Skip to main content
SpringerLink
Log in

Second adjointness for tempered admissible representations of a real group

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study second adjointness in the context of tempered admissible representations of a real reductive group. Compared to a recent result of Crisp and Higson, this generalizes from SL2 to a general group, but specializes to only considering admissible representations. We also discuss Casselman’s canonical pairing in this context, and the relation to Bernstein morphisms. Additionally, we take the opportunity to discuss some relevant functors and some of their relations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. van den Ban, Induced representations and the Langlands classification, in Representation Theory and Automorphic Forms (Edinburgh, 1996), Proceedings of Symposia in Pure Mathematics, Vol. 61, American Mathematical Society, Providence, RI, 1997, pp. 123–155.

    Chapter  Google Scholar 

  2. E. van den Ban and P. Delorme, Quelques propriétés des représentations sphériques pour les espaces symétriques réductifs, Journal fo Functional Analysis 80 (1988), 284–307.

    Article  Google Scholar 

  3. J. Bernstein, On the support of Plancherel measure, Journal of Geometry and Physics 5 (1988), 663–710.

    Article  MathSciNet  Google Scholar 

  4. W. Casselman, Canonical extensions of Harish-Chandra modules to representations of G, Canadian Journal of Mathematics 41 (1989), 385–438.

    Article  MathSciNet  Google Scholar 

  5. T.-H. Chen, D. Gaitsgory and A. Yom Din, On the Casselman-Jacquet functor, in Representations of Reductive Groups, Proceedings of Symposia in Pure Mathematics, Vol. 101, American Mathematical Society, Providence, RI, 2019, pp. 73–112.

    Chapter  Google Scholar 

  6. T. Crisp and N. Higson, A second adjoint theorem for SL(2, ℝ), in Around Langlands Correspondences, Contemporary Mathematics, Vol. 691, American Mathematical Society, Providence, RI, 2017, pp. 73–101.

    Chapter  Google Scholar 

  7. P. Delorme, F. Knop, B. Krötz and H. Schlichtkrull, Plancherel theory for real spherical spaces: Construction of the Bernstein morphisms, Journal of the American Mathematical Society, to appear, https://doi.org/10.1090/jams/971.

  8. D. Gaitsgory and A. Yom Din, An analog of the Deligne-Lusztig duality for (\(\mathfrak{g},K\))-modules, Advances in Mathematics 333 (2018), 212–265.

    Article  MathSciNet  Google Scholar 

  9. H. Hecht and W. Schmid, On the asymptotics of Harish-Chandra modules, Journal für die Reine und Angewandte Mathematik 343 (1983), 169–183.

    MathSciNet  MATH  Google Scholar 

  10. D. Milicic, Asymptotic behaviour of matrix coefficients of the discrete series, Duke Mathematical Journal 44 (1977), 59–88.

    Article  MathSciNet  Google Scholar 

  11. Y. Sakellaridis and A. Venkatesh, Periods and harmonic analysis on spherical varieties, Asterisque 396 (2017).

  12. N. Wallach, Asymptotic expansions of generalized matrix entries of representations of real reductive groups, in Lie Group Representations, I (College Park, Md., 1982/1983), Lecture Notes in Mathematics, Vol. 1024, Springer, Berlin, 1983, pp. 287–369.

    Chapter  Google Scholar 

  13. N. Wallach, Real Reductive Groups. I, Pure and Applied Mathematics, Vol. 132, Academic Press, Boston, MA, 1988.

    MATH  Google Scholar 

  14. N. Wallach, Real Reductive Groups. II, Pure and Applied Mathematics, Vol. 132-II, Academic Press, Boston, MA, 1992.

    MATH  Google Scholar 

  15. A. Yom Din, On properties of the Casselman-Jacquet functor, Ph.D. thesis, Tel Aviv University, https://arxiv.org/abs/1609.02523.

  16. A. Yom Din, Representation theory of real groups, course notes, http://math.huji.ac.il/∼ayomdin/Notes/RepRealGroups.pdf.

Download references

Author information

Authors and Affiliations

  1. Einstein Institute of Mathematics, The Hebrew University of Jeruaslem The Edmond J. Safra Campus, Givat Ram, Jerusalem, 9190401, Israel

    Alexander Yom Din

Authors
  1. Alexander Yom Din
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Yom Din.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Din, A.Y. Second adjointness for tempered admissible representations of a real group. Isr. J. Math. 244, 215–244 (2021). https://doi.org/10.1007/s11856-021-2178-1

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-021-2178-1

Search

Navigation