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MR2131140 (2006d:22018) 22E50 (11F70)
Aubert, AnneMarie (FCNRSMJ); Plymen, Roger (4MANCSM)
Plancherel measure for GL(n, F) and GL(m, D): explicit formulas and Bernstein
decomposition. (English summary)
J. Number Theory 112 (2005), no. 1, 2666.
Let F be a nonArchimedean local field. Let denote the Plancherel measure for GL(n, F). In this
paper, the authors first prove an explicit formula for the HarishChandra µfunction that appears
in the HarishChandra Plancherel theorem. Thus the value of µ at a discrete series representation
of a Levi subgroup depends only on the fundamental invariants yielded by the inducing data of a
string of irreducible supercuspidal representations of smaller general linear groups. The invariants
are the residue cardinality of F, the sizes of the smaller general linear groups, the exponents,
the torsion numbers, the formal degrees and the conductors of these supercuspidals. In the single
exponent case, µ is expressed as a function on a compact torus where the coordinates can be
thought of as generalized Satake parameters. The proofs of the formulas make use of F. Shahidi's
work on the Langlands conjecture on the Plancherel measure [Ann. of Math. (2) 132 (1990), no. 2,
273330; MR1070599 (91m:11095)] and HarishChandra's product formula for µ. The explicit
