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La formule de Poisson-Plancherel pour un groupe presque algébrique a radical abélien: Cas où le stabilisateur générique est réductif. (Poisson-Plancherel formula for quasi-algebraic groups with Abelian radical: The case of reductive generic stabilizer). (French) Zbl 0759.43004
Let \(V\) denote a finite dimensional real vector space and \(V^*\) be the dual vector space. Let \(L\) denote a lattice in \(V\) and \(L^*\) the dual lattice of \(L\) in \(V^*\). By a standard averaging process, the Plancherel formula for smooth functions on the compact torus \(T=V/L\) is equivalent to the Poisson summation formula for \(L\) and \(L^*\) for smooth compactly supported functions on \(V\). It is well known that the classical Poisson summation formula is a powerful tool in analysis and analytic number theory. It is less well known that due to the interpretation of \(T\) as a collapsed torus, the Poisson summation formula has a quantum mechanical background, too. Because the phase-space formulation of quantum mechanics relies on the coadjoint orbit method, it is not surprising that the Poisson-Plancherel formula which provides the Plancherel measure of a Lie group in a neighborhood of its neutral element needs the study of orbital integrals not only on the Lie algebra itself but also on its dual vector space. Based on the Poisson summation formula for a class of functions with singularities, the author establishes the Poisson-Plancherel formula for a class of quasi-algebraic groups. The construction depends on the existence of a class of polynomials which are invariant under the coadjoint action and on an \(L^ 1\) type a priori estimate for functions which are smooth on a Weyl chamber for a root subsystem.
Reviewer: W.Schempp (Siegen)

MSC:
43A80 Analysis on other specific Lie groups
22E46 Semisimple Lie groups and their representations
22E30 Analysis on real and complex Lie groups
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A05 Measures on groups and semigroups, etc.
22E40 Discrete subgroups of Lie groups
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