We show in this paper that the multiplicities of mixed representations are uniformly infinite or uniformly finite and bounded, in the setting of completely solvable Lie groups extending then the situation of nilpotent Lie groups. Necessary and sufficient conditions for these multiplicities to be finite are provided.
"The Multiplicity Function of Mixed Representations on Completely Solvable Lie Groups." Tokyo J. Math. 30 (1) 41 - 55, June 2007. https://doi.org/10.3836/tjm/1184963646