A fixed point localization formula for the Fourier transform of regular semisimple coadjoint orbits

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dc.creatorLibine, Matvei
dc.date2003-02-28
dc.date2003-09-25
dc.date.accessioned2026-07-07T04:55:39Z
dc.date.available2026-07-07T04:55:39Z
dc.descriptionLet G_R be a Lie group acting on an oriented manifold M, and let $ω$ be an equivariantly closed form on M. If both G_R and M are compact, then the integral $\int_M ω$ is given by the fixed point integral localization formula (Theorem 7.11 in [BGV]). Unfortunately, this formula fails when the acting Lie group G_R is not compact: there simply may not be enough fixed points present. A proposed remedy is to modify the action of G_R in such a way that all fixed points are accounted for. Let G_R be a real semisimple Lie group, possibly noncompact. One of the most important examples of equivariantly closed forms is the symplectic volume form $dβ$ of a coadjoint orbit $Ω$. Even if $Ω$ is not compact, the integral $\int_Ω dβ$ exists as a distribution on the Lie algebra g_R. This distribution is called the Fourier transform of the coadjoint orbit. In this article we will apply the localization results described in [L1] and [L2] to get a geometric derivation of Harish-Chandra's formula (9) for the Fourier transforms of regular semisimple coadjoint orbits. Then we will make an explicit computation for the coadjoint orbits of elements of G_R* which are dual to regular semisimple elements lying in a maximally split Cartan subalgebra of g_R.
dc.description19 pages, LaTeX, no figures, final version which will appear in the Jour. Func. Analysis
dc.identifierhttps://arxiv.org/abs/math/0302352
dc.identifierhttp://arxiv.org/abs/math/0302352
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/66653
dc.subjectDifferential Geometry
dc.subjectRepresentation Theory
dc.titleA fixed point localization formula for the Fourier transform of regular semisimple coadjoint orbits
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