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OPERATOR KERNELS FOR IRREDUCIBLE REPRESENTATIONS OF EXPONENTIAL LIE GROUPS

Published online by Cambridge University Press:  01 October 2008

DETLEV POGUNTKE*
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100 131, 33501 Bielefeld, Germany (email: [email protected])
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Abstract

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A nine-dimensional exponential Lie group G and a linear form on the Lie algebra of G are presented such that for all Pukanszky polarizations 𝔭 at the canonically associated unitary representation ρ=ρ(,𝔭) of G has the property that ρ(ℒ1(G)) does not contain any nonzero operator given by a compactly supported kernel function. This example shows that one of Leptin’s results is wrong, and it cannot be repaired.

Type
Research Article
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Abdennadher, J. and Molitor-Braun, C., ‘Operator kernels for irreducible unitary representations of solvable exponential Lie groups’, J. Lie Theory 16 (2006), 225238.Google Scholar
[2]Bernat, P., Représentations des Groupes de Lie Résoluble (Dunod, Paris, 1972).Google Scholar
[3]Howe, R., ‘On a connection between nilpotent groups and oscillator integrals associated to singularities’, Pacific J. Math. 73 (1977), 329363.Google Scholar
[4]Leptin, H., ‘Irreduzible darstellungen von exponentialgruppen und operatoren mit glatten Kernen’, J. Reine Angew. Math. 494 (1998), 134.CrossRefGoogle Scholar
[5]Leptin, H. and Ludwig, J., Unitary Representation Theory of Exponential Lie Groups, Expositions in Mathematics, 18 (de Gruyter, Berlin, 1994).CrossRefGoogle Scholar
[6]Ludwig, J., ‘Irreducible representations of exponential solvable Lie groups and operators with smooth kernels’, J. Reine Angew. Math. 339 (1983), 126.Google Scholar
[7]Ludwig, J. and Molitor-Braun, C., ‘Exponential actions, orbits and their kernels’, Bull. Austral. Math. Soc. 57 (1998), 497513.Google Scholar
[8]Molitor-Braun, C., ‘Actions exponentielles et noyaux d’opérateurs’ Travaux mathématiques, IX (Centre Universitaire, Luxembourg, 1997), pp. 23101.Google Scholar
[9]Poguntke, D., ‘Gewisse Segalsche Algebren Auf Lokalkompakten Gruppen’, Arch. Math. 33 (1979), 454460.Google Scholar
[10]Poguntke, D., ‘Nichtsymmetrische sechsdimensionale Liesche Gruppen’, J. Reine Angew. Math. 306 (1979), 154176.Google Scholar
[11]Poguntke, D., ‘Algebraically irreducible representations of L 1-algebras of exponential Lie groups’, Duke Math. J. 50 (1983), 10771106.Google Scholar