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MR2131140 (2006d:22018) 22E50 (11F70)
Aubert, Anne-Marie (F-CNRS-MJ); Plymen, Roger (4-MANC-SM)
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 non-Archimedean local field. Let denote the Plancherel measure for GL(n, F). In this
paper, the authors first prove an explicit formula for the Harish-Chandra -function that appears
in the Harish-Chandra Plancherel theorem. Thus the value of 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, 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 Harish-Chandra's product formula for . The explicit

  

Source: Anandavardhanan, U. K. - Department of Mathematics, Indian Institute of Technology Bombay

 

Collections: Mathematics