Let G be an exponential solvable Lie group, H and A two closed connected subgroups of G and σ a unitary and irreducible representation of H. We prove the orbital spectrum formula of the Up-Down representation ρ(G, H, A, σ) = IndH G σ|A. When G is nilpotent, the multiplicities of such representation turns out to be uniformly infinite or finite and bounded. A necessary and sufficient condition for the finiteness of the multiplicities is given. The same results are obtained when G is exponential solvable Lie group, H and A are invariant.
@article{urn:eudml:doc:41451,
title = {Sur les repr\'esentations mixtes des groupes de Lie r\'esolubles exponentiels.},
journal = {Publicacions Matem\`atiques},
volume = {46},
year = {2002},
pages = {179-199},
mrnumber = {MR1904862},
zbl = {1015.22004},
language = {fr},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41451}
}
Baklouti, Ali; Ghorbel, Amira; Hamrouni, Hatem. Sur les représentations mixtes des groupes de Lie résolubles exponentiels.. Publicacions Matemàtiques, Tome 46 (2002) pp. 179-199. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41451/