Orbit method: Difference between revisions

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For a [[Lie group]] <math>G</math>, the [[Kirillov orbit method]] gives a heuristic method in [[representation theory]]. It connects the [[Fourier transform]]s of [[coadjoint orbit]]s, which lie in the [[dual space]] of the [[Lie algebra]] of ''G'', to the [[infinitesimal character]]s of the [[irreducible representation]]s. The method got its name after the [[Russia]]n mathematician [[Alexandre Kirillov]].

At its simplest, it states that a character of a Lie group may be given by the [[Fourier transform]] of the [[Dirac delta function]] [[support (mathematics)|support]]ed on the coadjoint orbits, weighted by the square-root of the [[Jacobian matrix and determinant|Jacobian]] of the [[exponential map (Lie theory)|exponential map]], denoted by <math>j</math>. It does not apply to all Lie groups, but works for a number of classes of [[connected space|connected]] Lie groups, including [[nilpotent]], some [[Semisimple Lie group|semisimple]] groups, and [[compact group]]s.

==Special cases==