert, Anne-Marie
f1ab184a-28bd-49a7-bec5-d0abcec2fed0
Plymen, Roger
76de3dd0-ddcb-4a34-98e1-257dddb731f5

Aubert, Anne-Marie and Plymen, Roger (2005) Plancherel measure for GL(n,F) and GL(m,D): explicit formulas and Bernstein decomposition. Journal of Number Theory, 112 (1), 26-66. (doi:10.1016/j.jnt.2005.01.005).

Record type: Article

Abstract

Let F be a nonarchimedean local field, let D be a division algebra over F, let GL(n)=GL(n,F). Let ? denote Plancherel measure for GL(n). Let ? be a component in the Bernstein variety ?(GL(n)). Then ? 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 ?? 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).

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Published date: May 2005
Organisations: Pure Mathematics

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Local EPrints ID: 359950
URI: https://eprints.soton.ac.uk/id/eprint/359950
ISSN: 0022-314X
PURE UUID: 8e5451b6-5c79-429b-be29-a7b32c5341d2

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Date deposited: 19 Nov 2013 14:02
Last modified: 18 Jul 2017 03:16

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Author: Anne-Marie Aubert
Author: Roger Plymen

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