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

Choiy, Kwangho

Canadian journal of mathematics, Tome 66 (2014) no. 3, pp. 566-595

Résumé

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 ${{D}_{n}}$ .

Détail

DOI : 10.4153/CJM-2012-063-1

Mots-clés : 22E50, 11F70, 22E55, 22E35, Plancherel measure, inner form, local to global global argument, cuspidal automorphicrepresentation, Jacquet-Langlands correspondence