Sur les caractères des groupes de Lie résolubles. (On characters of soluble Lie groups.) (Q803297)

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.