Kirillov character formula

Character formula for compact Lie groups

Let \lambda \in \mathfrak{t}^* be the highest weight of an irreducible representation \pi , where \mathfrak{t}^* is the dual of the Lie algebra of the maximal torus, and let \rho be half the sum of the positive roots.

We denote by \mathcal{O}{\lambda + \rho} the coadjoint orbit through \lambda + \rho \in \mathfrak{t}^* and by \mu{\lambda + \rho} the G-invariant measure on \mathcal{O}{\lambda + \rho} with total mass \dim \pi = d\lambda, known as the Liouville measure. If \chi_\lambda is the character of the representation, the** Kirillov's character formula** for compact Lie groups is given by

: j^{1/2}(X) \chi_\lambda (\exp X) = \int_{\mathcal{O}{\lambda + \rho}} e^{i\beta (X)}d\mu{\lambda + \rho} (\beta), ; \forall ; X \in \mathfrak{g} ,

where j(X) is the Jacobian of the exponential map.

Example: SU(2)

For the case of SU(2), the highest weights are the positive half integers, and \rho = 1/2 . The coadjoint orbits are the two-dimensional spheres of radius \lambda + 1/2 , centered at the origin in 3-dimensional space.

By the theory of Bessel functions, it may be shown that

: \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},

and

: j(X) = \frac{\sin X/2}{X/2}

thus yielding the characters of SU(2):

: \chi_\lambda (\exp X) = \frac{\sin((2\lambda + 1)X)}{\sin X/2}

References

• Kirillov, A. A., Lectures on the Orbit Method, Graduate Studies in Mathematics, 64, AMS, Rhode Island, 2004.