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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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