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Langlands’ Conjecture on Plancherel Measures for p-Adic Groups

Chapter
Part of the Progress in Mathematics book series (PM, volume 101)

Abstract

One of the major achievements of Harish-Chandra was a derivation of the Plancherel formula for real and p-adic groups [9,10]. To have an explicit formula, one will have to compute the measures appearing in the formula; the so called Plancherel measures and formal degrees [12]. (For reasons stemming from L-indistinguishability, we would like to distinguish between the formal degrees for discrete series and the Plancherel measures for non-discrete tempered representations, cf. Proposition 9.3 of [29].) While for real groups the Plancherel measures are completely understood [1, 9, 22], until recently little was known in any generality for p-adic groups [29] (except for their rationality and general form due to Silberger [39]). On the other hand any systematic study of the non-discrete tempered spectrum of a p-adic group would very likely have to follow the path of Knapp and Stein [20, 21] and their theory of 72-groups. Since the basic reducibility theorems for p-adic groups are available [40, 41], it is the knowledge of Plancherel measures which would be necessary to determine the R-groups. This is particularly evident from the important and the fundamental work of Keys [16, 17, 18] and the work of the author [29, 30, 31].

Keywords

Parabolic Subgroup Discrete Series Adjoint Action Complementary Series Levi Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

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