## Abstract

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.

This is a preview of subscription content, access via your institution.

## References

- 1.
Blanc, P., Delorme, P.: Vecteurs distributions \(H\)-invariants de représentations induites, pour un espace symétrique réductif \(p\)-adique \(G/H\). Ann. Inst. Fourier (Grenoble)

**58**(1), 213–261 (2008) - 2.
Bernstein, I.N., Zelevinsky, A.V.: Representations of the group \(GL(n, F),\) where \(F\) is a local non-Archimedean field. Uspehi Mat. Nauk

**31**(3(189)), 5–70 (1976) - 3.
Casselman, W.: Introduction to the theory of admissible representations of \(p\)-adic reductive groups. Unpublished manuscript, draft prepared by the Séminaire Paul Sally (1995)

- 4.
Carmona, J., Delorme, P.: Constant term of Eisenstein integrals on a reductive \(p\)-adic symmetric space. Trans. Am. Math. Soc.

**366**(10), 5323–5377 (2014) - 5.
Delorme, P.: Constant term of smooth \(H_\psi \)-spherical functions on a reductive \(p\)-adic group. Trans. Am. Math. Soc.

**362**(2), 933–955 (2010) - 6.
Delorme, P., Sécherre, V.: An analogue of the Cartan decomposition for \(p\)-adic symmetric spaces of split \(p\)-adic reductive groups. Pac. J. Math.

**251**(1), 1–21 (2011) - 7.
Gurevich, M., Offen, O.: A criterion for integrability of matrix coefficients with respect to a symmetric space. J. Funct. Anal.

**270**(12), 4478–4512 (2016) - 8.
Helminck, A.G.: Tori invariant under an involutorial automorphism. I. Adv. Math.

**85**(1), 1–38 (1991) - 9.
Helminck, A.G., Helminck, G.F.: A class of parabolic \(k\)-subgroups associated with symmetric \(k\)-varieties. Trans. Am. Math. Soc.

**350**(11), 4669–4691 (1998) - 10.
Helminck, A.G., Wang, S.P.: On rationality properties of involutions of reductive groups. Adv. Math.

**99**(1), 26–96 (1993) - 11.
Kato, S.-I., Takano, K.: Subrepresentation theorem for \(p\)-adic symmetric spaces. Int. Math. Res. Not. IMRN No. 11 (2008)

- 12.
Kato, S.-I., Takano, K.: Square integrability of representations on \(p\)-adic symmetric spaces. J. Funct. Anal.

**258**(5), 1427–1451 (2010) - 13.
Lagier, N.: Terme constant de fonctions sur un espace symétrique réductif \(p\)-adique. J. Funct. Anal.

**254**(4), 1088–1145 (2008) - 14.
Offen, O.: Residual spectrum of \({{{\rm GL}}}_{2n}\) distinguished by the symplectic group. Duke Math. J.

**134**(2), 313–357 (2006) - 15.
Offen, O.: On parabolic induction associated with a \(p\)-adic symmetric space. J. Number Theory

**170**, 211–227 (2017) - 16.
Smith, J.M.: Construction of relative discrete series representations for \(p\)-adic \({{\mathbf{GL}}}_n\). Ph.D. thesis, University of Toronto (2017)

- 17.
Smith, J.M.: Local unitary periods and relative discrete series. Pac. J. Math.

**297**(1), 225–256 (2018) - 18.
Smith, J.M.: Relative discrete series representations for two quotients of \(p\)-adic \({{ GL}}_n\). Can. J. Math.

**70**(6), 1339–1372 (2018)

## Acknowledgements

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.

## Author information

### Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

## About this article

### Cite this article

Smith, J.M. The support of closed orbit relative matrix coefficients.
*manuscripta math.* **164, **95–117 (2021). https://doi.org/10.1007/s00229-020-01182-6

Received:

Accepted:

Published:

Issue Date:

### Mathematics Subject Classification

- Primary 22E50
- Secondary 22E35