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

TL;DR

This paper demonstrates that Plancherel measures for unitary supercuspidal representations transfer between p-adic inner forms under a local Jacquet-Langlands correspondence, extending previous results to broader group classes.

Contribution

It proves the transfer of Plancherel measures for supercuspidal representations between groups and their inner forms, assuming invariance of these measures on certain sets.

Findings

Plancherel measures are transferred under the local Jacquet-Langlands correspondence.

The result extends to groups of types E6, E7, A_n, B_n, C_n, D_n.

Assumes invariance of Plancherel measures on specific sets.

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 \textit{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 Mui{\'c} 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 $D_n$.