Generalized Duistermaat–Heckman formula for compact groups

Let $G$ be a compact group, with Lie algebra containing a Cartan subalgebra and a chosen Borel subalgebra ${\mathfrak b}$. Let ${\mathfrak n}_+=[{\mathfrak b},{\mathfrak b}]$ be the sum of the positive-root spaces, let ${\cal W}$ be the Weyl group, let $X,Y$ be elements of the Cartan subalgebra, and let $F$ be a suitable function of two matrix arguments. For $w\in{\cal W}$, write $w(Y)$ for the Weyl-group action and let $\varepsilon_w$ denote its sign. The measures and normalization constant are those appropriate to the chosen compact group.

Generalized Duistermaat–Heckman conjecture. An analogous correlation-function formula should hold for every compact group, with the triangular integration domain identified with ${\mathfrak n}_+=[{\mathfrak b},{\mathfrak b}]$; in particular, the integral over $G$ should be expressible as a Weyl-group sum of integrals over ${\mathfrak n}_+$ of the displayed form.

The conjecture extends the formulas established in the paper for the orthogonal and symplectic groups, together with the corresponding unitary-group result. The source gives no resolution or further hypotheses for the proposed formula, so its status remains open.