Kirillov character formula

In mathematics, for a Lie group ${\displaystyle G}$, the

.

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 ${\displaystyle j}$. It does not apply to all Lie groups, but works for a number of classes of

.

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 ${\displaystyle \lambda \in {\mathfrak {t}}^{*}}$ be the

${\displaystyle \pi }$, where ${\displaystyle {\mathfrak {t}}^{*}}$ is the , and let ${\displaystyle \rho }$ be half the sum of the positive roots.

We denote by ${\displaystyle {\mathcal {O}}_{\lambda +\rho }}$ the coadjoint orbit through ${\displaystyle \lambda +\rho \in {\mathfrak {t}}^{*}}$ and by ${\displaystyle \mu _{\lambda +\rho }}$ the ${\displaystyle G}$-invariant measure on ${\displaystyle {\mathcal {O}}_{\lambda +\rho }}$ with total mass ${\displaystyle \dim \pi =d_{\lambda }}$, known as the

. If ${\displaystyle \chi _{\lambda }}$ is the character of the representation, the Kirillov's character formula for compact Lie groups is given by

${\displaystyle j^{1/2}(X)\chi _{\lambda }(\exp X)=\int _{{\mathcal {O}}_{\lambda +\rho }}e^{i\beta (X)}d\mu _{\lambda +\rho }(\beta ),\;\forall \;X\in {\mathfrak {g}}}$,

where ${\displaystyle j(X)}$ is the Jacobian of the exponential map.

Example: SU(2)

For the case of

are the positive half integers, and ${\displaystyle \rho =1/2}$. The coadjoint orbits are the two-dimensional of radius ${\displaystyle \lambda +1/2}$, centered at the origin in 3-dimensional space.

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

${\displaystyle \int _{{\mathcal {O}}_{\lambda +1/2}}e^{i\beta (X)}d\mu _{\lambda +1/2}(\beta )={\frac {\sin((2\lambda +1)X)}{X/2}},\;\forall \;X\in {\mathfrak {g}},}$

and

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

thus yielding the characters of SU(2):

${\displaystyle \chi _{\lambda }(\exp X)={\frac {\sin((2\lambda +1)X)}{\sin X/2}}}$

See also

• Weyl character formula
• Localization formula for equivariant cohomology

References

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