Plancherel Decompositions and Distinctions for Sp2nbackslash Gl2n+1 and P(Sp2n)backslash Gl2n
Abstract
Let F be a non-Archimedean local field of characteristic zero. We determine the Plancherel decompositions of Sp2n(F)backslashGL2n+1(F) and P(Sp2n)(F)backslashGL2n(F), where P(Sp2n) denotes the mirabolic subgroup of Sp2n, and study smooth distinction for both spaces. We carry out the local unfolding procedure of arXiv:1203.0039, Section 9.5, constructing unitary isomorphisms with degenerate Whittaker models by successive Fourier transforms. The isomorphism for the even model is already stated there in Example 9.5.17. The resulting Plancherel formulas are direct integrals of normalized parabolic inductions from the tempered duals of GLn(F)×GLn+1(F) and GLn(F)×GLn(F), respectively, with product Whittaker-Plancherel measures. In the odd case, the parameter map agrees with the prediction of arXiv:2409.04677. Using closed parabolic orbits and local Rankin-Selberg integrals, we prove that all representations induced from these tempered data are distinguished for the corresponding model. We also prove that essentially square-integrable representations are not distinguished for the odd model, and that supercuspidal representations are not distinguished for the even model. For the even model with n=1,2, we construct distinguished square-integrable representations outside the Plancherel support. Thus, the smooth tempered distinguished spectrum can be strictly larger than the corresponding L2-spectrum. For GL5(F), however, we prove that an irreducible tempered representation is Sp4(F)-distinguished if and only if it is induced from tempered representations of GL2(F) and GL3(F).
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BibTeX
@misc{mukherjee2026plancherel,
title = {{Plancherel Decompositions and Distinctions for $\mathrmSp_2n\backslash \mathrmGl_2n+1$ and $P(\mathrmSp_2n)\backslash \mathrmGl_2n$}},
author = {Sagnik Mukherjee},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23737},
eprint = {2610.23737},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23737},
}
RIS
TY - UNPB
TI - Plancherel Decompositions and Distinctions for Sp_2nbackslash Gl_2n+1 and P(Sp_2n)backslash Gl_2n
AU - Sagnik Mukherjee
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23737
UR - https://arxiv.org/abs/2610.23737
AB - Let F be a non-Archimedean local field of characteristic zero. We determine the Plancherel decompositions of Sp_2n(F)backslashGL_2n+1(F) and P(Sp_2n)(F)backslashGL_2n(F), where P(Sp_2n) denotes the mirabolic subgroup of Sp_2n, and study smooth distinction for both spaces. We carry out the local unfolding procedure of arXiv:1203.0039, Section 9.5, constructing unitary isomorphisms with degenerate Whittaker models by successive Fourier transforms. The isomorphism for the even model is already stated there in Example 9.5.17. The resulting Plancherel formulas are direct integrals of normalized parabolic inductions from the tempered duals of GL_n(F)×GL_n+1(F) and GL_n(F)×GL_n(F), respectively, with product Whittaker-Plancherel measures. In the odd case, the parameter map agrees with the prediction of arXiv:2409.04677. Using closed parabolic orbits and local Rankin-Selberg integrals, we prove that all representations induced from these tempered data are distinguished for the corresponding model. We also prove that essentially square-integrable representations are not distinguished for the odd model, and that supercuspidal representations are not distinguished for the even model. For the even model with n=1,2, we construct distinguished square-integrable representations outside the Plancherel support. Thus, the smooth tempered distinguished spectrum can be strictly larger than the corresponding L^2-spectrum. For GL_5(F), however, we prove that an irreducible tempered representation is Sp_4(F)-distinguished if and only if it is induced from tempered representations of GL_2(F) and GL_3(F).
ER -