Kirillov character formula

Remarks

In words, the character of $\pi$, pulled back to $\mathfrak g$ by $\exp$, is the Fourier transform of the orbital measure $\beta$. The proof goes by induction on $\dim G$. In the induction step, either $\pi$ is induced from a normal subgroup of codimension one, or $\pi$ factors through the quotient of $G$ by a central one-parameter subgroup. The sign in $e^{i \ell(X)}$ matches the convention in which $\pi$ is induced from the character $\exp Y \mapsto e^{i \ell(Y)}$ of a polarizing subgroup.

Example: for $G = \R^n$ the orbits are points $\ell$, the representations are the characters $x \mapsto e^{i \ell(x)}$, and the formula reduces to $\tr \pi(f) = \int f(x) e^{i \ell(x)} \, dx$. For the Heisenberg group and an orbit $\set{\ell : \ell(Z) = \lambda}$ with $\lambda \ne 0$, the formula shows that the character of the Schrödinger representation is a distribution supported on the centre.

For some other Lie groups and some of their representations a version of Kirillov's formula holds with a correction: near $0$ in $\mathfrak g$ one multiplies by $j(X)^{-1/2}$, where $j(X) = \det \big( \sinh(\mathrm{ad} X / 2) / (\mathrm{ad} X / 2) \big)$. For nilpotent $\mathfrak g$, $\mathrm{ad} X$ is nilpotent and $j \equiv 1$, so no correction is needed.

Going deeper

For a student

The formula is the strongest form of the orbit method for nilpotent groups. It makes the bijection between orbits and representations compatible with characters. It also identifies the Plancherel measure. Integrating the formula over all orbits gives Fourier inversion on $\mathfrak g^*$, so the Plancherel measure is the image of Lebesgue measure on $\mathfrak g^*$ in the space of orbits. For compact connected Lie groups an analogous formula holds with the correction factor $j^{-1/2}$. There the representation of highest weight $\lambda$ uses the orbit through $\lambda + \rho$, and the formula is equivalent to the Weyl character formula.