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 {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 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 γ / N {G_\gamma }/N , where G γ {G_\gamma } is the stability subgroup at γ \gamma in N. When both N and G γ / N {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.