A Plancherel formula for idyllic nilpotent Lie groups

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.