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| In [[mathematics]], 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]].
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| 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]] of the [[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.
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| The Kirillov orbit method has led to a number of important developments in Lie theory, including the [[Duflo isomorphism]] and the [[wrapping map]].
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| == Character formula for compact Lie groups ==
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| Let <math>\lambda</math> be the [[highest weight]] of an [[Group representation|irreducible representation]] in the [[dual]] of the [[Lie algebra]] of the [[maximal torus]], denoted by <math>\mathfrak{t}^*</math>, and <math>\rho</math> half the sum of the positive [[root of a function|roots]].
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| We denote by
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| :<math>\mathcal{O}_{\lambda + \rho}</math>
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| the coadjoint orbit through
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| :<math>\lambda + \rho \in \mathfrak{t}^*</math>
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| and
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| :<math>\mu_{\lambda + \rho}</math>
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| is the <math>G</math>-invariant [[Measure (mathematics)|measure]] on | |
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| :<math>\mathcal{O}_{\lambda + \rho}</math>
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| with total mass
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| :<math>\dim \pi = d_\lambda</math>,
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| known as the [[Liouville measure]]. If <math>\chi_\pi = \chi_\lambda</math> is the character of a [[group representation|representation]], then''' Kirillov's character formula''' for compact Lie groups is then given by
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| :<math> j(X) \chi_\lambda (\exp X) = \int_{\mathcal{O}_{\lambda + \rho}} e^{i\beta (X)}d\mu_{\lambda + \rho} (\beta), \; \forall \; X \in \mathfrak{g} </math>
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| == Example: SU(2) ==
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| For the case of [[SU(2)]], the [[highest weight]]s are the positive half integers, and <math> \rho = 1/2 </math>. The coadjoint orbits are the two-dimensional [[spheres]] of radius <math> \lambda + 1/2 </math>, centered at the origin in 3-dimensional space.
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| By the theory of [[Bessel function]]s, it may be shown that
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| :<math> \int_{\mathcal{O}_{\lambda + 1/2}} e^{i\beta (X)}d\mu_{\lambda + 1/2} (\beta) = \frac{\sin((2\lambda + 1)X)}{X/2}, \; \forall \; X \in \mathfrak{g}, </math>
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| and
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| :<math> j(X) = \frac{\sin X/2}{X/2} </math>
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| thus yielding the characters of ''SU''(2):
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| :<math> \chi_\lambda (\exp X) = \frac{\sin((2\lambda + 1)X)}{\sin X/2} </math>
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| == References ==
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| *Kirillov, A. A., ''Lectures on the Orbit Method'', Graduate studies in Mathematics, 64, AMS, Rhode Island, 2004.
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| [[Category:Representation theory of Lie groups]]
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Let me initial begin by introducing myself. My title is Boyd Butts even though it is not the title on my beginning certificate. His family life in South Dakota but his spouse desires them to transfer. I am a meter reader but I strategy on changing it. He is really fond of doing ceramics but he is having difficulties to discover time for it.
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