Abstract
We determine the space of primary ideals in the group algebra \(L^{1}(G) \) of a connected nilpotent Lie group by identifying for every \(\pi \in \widehat{G} \) the family \(\mathcal I^\pi \) of primary ideals with hull \(\{\pi \} \) with the family of invariant subspaces of a certain finite dimensional sub-space \(\mathcal P_Q^\pi \) of the space of polynomials \(\mathcal P(G) \) on G.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Boidol, J., Leptin, H., Schürman, J., Vahle, D.: Räume primitiver Ideale von Gruppenalgebren. Math. Ann. 236(1), 1–13 (1978)
Brown, I.D.: Dual topology of a nilpotent Lie group. Ann. Sci. École Norm. Sup. (4) 6, 407–411 (1973)
Corwin, L.J., Greenleaf, F.P.: Representations of Nilpotent Lie Groups and Their Applications. Part I. Basic Theory and Examples. Cambridge Studies in Advanced Mathematics, vol. 18. Cambridge University Press, Cambridge (1990)
Dixmier, J.: Opérateurs de rang fini dans les représentations unitaires. Publ. Mathé. de l’IHÉS 6(1), 13–25 (1960)
Dixmier, J.: Enveloping Algebras: North-Holland Mathematical Library, vol. 14. North-Holland Publishing Co., Amsterdam (1977)
du Cloux, F.: Jets de fonctions différentiables sur le dual d’un groupe de Lie nilpotent. Invent. Math. 88(2), 375–394 (1987)
Hauenschild, W., Ludwig, J.: The injection and the projection theorem for spectral sets. Mon. Math. 92(3), 167–177 (1981)
Howe, R.E.: On a connection between nilpotent groups and oscillatory integrals associated to singularities. Pac. J. Math. 73(2), 329–363 (1977)
Kirillov, A.A.: Unitary representations of nilpotent Lie groups. Uspehi Mat. Nauk. 17(4(106)), 57–110 (1962)
Ludwig, J.: Polynomial growth and ideals in group algebras. Manuscr. Math. 30(3), 215–221 (1980)
Ludwig, J.: On the spectral synthesis problem for points in the dual of a nilpotent Lie group. Ark. Mat. 21(1), 127–144 (1983)
Ludwig, J.: On primary ideals in the group algebra of a nilpotent Lie group. Math. Ann. 262(3), 287–304 (1983)
Ludwig, J.: On the Hilbert–Schmidt seminorms of \(L^1\) of a nilpotent Lie group. Math. Ann. 273(3), 383–395 (1986)
Ludwig, J., Molitor-Braun, C., Poguntke, D.: Spectral synthesis for flat orbits in the dual space of weighted group algebras of nilpotent Lie groups. Trans. Am. Math. Soc. 365(8), 4433–4473 (2013)
Ludwig, J., Molitor-Braun, C.: Flat orbits, minimal ideals and spectral synthesis. Mon. Math. 160(3), 271–312 (2010)
Pedersen, N.V.: On the infinitesimal kernel of irreducible representations of nilpotent Lie groups. Bull. Soc. Math. Fr. 112(4), 423–467 (1984)
Pedersen, N.V.: Matrix coefficients and a Weyl correspondence for nilpotent Lie groups. Invent. Math. 118(1), 1–36 (1994)
Pukanszky, L.: On Kirillov’s character formula. J. Reine Angew. Math. 311(312), 408–440 (1979)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author has been partially supported by the Grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0131.
Rights and permissions
About this article
Cite this article
Beltiţă, I., Ludwig, J. Spectral synthesis for coadjoint orbits of nilpotent Lie groups. Math. Z. 284, 1111–1136 (2016). https://doi.org/10.1007/s00209-016-1691-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-016-1691-0