abelian}
  \operatorname{DH}^T_M = (\Phi_T)_* (\mu_M) = (\pi_{K,T})_* (\Phi_K)_* (\mu_M).$$

Throughout this paper we shall assume for simplicity that the Abelian moment polytope $\Delta_T(M)$ is of maximal dimension. This can always be arranged for by replacing $T$ by the quotient $T / \bigcap_{m \in M} T_m$, where $T_m$ denotes the $T$-stabilizer of a point $m \in M$. If $T$ is the maximal torus of a semisimple Lie group, it follows already from that $\Delta_T(M)$ is of maximal dimension. As a consequence, $T$ acts locally freely on a dense, open subset whose complement has Liouville measure zero [@duistermaatheckman82 Lemma 3.1]. In particular, generic points in $M$ are regular for the Abelian moment map. Therefore the Abelian Duistermaat–Heckman measure is absolutely continuous with respect to Lebesgue measure on $\mathfrak t^*$. By the Duistermaat–Heckman Theorem, $\operatorname{DH}^T_M$ in fact has a polynomial density function of degree at most $n-r$ on each connected component of the set of regular values [@duistermaatheckman82 Corollary 3.3]. We shall call these components the *regular chambers*, and one can show that, except for the unbounded one, every such chamber is an open convex polytope. If the closures of two regular chambers have a common boundary of maximal dimension (i.e., of codimension 