The support of closed orbit relative matrix coefficients


Let F be a nonarchimedean local field with odd residual characteristic and let G be the F-points of a connected reductive group defined over F. Let \(\theta \) be an F-involution of G. Let H be the subgroup of \(\theta \)-fixed points in G. Let \(\chi \) be a quasi-character of H. A smooth complex representation \((\pi ,V)\) of G is \((H,\chi )\)-distinguished if there exists a nonzero element \(\lambda \) in \({\text {Hom}}_H(\pi ,\chi )\). We generalize a construction of descended invariant linear forms on Jacquet modules first carried out independently by Kato and Takano (Int Math Res Not, IMRN no 11, 2008), and Lagier (J Funct Anal 254(4):1088–1145, 2008) to the setting of \((H,\chi )\)-distinction. We follow the methods of Kato and Takano, providing a new proof of similar results of Delorme (Trans Am Math Soc 362(2):933–955, 2010). Moreover, we give an \((H,\chi )\)-analogue of Kato and Takano’s relative version of the Jacquet Subrepresentation Theorem. In the case that \(\chi \) is unramified, \(\pi \) is parabolically induced from a \(\theta \)-stable parabolic subgroup of G, and \(\lambda \) arises via the closed orbit in \(Q\backslash G / H\), we study the (non)vanishing of the descended forms via the support of \(\lambda \)-relative matrix coefficients.

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Thank you to the anonymous referee for their careful reading and many suggestions that improved this paper. The author would also like to thank the Automorphic Representations Research Group at the University of Calgary for several helpful discussions. Thank you to Fiona Murnaghan for suggesting this project. Thank you to Omer Offen for his encouragement and many helpful suggestions. Finally, thank you to Shuichiro Takeda for informing the author of his independent work on Theorems 2.3.1 and 2.4.2.

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Correspondence to Jerrod Manford Smith.

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Smith, J.M. The support of closed orbit relative matrix coefficients. manuscripta math. 164, 95–117 (2021).

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Mathematics Subject Classification

  • Primary 22E50
  • Secondary 22E35