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Sur les caractères des groupes de Lie résolubles. (On characters of soluble Lie groups.). (French) Zbl 0727.22006
We consider a connected, solvable, unimodular Lie group G. Let $${\mathfrak g}$$ be the Lie algebra of G. Let $${\mathfrak l}$$ be in the dual of $${\mathfrak g}$$. Under the assumption that $${\mathfrak g}({\mathfrak l})$$ is reductive in $${\mathfrak g}$$, we construct a map $$\phi \to F_{{\mathfrak l},\phi}$$ from D(G) to the space of $$C^{\infty}$$ functions on an open dense subset of G($${\mathfrak l})$$. Using this map we give a formula for the trace of the operator n($${\mathfrak l},G)(\phi)$$, where n($${\mathfrak l},G)$$ is the unitary representation of G associated to $${\mathfrak l}$$. This formula applies to the square-integrable representations modulo Z(G) of the group G.

##### MSC:
 22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
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##### References:
 [1] N. H. ANH, Classification of connected unimodular Lie groups with discrete series, Ann. Inst. Fourier, Grenoble, 30, 1 (1980), 159-192. · Zbl 0418.22010 [2] L. AUSLANDER, B. KOSTANT, Polarization and unitary representations of solvable Lie groups, Inv. Math., 14 (1971), 255-354. · Zbl 0233.22005 [3] P. BERNAT et al., Représentations des groupes de Lie résolubles, Paris, Dunod, 1972. · Zbl 0248.22012 [4] N. BOURBAKI, Algèbre, Chapitre 9, Hermann, Paris, 1959. · Zbl 0102.25503 [5] J. Y. CHARBONNEL, La formule de Plancherel pour un groupe de Lie résoluble connexe, Lecture Notes in Mathematics, Springer-Verlag, 587 (1977), 32-76. · Zbl 0365.22009 [6] C. CHEVALLEY, Théorie des groupes de Lie, vol. II, Groupes algébriques, Hermann, Paris, 1951. · Zbl 0054.01303 [7] HARISH-CHANDRA, A formula for semisimple Lie groups, Am. J. of Math., 79 (1957), 733-760. · Zbl 0080.10201 [8] G. HOCHSCHILD, The structure of Lie groups, San Francisco, Holden-Day, 1965. · Zbl 0131.02702 [9] G. W. MACKEY, Induced representations of locally compact groups I, Ann. Math., 55 (1952), 101-139. · Zbl 0046.11601
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