the interior of the positive Weyl chamber. Its support is equal to the moment polytope. Note that $p_K(\lambda)$ is equal to the Liouville volume of a maximal-dimensional coadjoint orbit $\mathcal O_\lambda$ [@berlinegetzlervergne03 Proposition 7.26]. Therefore, the above is a natural definition to use in our context: It is normalized so that the Duistermaat–Heckman measure associated with the action of $K$ on a generic coadjoint orbit $\mathcal O_\lambda$ is a probability distribution concentrated at the point $\lambda$.

If $H$ is another compact, connected Lie group, with Lie algebra $\mathfrak h$, acting on $M$ via a group homomorphism ${\varphi}\colon H \rightarrow K$, then this action is also Hamiltonian, with a moment map given by the composition $$\label{restriction to subgroups}
  \Phi_H = (d{\varphi})^* \circ \Phi_K \colon M \rightarrow \mathfrak k^* \rightarrow \mathfrak h^*.$$ This in turn determines a Duistermaat–Heckman measure $\operatorname{DH}^H_M$. In particular, we can associate moment maps and Duistermaat–Heckman measures with all closed subgroups of $K$. In the case of the maximal torus $T \subseteq K$, we shall call $\Phi_T$ the *Abelian moment map* and $\operatorname{DH}^T_M$ the *Abelian Duistermaat–Heckman measure*, in order to distinguish them from the *non-Abelian moment map* $\Phi_K$ and the *non-Abelian Duistermaat–Heckman measure* $\operatorname{DH}^K_M$, respectively. Explicitly, $$\label{definition duistermaat-heckman measure 