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Square integrable representations and a Plancherel theorem for parabolic subgroups


Author: Frederick W. Keene
Journal: Trans. Amer. Math. Soc. 243 (1978), 61-73
MSC: Primary 22E45
DOI: https://doi.org/10.1090/S0002-9947-1978-0498983-X
MathSciNet review: 0498983
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Abstract: Let G be a semisimple Lie group with Iwasawa decomposition $ G\, = \,KAN$. Let $ {\mathfrak{g}_0} \,=\, {\mathcal{f}_0} \,+\, \mathfrak{a} \,+\, \mathfrak{n}$ be the corresponding decomposition of the Lie algebra of G. Then the nilpotent subgroup N has square integrable representations if and only if the reduced restricted root system is of type $ {A_1}$ or $ {A_2}$. The Plancherel measure for N can be found explicitly in these cases. We then prove the Plancherel theorem in the $ {A_1}$ case for the solvable subgroup NA by combining Mackey's ``Little Group'' method with an idea due to C. C. Moore: we find an operator D, defined on the $ {C^\infty }$ functions on NA with compact support, such that

$\displaystyle \phi (e) = \int_{{{(NA)}^\wedge}} {{\text{tr}}} (D\pi (\phi ))d\mu (\pi )$

where $ {(NA)^ \wedge }$ is the unitary dual, e is the identity, and $ \mu $ is the Plancherel measure for NA, and D is an unbounded selfadjoint operator. In the $ {A_1}$ case, D involves fractional powers of the Laplace operator and hence is not a differential operator.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0498983-X
Keywords: Parabolic subgroup, semisimple Lie group, Plancherel theorem, nonunimodular group, square integrable representations, solvable subgroups of type AN
Article copyright: © Copyright 1978 American Mathematical Society

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