Square integrable representations and a Plancherel theorem for parabolic subgroups
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- by Frederick W. Keene PDF
- Trans. Amer. Math. Soc. 243 (1978), 61-73 Request permission
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 \[ \phi (e) = \int _{{{(NA)}^\wedge }} {{\text {tr}}} (D\pi (\phi ))d\mu (\pi )\] 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.References
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Additional Information
- © Copyright 1978 American Mathematical Society
- 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