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Disintegration of monomial representations of nilpotent Lie groups. (Désintégration des représentations monomiales des groupes de Lie nilpotents.) (French) Zbl 0921.22006
Disintegration is one of the main problems in the theory of representations. It leads us to the harmonic analysis of various spaces of functions or distributions on certain groups or their homogeneous spaces. In the course of a study we often need to concrete an abstract theory. This paper concerns induced representations of a connected, simply connected nilpotent Lie group $$G$$. Their direct integral decompositions are well known [cf. L. Corwin, F. P. Greenleaf and G. Grélaud, Trans. Am. Math. Soc. 304, 549-583 (1987; Zbl 0629.22005); R. Lipsman, Trans. Am. Math. Soc. 313, 433-473 (1989; Zbl 0683.22009)]. Here, the authors construct explicitly an intertwining operator and its inverse between an induced representation and its disintegration. For this purpose they make delicate choices of various vectors to parametrize the base space and its measure for the disintegration, and of polarizations to realize the irreducible unitary representations appearing there. It is interesting to note that their results are independent of whether the multiplicities are finite or not.

##### MSC:
 2.2e+28 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 2.2e+26 Nilpotent and solvable Lie groups
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