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

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 ${\Pi }_{i=1}^{r}S{L}_{n_i}\subseteq M\subseteq {\Pi }_{i=1}^{r}G{L}_{n_i}$ 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{ć} and Savin (2000) for Siegel Levi subgroups of the groups $S{O}_{4n}$ and $S{p}_{4n}$ 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 $D_n$.

arXiv commentto appear in Canadian Journal of Mathematics (Published electronically on February 21, 2013 at http://dx.doi.org/10.4153/CJM-2012-063-1)