Skip to main content
SpringerLink
Log in

Stepwise square integrable representations of nilpotent Lie groups

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We study the conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable (relative discrete series) unitary representations, that fit together to form a filtration by normal subgroups. Then we use that filtration to construct a class of “stepwise square integrable” representations on which Plancherel measure is concentrated. Further, we work out the character formulae for those stepwise square integrable representations, and we give an explicit Plancherel formula. Next, we use some structure theory to check that all these constructions and results apply to nilradicals of minimal parabolic subgroups of real reductive Lie groups. Finally, we develop multiplicity formulae for compact quotients \(N/\varGamma \) where \(\varGamma \) respects the filtration.

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. Auslander, L. et al.: Flows on Homogeneous Spaces. Ann. Math. Stud. 53 (1963)

  2. Barberis, M.L.: Abelian hypercomplex structures on central extensions of H-type Lie algebras. J. Pure Appl. Algebra 158, 1523 (2001)

    MathSciNet  Google Scholar 

  3. Barberis, M.L., Dotti, I.: Abelian complex structures on solvable Lie algebras. J. Lie Theory 14, 25–34 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Barberis, M.L., Dotti, I.: Private communication

  5. Kaplan, A.: Riemannian nilmanifolds attached to Clifford modules. Geom. Dedicata 11, 127–136 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kirillov, A.A.: Unitary representations of nilpotent Lie groups, Uspekhi Math. Nauk 17 (1962), 57–110 (English. Russian Math. Surveys 17, 53–104 (1962))

  7. Moore, C.C.: Decomposition of unitary representations defined by discrete subgroups of nilpotent groups. Ann. Math. 82, 146–182 (1965)

    Article  MATH  Google Scholar 

  8. Moore, C.C., Wolf, J.A.: Square integrable representations of nilpotent groups. Trans. Am. Math. Soc. 185, 445–462 (1973)

    Article  MathSciNet  Google Scholar 

  9. Pukánszky, L.: On characters and the Plancherel formula of nilpotent groups. J. Funct. Anal. 1, 255–280 (1967)

    Article  MATH  Google Scholar 

  10. Raghunathan, M.S.: Discrete Subgroups of Lie Groups. Ergebnisse der Mathematik und ihrer Grenzgebeite, vol. 68 (1972)

  11. Wolf, J.A.: Classification and Fourier Inversion for Parabolic Subgroups with Square Integrable Nilradical, vol. 225. Memoirs of the American Mathematical Society (1979)

  12. Wolf, J.A.: Harmonic Analysis on Commutative Spaces. Mathematical Surveys and Monographs, vol. 142. American Mathematical Society (2007)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Joseph A. Wolf

Authors
  1. Joseph A. Wolf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joseph A. Wolf.

Additional information

Research partially supported by the Simons Foundation.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wolf, J.A. Stepwise square integrable representations of nilpotent Lie groups. Math. Ann. 357, 895–914 (2013). https://doi.org/10.1007/s00208-013-0925-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-013-0925-2

Keywords

Search

Navigation