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Orbital decompositions of representations of non-simply connected nilpotent groups

Published online by Cambridge University Press:  17 April 2009

Ronald L. Lipsman
Ronald L. Lipsman
Affiliation:
Department of Mathematics, University of Maryland College Park, MD 20742, United States of America
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Abstract

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An orbital integral formula is proven for the direct integral decomposition of an induced representation of a connected nilpotent Lie group. Previous work required simple connectivity. An explicit description of the spectral measure and spectral multiplicity function is derived in terms of orbital parameters. It is also proven that connected (but not necessarily simply connected) exponential solvable symmetric spaces are multiplicity free. Finally, the qualitative properties of the spectral multiplicity function are examined via several illuminating examples.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 42 , Issue 2 , October 1990 , pp. 293 - 306
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Benoist, Y., ‘Multiplicité un pour les espaces symetriques exponentiels’, Mem. Soc. Math. France, 15 (1984), 1–37.Google Scholar
[2]Bernat, P., ‘Sur les représentations unitaires des groupes de Lie résolubles’, Ann. Sci. École Norm. Sup. 82 (1965), 37–99.Google Scholar
[3]Corwin, L. and Greenleaf, F., ‘Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups’, Trans. Amer. Math. Soc. 304 (1987), 549–583.CrossRefGoogle Scholar
[4]Corwin, L. and Greenleaf, F., ‘Complex algebraic grometry and calculation of multiplicities for induced representations of nilpotent Lie groups’, Trans. Amer. Math. Soc. 305 (1988), 601–622.Google Scholar
[5]Fujiwara, H., ‘Repreśentations monomiales des groupes de Lie résolubles exponentiels’, (peprint).Google Scholar
[5]Kirillov, A., ‘Unitary representations of nilpotent Lie groups’, Russian Math. Surveys 17 (1962), 53–104.CrossRefGoogle Scholar
[7]Lipsman, R., ‘Orbital parameters for induced and restricted representations’, Trans. Amer. Math. Soc. 313 (1989), 433–473.CrossRefGoogle Scholar
[8]Lipsman, R., ‘Induced representations of completely solvable Lie groups’, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 (to appear).Google Scholar
[9]Stephenson, L., ‘Representation theory of nilpotent groups over local fields of characteristic zero’, Thesis. (University of Maryland, 1988).Google Scholar