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

Published:2013-02-21
Printed: Jun 2014
• Kwangho Choiy,
Mathematics Department, Purdue University, West Lafayette, IN 47907, U.S.A.
 Format: LaTeX MathJax PDF

## Abstract

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 $\Pi_{i=1}^{r} SL_{n_i} \subseteq M \subseteq \Pi_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ 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 $SO_{4n}$ and $Sp_{4n}$ 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: Plancherel measure, inner form, local to global global argument, cuspidal automorphic representation, Jacquet-Langlands correspondence
 MSC Classifications: 22E50 - Representations of Lie and linear algebraic groups over local fields [See also 20G05] 11F70 - Representation-theoretic methods; automorphic representations over local and global fields 22E55 - Representations of Lie and linear algebraic groups over global fields and adele rings [See also 20G05] 22E35 - Analysis on $p$-adic Lie groups

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