The up-down formula for nil-homogeneous spaces (Q1342830)

A multiflow on a nil-homogeneous space, i.e. the action of a vector group \(A\) on a quotient \(N/H\), is considered, where \(N\) is a simply connected nilpotent Lie group and \(H\) is a closed connected subgroup. If \(A\) is also a subgroup of \(N\), then the corresponding unitary action of \(A\) on \(L^2(N/H)\) is an up-down representation -- a succession of induced and restricted representations. An explicit orbital formula for the direct integral decomposition of this unitary representation is obtained. A simple sufficient condition for finite multiplicity is derived as a consequence. Similarities to ergodic actions are indicated.