Up-down representations and ergodic theory in nilpotent Lie groups (Q1003143)

This paper gives a new necessary and sufficient condition for a nilflow \((G/\Gamma,A)\) to be ergodic where \(G\) is a simply connected nilpotent Lie group and \(\Gamma\) is a lattice subgroup of \(G\). \(A\) is a connected one-parameter subgroup of \(G\). The results are simpler than similar results previously obtained by Auslander, Green and Hahn. The author gives an explicit decomposition into irreducibles of the up-down representation \((\text{Ind}^G_\Gamma 1)_A\).