In mathematics, for a Lie group
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
The Kirillov orbit method has led to a number of important developments in Lie theory, including the Duflo isomorphism and the wrapping map.
be the highest weight of an irreducible representation
is the dual of the Lie algebra of the maximal torus, and let
be half the sum of the positive roots.
is the character of the representation, the Kirillov's character formula for compact Lie groups is given by where
is the Jacobian of the exponential map.
For the case of SU(2), the highest weights are the positive half integers, and
The coadjoint orbits are the two-dimensional spheres of radius
, centered at the origin in 3-dimensional space.
By the theory of Bessel functions, it may be shown that and thus yielding the characters of SU(2):