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Certaines représentations monomiales d’un groupe de Lie résoluble exponentiel. (Certain monomial representations of a solvable exponential Lie group). (French) Zbl 0706.22008

Representations of Lie groups: analysis on homogeneous spaces and representations of Lie groups, Proc. Symp., Kyoto/Jap. and Hiroshima/Jap. 1986, Adv. Stud. Pure Math. 14, 153-190 (1988).
[For the entire collection see Zbl 0694.00014.]
This paper extends to a solvable exponential Lie group G two results already known for nilpotent groups. Let f be a linear form on the Lie algebra of G, and let H be the analytic subgroup of G corresponding to a real polarization at f. The first main result is a reciprocity theorem (due to R. Howe in the nilpotent case), stating that, under Pukanszky’s condition, certain spaces of H-semi-invariant generalized vectors have dimension 0 or 1.
The second main theorem is an explicit Plancherel formula for the monomial representation induced from H to G by the unitary character given by f. The corresponding abstract formula for nilpotent groups was due to R. Penney. Both results are proved by induction on the dimension of G. Several low dimensional examples are considered at the end of the paper.
Reviewer: F.Rouvière

MSC:

22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)

Citations:

Zbl 0694.00014