Quantizations on Nilpotent Lie Groups and Algebras Having Flat Coadjoint Orbits

  • M. MăntoiuEmail author
  • M. Ruzhansky


For a connected simply connected nilpotent Lie group \(\textsf {G}\) with Lie algebra \({{\mathfrak {g}}}\) and unitary dual \(\widehat{{\textsf {G}}}\) one has (a) a global quantization of operator-valued symbols defined on \(\textsf {G}\times \widehat{{\textsf {G}}}\), involving the representation theory of the group, (b) a quantization of scalar-valued symbols defined on \(\textsf {G}\times {{\mathfrak {g}}}^*\), taking the group structure into account and (c) Weyl-type quantizations of all the coadjoint orbits \(\big \{\Omega _\xi \mid \xi \in \widehat{{\textsf {G}}}\big \}\). We show how these quantizations are connected, in the case when flat coadjoint orbits exist. This is done by a careful new analysis of the composition of two different types of Fourier transformations, interesting in itself. We also describe the concrete form of the operator-valued symbol quantization, by using Kirillov theory and the Euclidean version of the unitary dual and Plancherel measure. In the case of the Heisenberg group, this corresponds to the known picture, presenting the representation theoretical pseudo-differential operators in terms of families of Weyl operators depending on a parameter. For illustration, we work out a couple of examples and put into evidence some specific features of the case of Lie algebras with one-dimensional center. When \(\textsf {G}\) is also graded, we make a short presentation of the symbol classes \(S^m_{\rho ,\delta }\), transferred from \(\textsf {G}\times \widehat{{\textsf {G}}}\) to \(\textsf {G}\times {{\mathfrak {g}}}^*\) by means of the connection mentioned above.


Nilpotent group Lie algebra Coadjoint orbit Pseudo-differential operator Symbol Weyl calculus 

Mathematics Subject Classification

Primary 22E25 47G30 Secondary 22E30 


  1. 1.
    Bahouri, H., Fermanian-Kammerer, C., Gallagher, I.: Phase space analysis and pseudodifferential calculus on the Heisenberg group. Asterisque 342 (2012)Google Scholar
  2. 2.
    Beltiţă, I., Beltiţă, D.: Smooth vectors and Weyl–Pedersen calculus for representations of nilpotent Lie groups. Ann. Univ. Buchar. (Mathematical Series) 1 (LIX) 1, 17–46 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Beltiţă, I., Beltiţă, D.: Continuity of magnetic Weyl calculs. J. Funct. Anal. 260(7), 1944–1968 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Beltiţă, I., Beltiţă, D.: Boundedness for Weyl–Pedersen calculus on flat coadjoint orbits. Int. Math. Res. Not. 2015, 787–816 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brown, I.D.: Dual topology of a nilpotent Lie group. Ann. Sci. Ecole Norm. Sup. 4(6), 407–411 (1973)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Burde, D.: Characteristically nilpotent Lie algebras and symplectic structures. Forum Math. 18(5), 769–787 (2006)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Corwin, L.J., Greenleaf, F.P.: Representations of Nilpotent Lie Groups and Applications. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  8. 8.
    Christ, M., Geller, D., Glowacki, P., Polin, D.: Pseudodifferential operators on groups with dilations. Duke Math. J. 68, 31–65 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dixmier, J.: Les \(C^*\)-algèbres et leurs représentations. Cahiers scientifiques, XXIX. Gauthier-Villars Éditeurs, Paris (1969)Google Scholar
  10. 10.
    Dykema, K., Noles, J., Sukochev, F., Zanin, D.: On reduction theory and brown measure for closed unbounded operators. arXiv:1509.03362v1
  11. 11.
    Fischer, V., Ruzhansky, M.: A Pseudo-differential Calculus on Graded Nilpotent Groups, in Fourier Analysis. Trends in Mathematics, pp. 107–132. Birkhäuser, Boston (2014). arXiv:1209.2621v2 Google Scholar
  12. 12.
    Fischer, V., Ruzhansky, M.: A pseudo-differential calculus on the Heisenberg group. C. R. Acad. Sci. Paris Ser I 352, 197–204 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fischer, V., Ruzhansky, M.: Quantization on Nilpotent Lie Groups, Progress in Mathematics, vol. 314. Birkhäuser, Boston (2016)CrossRefGoogle Scholar
  14. 14.
    Fischer, V., Ruzhansky, M.: Sobolev spaces on graded groups. Ann. Inst. Fourier (Grenoble) 67, 1671–1723 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Folland, G.B.: Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol. 122. Princeton University Press, Princeton (1989)Google Scholar
  16. 16.
    Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups, Mathematical Notes, vol. 28. Princeton University Press, Princeton (1982)zbMATHGoogle Scholar
  17. 17.
    Führ, H.: Abstract Harmonic Analysis of Continuous Wavelet Transforms, L.N.M, vol. 1863. Springer, Berlin (2005)CrossRefGoogle Scholar
  18. 18.
    Geller, D.: Fourier analysis on the Heisenberg group. I. Schwartz space. J. Funct. Anal. 36, 205–254 (1980)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Głowacki, P.: A symbolic calculus and \(L^2\)-boundedness on nilpotent Lie groups. J. Funct. Anal. 206, 233–251 (2004)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Głowacki, P.: The Melin calculus for general homogeneous groups. Ark. Mat. 45(1), 31–48 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Głowacki, P.: Invertibility of convolution operators on homogeneous groups. Rev. Mat. Iberoam. 28(1), 141–156 (2012)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Helffer, B., Nourrigat, J.: Caracterisation des opérateurs hypoelliptiqes homogènes invariants à gauche sur un groupe de Lie nilpotent gradué. Commun. Partial Differ. Equ. 4(8), 899–958 (1979)CrossRefGoogle Scholar
  23. 23.
    Howe, R.: On the connection between nilpotent groups and oscillatory integrals associated to singularities. Pac. J. Math. 73, 329–363 (1977)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kirillov, A.A.: Lectures on the Orbit Method, Graduate Studies in Mathematics, vol. 64. American Mathematical Society, Providence (2004)Google Scholar
  25. 25.
    Ludwig, J.: Good ideals in the group algebra of a nilpotent group. Math. Zeitsch. 16, 195–210 (1978)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ludwig, J., Molitor-Braun, C., Scuto, L.: On Fourier’s inversion theorem in the context of nilpotent Lie groups. Acta Sci. Math. 3–4(73), 547–591 (2007)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Manchon, D.: Formule de Weyl pour les groupes de Lie nilpotentes. J. Reine Angew. Math. 418, 77–129 (1991)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Manchon, D.: Calcul symbolyque sur les groupes de Lie nilpotentes et applications. J. Funct. Anal. 102(2), 206–251 (1991)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Măntoiu, M., Ruzhansky, M.: Pseudo-differential operators, Wigner transform and Weyl systems on type I locally compact groups. Doc. Math. 22, 1539–1592 (2017)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Melin, A.: Parametrix constructions for right invariant differential operators on nilpotent groups. Ann. Global Anal. Geom. 1(1), 79–130 (1983)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Miller, K.: Invariant pseudodifferential operators on two step nilpotent Lie groups. Mich. Math. J. 29, 315–328 (1982)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Miller, K.: Inverses and parametrices for right-invariant pseudodifferential operators on two-step nilpotent Lie groups. Trans. AMS 280(2), 721–736 (1983)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Moore, C.C., Wolf, J.: Square integrable representations of nilpotent groups. Trans. AMS 185, 445–462 (1973)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Nussbaum, A.E.: Reduction theory for unbounded closed operators in Hilbert space. Duke Math. J. 31, 33–44 (1964)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Pedersen, N.V.: Matrix coefficients and a Weyl correspondence for nilpotent Lie groups. Invent. Math. 118, 1–36 (1994)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Ricci, F.: Sub-Laplacians on Nilpotent Lie Groups, Unpublished Lecture Notes.
  37. 37.
    Ruzhansky, M., Turunen, V.: Pseudodifferential Operators and Symmetries. Pseudo-Differential Operators: Theory and Applications, vol. 2. Birkhäuser, Boston (2010)CrossRefGoogle Scholar
  38. 38.
    Ruzhansky, M., Turunen, V.: Global quantization of pseudo-differential operators on compact Lie groups, SU(2) and 3-sphere. Int. Math. Res. Not. IMRN 11, 2439–2496 (2013)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Street, B.: Multiparameter Singular Integrals. Annals of Mathematical Studies, vol. 189. Princeton University Press, Princeton (2014)Google Scholar
  40. 40.
    Taylor, M.: Noncommutative microlocal analysis. Mem. AMS 52, 313 (1984)MathSciNetzbMATHGoogle Scholar
  41. 41.
    van Dijk, G., Neeb, K.-H., Salmasian, H., Zellner, C.: On the characterization of trace class representations and Schwartz operators. arXiv:1512.02451

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© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Universidad de Chile Facultad de CienciasSantiagoChile
  2. 2.Imperial College LondonLondonUK
  3. 3.Ghent UniversityGentBelgium
  4. 4.Queen Mary University of LondonLondonUK

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