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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Canonical semi-invariants and the Plancherel formula for parabolic groups


Authors: Ronald L. Lipsman and Joseph A. Wolf
Journal: Trans. Amer. Math. Soc. 269 (1982), 111-131
MSC: Primary 22E30; Secondary 22E46, 43A80
MathSciNet review: 637031
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Abstract: A parabolic subgroup of a reductive Lie group is called "good" if the center of the universal enveloping algebra of its nilradical contains an element that is semi-invariant of weight proportional to the modular function. The "good" case is characterized here by invariance of the set of simple roots defining the parabolic, under the negative of the opposition element of the Weyl group. In the "good" case, the unbounded Dixmier-Pukanszky operator of the parabolic subgroup is described, the conditions under which it is a differential operator rather than just a pseudodifferential operator are specified, and an explicit Plancherel formula is derived for that parabolic.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0637031-6
PII: S 0002-9947(1982)0637031-6
Article copyright: © Copyright 1982 American Mathematical Society