coadjoint orbit $\mathcal O_\lambda$ to the set of Hermitian operators with eigenvalues $$\label{spectra to positive weyl chamber}
  \hat\lambda_j = \frac 1 {\dim \mathcal H} + i \lambda_j
  \quad
  (j=1,\ldots,\dim \mathcal H),$$ where the $\lambda_j$ are the eigenvalues of $\lambda$ (eigenvalues are repeated according to their multiplicity). Let us choose coordinates $\mathcal H \cong {\mathbb C}^d$ and identify $\operatorname{SU}(\mathcal H) \cong \operatorname{SU}(d)$ accordingly. Then the eigenvalues $\lambda_i$ are just the diagonal entries of the matrix $\lambda \in \mathfrak t^*_+$ labeling the coadjoint orbit (see for our conventions). It follows that defines a bijection between the positive Weyl chamber $\mathfrak t^*_+$ and the set of eigenvalue spectra of trace-one Hermitian operators, which we think of as elements of the set $\{ \hat\lambda \in \mathbf R^d : \hat\lambda_1 \geq \ldots \geq \hat\lambda_d, \sum_j \hat\lambda_j = 1 \}$.

Note that the set of pure states is identified with the coadjoint orbit through the highest weight of the defining representation of $\operatorname{SU}(\mathcal H)$, that is, with projective space ${\mathbb P}(\mathcal H)$: The density matrix corresponding to a point $[\psi] \in {\mathbb P}(\mathcal H)$ is simply the orthogonal projection $P_\psi$ onto ${\mathbb C}\psi$.

Moreover, the Liouville measure on a coadjoint orbit $\mathcal O_\lambda$ is identified via 