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

 

Square integrable representations of nilpotent groups


Authors: Calvin C. Moore and Joseph A. Wolf
Journal: Trans. Amer. Math. Soc. 185 (1973), 445-462
MSC: Primary 22E45; Secondary 22E25
MathSciNet review: 0338267
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Abstract: We study square integrable irreducible unitary representations (i.e. matrix coefficients are to be square integrable mod the center) of simply connected nilpotent Lie groups N, and determine which such groups have such representations. We show that if N has one such square integrable representation, then almost all (with respect to Plancherel measure) irreducible representations are square integrable. We present a simple direct formula for the formal degrees of such representations, and give also an explicit simple version of the Plancherel formula. Finally if $ \Gamma $ is a discrete uniform subgroup of N we determine explicitly which square integrable representations of N occur in $ {L_2}(N/\Gamma )$, and we calculate the multiplicities which turn out to be formal degrees, suitably normalized.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0338267-9
PII: S 0002-9947(1973)0338267-9
Article copyright: © Copyright 1973 American Mathematical Society