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A Plancherel formula for idyllic nilpotent Lie groups

Author: Eloise Carlton
Journal: Trans. Amer. Math. Soc. 224 (1976), 1-42
MSC: Primary 22E25
MathSciNet review: 0425014
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Abstract: A procedure is developed which can be used to compute the Plancherel measure for a certain class of nilpotent Lie groups, including the Heisenberg groups, free groups, two-and three-step groups, the nilpotent part of an Iwasawa decomposition of the R-split form of the classical simple groups $ {A_l},{C_l},{G_2}$.

Let G be a connected, simply connected nilpotent Lie group. The Plancherel formula for G can be expressed in terms of Plancherel measure of a normal subgroup N and projective Plancherel measures of certain subgroups of $ G/N$. To get an explicit measure for G, we need an explicit formula for (1) the disintegration of Plancherel measure of N under the action of G on N, and (2) projective Plancherel measures of $ {G_\gamma }/N$, where $ {G_\gamma }$ is the stability subgroup at $ \gamma $ in N. When both N and $ {G_\gamma }/N$ are abelian, the measures (1) and (2) are obtained as special cases of more general problems. These measures combine into Plancherel measure for G.

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Keywords: Connected, simply connected real nilpotent Lie group, nilpotent Lie algebra, Jordan-Hölder basis, unipotent action, contragredient action, G-invariant measure, Zariski open set, disintegration of measures, idyll, coadjoint representation, stability subgroup, multipler representation, irreducible representation, projective Plancherel measure, Plancherel formula for group extensions
Article copyright: © Copyright 1976 American Mathematical Society