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Transfer of Plancherel Measures for Unitary Supercuspidal Representations between p-adic Inner Forms

Published online by Cambridge University Press:  20 November 2018

Kwangho Choiy
Kwangho Choiy*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, U.S.A Department of Mathematics, Oklahoma State University, Stillwater, OK 74078-1058, USA. e-mail: kchoiy@math.purdue.edu, kwangho.choiy@okstate.edu
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Abstract

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Let $F$ be a $p$-adic field of characteristic 0, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $\Pi _{i=1}^{r}\,\text{S}{{\text{L}}_{ni}}\,\subseteq \,M\,\subseteq \,\Pi _{i=1}^{r}\,\text{G}{{\text{L}}_{ni}}$ for positive integers $r$ and ${{n}_{i}}$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M\left( F \right)$ is identically transferred under the local Jacquet–Langlands type correspondence between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Muić and Savin (2000) for Siegel Levi subgroups of the groups $\text{S}{{\text{O}}_{4n}}$ and $\text{S}{{\text{p}}_{4n}}$ under the local Jacquet–Langlands correspondence. It can be applied to a simply connected simple $F$-group of type ${{E}_{6}}$ or ${{E}_{7}}$, and a connected reductive $F$-group of type ${{A}_{n}},\,{{B}_{n}},\,{{C}_{n}}$ or ${{D}_{n}}$.

Keywords

22E5011F7022E5522E35Plancherel measureinner formlocal to global global argumentcuspidal automorphic representationJacquet-Langlands correspondence
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 3 , 01 June 2014 , pp. 566 - 595
Copyright
Copyright © Canadian Mathematical Society 2014

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