Kirillov character formula
In mathematics, for a Lie group , the Kirillov orbit method gives a heuristic method in representation theory. It connects the Fourier transforms of coadjoint orbits, which lie in the dual space of the Lie algebra of G, to the infinitesimal characters of the irreducible representations. The method got its name after the Russian 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 supported on the coadjoint orbits, weighted by the square-root of the Jacobian of the exponential map, denoted by . It does not apply to all Lie groups, but works for a number of classes of connected Lie groups, including nilpotent, some semisimple groups, and compact groups. The Kirillov orbit method has led to a number of important developments in Lie theory, including the Duflo isomorphism and the wrapping map. Character formula for compact Lie groups Let be the highest weight of an irreducible representation , where is the dual of the Lie algebra of the maximal torus, and let be half the sum of the positive roots. We denote by the coadjoint orbit through and by the -invariant measure