with the unique $\operatorname{SU}(\mathcal H)$-invariant measure on the set of Hermitian matrices with spectrum $\hat\lambda$, normalized to total volume $$\label{volume coadjoint su orbit}
  p_{\operatorname{SU}(\mathcal H)}(\lambda) =
  \prod_{j < k} \frac{i (\lambda_j - \lambda_k)}{k - j} =
  \prod_{j < k} \frac{(\hat\lambda_j - \hat\lambda_k)}{k - j}.$$ The non-Abelian Duistermaat–Heckman measure of $\mathcal O_\lambda$ corresponds to the Dirac measure at $\hat\lambda$, while the Abelian Duistermaat–Heckman measure corresponds to the distribution of the diagonal entries of a density matrix with spectrum $\hat\lambda$ chosen according to the invariant measure.

\[bloch ball\] The Lie algebra ${\mathfrak{su}}(2)$ is three-dimensional, generated by $-\frac i 2$ times the Pauli matrices $\sigma_x, \sigma_y, \sigma_z$. Functionals in its dual ${\mathfrak{su}}(2)^*$ can be identified with points in ${\mathbb R}^3$ by evaluating them at these generators. In this picture, the inverse of associates to a vector $\vec r = (x,y,z)$ the Hermitian matrix $\rho(\vec r) = \frac 1 2 ( {\mathbf 1}+ \vec r \cdot \vec\sigma )$, where $\vec\sigma$ is the Pauli vector $(\sigma_x, \sigma_y, \sigma_z)$. The $z$-axis is identified with the Lie algebra of the maximal torus, ${\mathbb R}\sigma_z$, its positive half-axis with our choice of positive Weyl chamber $\mathfrak t^*_+$, and $(0,0,2)$ with the corresponding positive root $\alpha > 0$. The 