# Homological multiplicities in representation theory of *p*-adic groups

Article

First Online:

- 11 Downloads

## Abstract

For a spherical variety are finite. The latter are known to be finite for symmetric varieties and for many other spherical varieties and conjectured to be finite for all spherical varieties. Furthermore, we show that the Euler–Poincaré characteristic is constant in families parabolically induced from finite length representations of a Levi subgroup \(M<G.\) In the case when \(M=G\) we compute these multiplicities more explicitly.

*X*of a reductive group*G*over a non-archimedean local field*F*, and for a smooth representation \(\pi \) of*G*we study homological multiplicities \(\dim {\text {Ext}}_{G}^{*}(\mathcal {S}(X),\pi )\). Based on Bernstein’s decomposition of the category of smooth representations of*G*, we introduce a sheaf that measures these multiplicities. We show that these multiplicities are finite whenever the usual mutliplicities$$\begin{aligned} \dim {\text {Hom}}_{G}(\mathcal {S}(X),\pi ) \end{aligned}$$

## Keywords

Branching laws Homological multiplicities Spherical spaces## Mathematics Subject Classification

22E45 20G25## Notes

### Acknowledgements

We thank Dipendra Prasad for a number of inspiring lectures on Ext braching laws. We also thank Joseph Bernstein and Dmitry Gourevitch for many useful discussions and Yotam Hendel for his careful proof reading. Finally, we thank the referee for his remarks and questions, and for his many useful suggestions that improved the readability of the paper. This project was conceived while both authors were in Bonn as part of the program “Multiplicity problems in harmonic analysis” in the Hausdorff Research Institute for Mathematics. A.A. was partially supported by ISF Grant 687/13, and a Minerva foundation Grant. E.S. was partially supported by ISF 1138/10 and ERC 291612.

## References

- 1.Aizenbud, A., Avni, N., Gourevitch, D.: Spherical pairs over close fields. Comment. Math. Helvetici
**87**(4) (2012). See also arXiv:0910.3199 [math.RT] - 2.Aizenbud, A., Gourevitch, D.: Generalized Harish–Chandra descent, Gelfand pairs and an Archimedean analog of Jacquet–Rallis’ Theorem. Duke Math. J.
**149**(3) (2009). See also arxiv:0812.5063 [math.RT] - 3.Aizenbud A., Gourevitch D.: Multiplicity one theorem for \((GL_{n+1}(\mathbb{R}),GL_{n}(\mathbb{R}))\). Sel. Math.
**15**(2) (2009). See also arXiv:0808.2729 [math.RT] - 4.Aizenbud, A., Gourevitch, D., Rallis, S., Schiffmann, G.: Multiplicity one theorems. Ann. Math.
**172**(2) (2010). See also arXiv:0709.4215 [math.RT] - 5.Aizenbud A., Gourevitch, D., Sayag, E.: \((GL_{n+1}(F),GL_n(F))\) is a Gelfand pair for any local field \(F\). Compos. Math.
**144**(2008). See postprint arXiv:0709.1273 [math.RT] - 6.Aizenbud, A., Gourevitch, D., Sayag, E.: Z-finite distributions on p-adic groups. Adv. Math.
**285**(5) (2015). See also ArXiv: 1405.2540 - 7.Bernstein, J.N.: Second adjointness theorem for representations of p-adic groups (1987)Google Scholar
- 8.Bernstein, J.N.: Le centre de Bernstein (edited by P. Deligne). In: Representations des Groupes Reductifs sur un Corps Local, Paris, pp. 1–32 (1984)Google Scholar
- 9.Bernstein J.N., Rumelhart, K.: Lectures on \(p\)-adic groups. http://www.math.tau.ac.il/~bernstei/Publication_list/ (Unpublished)
- 10.Bernstein, J., Zelevinsky, A.: Representations of \(GL(n, F)\) where \(F\) is a non-Archimedean local field. Russ. Math. Surv.
**31**(3), 1–68 (1976)CrossRefGoogle Scholar - 11.Casselman, W.: A new nonunitarity argument for \(p\)-adic representations. J. Fac. Sci. Univ. Tokyo Sect. IA Math.
**28**(3), 907–928 (1981) [MR0656064 (1984e:22018)]Google Scholar - 12.Carmona, J., Delorme, P.: Base méromorphe de vecteurs distributions H-invariants pour les séries principales généralisées déspaces symétriques réductifs: equation fonctionnelle. J. Funct. Anal.
**122**(1) (1994)Google Scholar - 13.Delorme, P.: Constant term of smooth \(H_\psi \)-spherical functions on a reductive \(p\)-adic group. Trans. Am. Math. Soc.
**362**, 933–955 (2010). http://iml.univ-mrs.fr/editions/publi2009/files/delorme_fTAMS.pdf - 14.Gan, W.T., Gross, B., Prasad, D.: Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Asterisque
**346**, (2012). arxiv:0909.2999 - 15.Gelfand, I.M., Kazhdan, D.: Representations of the group \({{\rm GL}}(n,K)\) where \(K\) is a local field, Lie groups and their representations (Proc. Summer School, Bolyai Janos Math. Soc., Budapest, 1971), pp. 95–118. Halsted, New York (1975)Google Scholar
- 16.Gourevitch, D., Sahi, S., Sayag, E.: Invariant functionals on Speh representations. Transform. Groups
**20**(4) (2015)Google Scholar - 17.Hakim, J., Murnaghan, F.: Distinguished tame supercuspidal representations. Int. Math. Res. Pap. IMRP 2008, no. 2. See also arxiv:0709.3506
- 18.Jacquet, H., Rallis, S.: Uniqueness of linear periods. Compos. Math.
**102**(1) (1996)Google Scholar - 19.Lenzing, H.: Endlich prasentierbare Moduln. Arch. Math. (Basel)
**20**, 262–266 (1969)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Offen, O., Sayag, E.: Global mixed periods and local Klyachko models for the general linear group. Int. Math. Res. Not. IMRN. (2008)/1Google Scholar
- 21.Offen, O., Sayag, E.: Uniqueness and disjointness of Klyachko models. J. Funct. Anal.
**254**(11) (2008). See also arxiv:0711.2884 - 22.Prasad, D.: Trilinear forms for representations of \(GL(2)\) and local \(\varepsilon \)-factors. Compos. Math.
**75**, 1 (1990)MathSciNetzbMATHGoogle Scholar - 23.Prasad, D.: Ext-analogues of Branching laws. Proc. Int. Cong. Math. Rio de Janeiro, Vol. 1 (2130). arXiv:1306.2729 [math.RT] (2018)
- 24.Sakellaridis , Y., Venkatesh, A.: Periods and harmonic analysis on spherical varieties. Asterisque
**396**(2018). arXiv:1203.0039 - 25.Shalika, J.A.: Multiplicity one theorem for \(GL_{n}\). Ann. Math.
**100**, 171–193 (1974)MathSciNetCrossRefzbMATHGoogle Scholar - 26.Sun, B., Zhu, C.-B.: Multiplicity one theorems: the Archimedean case. Ann. Math.
**175**(1) (2012). arXiv:0903.1413 [math.RT] - 27.van den Ban, E.P.: The principal series for a reductive symmetric space. I. \(H\)-fixed distribution vectors. Ann. Sci. École Norm. Sup.
**21**(3), 4 (1988)MathSciNetGoogle Scholar - 28.van Dijk, G.: Gelfand pairs and beyond. COE Lecture Note, 11. Math-for-Industry Lecture Note Series, p. ii+60. Kyushu University, Faculty of Mathematics, Fukuoka (2008)Google Scholar

## Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019