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Plancherel Decompositions and Distinctions for $\mathrm{Sp}_{2n}\backslash \mathrm{Gl}_{2n+1}$ and $P(\mathrm{Sp}_{2n})\backslash \mathrm{Gl}_{2n}$
Let $F$ be a non-Archimedean local field of characteristic zero. We determine the Plancherel decompositions of $\mathrm{Sp}_{2n}(F)\backslash\mathrm{GL}_{2n+1}(F)$ and $P(\mathrm{Sp}_{2n})(F)\backslash\mathrm{GL}_{2n}(F)$, where $P(\mathrm{Sp}_{2n})$ denotes the mirabolic subgroup of $\mathrm{Sp}_{2
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