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Homological multiplicities in representation theory of p-adic groups

  • Avraham Aizenbud
  • Eitan SayagEmail author
Article
  • 11 Downloads

Abstract

For a spherical variety 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
$$\begin{aligned} \dim {\text {Hom}}_{G}(\mathcal {S}(X),\pi ) \end{aligned}$$
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.

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. 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. 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. 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. 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. 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. 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. 7.
    Bernstein, J.N.: Second adjointness theorem for representations of p-adic groups (1987)Google Scholar
  8. 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. 9.
    Bernstein J.N., Rumelhart, K.: Lectures on \(p\)-adic groups. http://www.math.tau.ac.il/~bernstei/Publication_list/ (Unpublished)
  10. 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. 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. 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. 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. 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. 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. 16.
    Gourevitch, D., Sahi, S., Sayag, E.: Invariant functionals on Speh representations. Transform. Groups 20(4) (2015)Google Scholar
  17. 17.
    Hakim, J., Murnaghan, F.: Distinguished tame supercuspidal representations. Int. Math. Res. Pap. IMRP 2008, no. 2. See also arxiv:0709.3506
  18. 18.
    Jacquet, H., Rallis, S.: Uniqueness of linear periods. Compos. Math. 102(1) (1996)Google Scholar
  19. 19.
    Lenzing, H.: Endlich prasentierbare Moduln. Arch. Math. (Basel) 20, 262–266 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 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. 21.
    Offen, O., Sayag, E.: Uniqueness and disjointness of Klyachko models. J. Funct. Anal. 254(11) (2008). See also arxiv:0711.2884
  22. 22.
    Prasad, D.: Trilinear forms for representations of \(GL(2)\) and local \(\varepsilon \)-factors. Compos. Math. 75, 1 (1990)MathSciNetzbMATHGoogle Scholar
  23. 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. 24.
    Sakellaridis , Y., Venkatesh, A.: Periods and harmonic analysis on spherical varieties. Asterisque 396 (2018). arXiv:1203.0039
  25. 25.
    Shalika, J.A.: Multiplicity one theorem for \(GL_{n}\). Ann. Math. 100, 171–193 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sun, B., Zhu, C.-B.: Multiplicity one theorems: the Archimedean case. Ann. Math. 175(1) (2012). arXiv:0903.1413 [math.RT]
  27. 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. 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

Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesWeizmann Institute of ScienceRehovotIsrael
  2. 2.Department of MathematicsBen-Gurion University of the NegevBe’er ShevaIsrael

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