Acta Mathematica Sinica, English Series

, Volume 34, Issue 3, pp 341–370 | Cite as

Harmonic Analysis for Real Spherical Spaces

  • Bernhard Krötz
  • Henrik Schlichtkrull


We give an introduction to basic harmonic analysis and representation theory for homogeneous spaces Z = G/H attached to a real reductive Lie group G. A special emphasis is made to the case where Z is real spherical.


Real spherical spaces harmonic analysis homogeneous spaces representation theory 

MR(2010) Subject Classification

22F30 20G05 22E46 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



These lecture notes were originally prepared for a mini-course given at the Tsinghua Sanya International Mathematics Forum in November 2016. We are grateful to Michel Brion and Baohua Fu for providing the opportunity to present the material on that occasion.


  1. [1]
    Aizenbud, A., Gourevitch, D., Krötz, B.: Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture. Math. Z., 283, 979–992 and 993–994 (2016)Google Scholar
  2. [2]
    van den Ban, E.: Asymptotic behaviour of matrix coefficients related to reductive symmetric spaces. Nederl. Akad. Wetensch. Indag. Math., 49 (3), 225–249 (1987)MathSciNetMATHGoogle Scholar
  3. [3]
    Bernstein, J.: On the support of Plancherel measure. J. Geom. Phys., 5 (4), 663–710 (1988)MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    Bernstein, J., Krötz, B.: Smooth Fréchet globalizations of Harish–Chandra modules. Israel J. Math., 199, 45–111 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Brion, M.: Classification des espaces homogènes sphériques. Compositio Math., 63 (2), 189–208 (1987)MathSciNetMATHGoogle Scholar
  6. [6]
    Casselman, W.: Jacquet modules for real reductive groups. Proceedings of the International Congress of Mathematicians (Helsinki, 1978). pp. 557–563, Acad. Sci. Fennica, Helsinki, 1980Google Scholar
  7. [7]
    Delorme, P.: Injection de modules sphériques pour les espaces symétriques réductifs dans certaines représentations induites. Noncommutative harmonic analysis and Lie groups (Marseille-Luminy, 1985), pp. 108–143, Lecture Notes in Math. 1243, Springer, Berlin, 1987Google Scholar
  8. [8]
    Grothendieck, A.: Topological Vector Spaces, Gordon and Breach, Science Publishers, New York, 1973Google Scholar
  9. [9]
    Knop, F., Krötz, B.: Reductive group actions, arXiv: 1604.01005Google Scholar
  10. [10]
    Knop, F., Krötz, B., Pecher, T., et al.: Classification of reductive real spherical pairs I. The simple case. arXiv: 1609.00963Google Scholar
  11. [11]
    Knop, F., Krötz, B., Pecher, T., et al.: Classification of reductive real spherical pairs II. The semisimple case. arXiv: 1703.08048Google Scholar
  12. [12]
    Knop, F., Krötz, B., Schlichtkrull, H.: The local structure theorem for real spherical spaces. Compositio Mathematica, 151, 2145–2159 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Knop, F., Krötz, B., Schlichtkrull, H.: The tempered spectrum of a real spherical space, arXiv: 1509.03429Google Scholar
  14. [14]
    Knop, F., Krötz, B., Sayag, E., et al.: Simple compactifications and polar decomposition of homogeneous real spherical spaces. Selecta Math., N. S., 21 (3), 1071–1097 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Knop, F., Krötz, B., Sayag, E., et al.: Volume growth, temperedness and integrability of matrix coefficients on a real spherical space. J. Funct. Anal., 271, 12–36 (2016)MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    Kobayashi, T., Oshima, T.: Finite multiplicity theorems for induction and restriction. Advances in Math., 248, 921–944 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Krämer, M.: Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. Compositio math., 38 (2), 129–153 (1979)MathSciNetMATHGoogle Scholar
  18. [18]
    Krötz, B., Sayag, E., Schlichtkrull, H.: Vanishing at infinity on homogeneous spaces of reductive type. Compositio Math., 152 (7), 1385–1397 (2017)MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Krötz, B., Sayag, E., Schlichtkrull, H.: Decay of matrix coefficients on reductive homogeneous spaces of spherical type. Math. Z., 278, 229–244 (2014)MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    Krötz, B., Schlichtkrull, H.: Multiplicity bounds and the subrepresentation theorem for real spherical spaces. Trans. Amer. Math. Soc., 368, 2749–2762 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    Mikityuk, I. V.: On the integrability of Hamiltonian systems with homogeneous configuration spaces. Math. USSR Sbornik, 57 (2), 527–546 (1987)CrossRefMATHGoogle Scholar
  22. [22]
    Sakellaridis, Y., Venkatesh, A.: Periods and harmonic analysis on spherical varieties. Asterisque, to appearGoogle Scholar
  23. [23]
    Wallach, N.: Real Reductive Groups I, Academic Press, San Diego, 1988MATHGoogle Scholar
  24. [24]
    Wallach, N.: Real Reductive Groups II, Academic Press, San Diego, 1992MATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PaderbornPaderbornGermany
  2. 2.Department of MathematicsUniversity of CopenhagenCopenhagen ØDenmark

Personalised recommendations