Abstract
Let G be a connected and simply connected nilpotent Lie group and A a closed connected subgroup of G. Let Γ be a discrete cocompact subgroup of G. In the first part of this paper we give the direct integral decomposition of the up–down representation \({({\rm Ind}_\Gamma^G 1)\vert_A}\) . As a consequence, we establish a necessary and sufficient condition for A to act ergodically on G/Γ in the case when Γ is a lattice subgroup of G and A is a one-parameter subgroup of G.
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Hamrouni, H. Up–down representations and ergodic theory in nilpotent Lie groups. manuscripta math. 127, 511–519 (2008). https://doi.org/10.1007/s00229-008-0212-9
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DOI: https://doi.org/10.1007/s00229-008-0212-9