An integral formula for Lie groups, and the Mathieu conjecture reduced to Abelian non-Lie conjectures

We present an explicit integration formula for the Haar integral on a compact connected Lie group. This formula relies on a known decomposition of a compact connected simple Lie group into symplectic leaves, when one views the group as a Poisson Lie group. In this setting the Haar integral is constructed using the Kostant harmonic volume form on the corresponding flag manifold, and explicit coordinates are known for these invariant differential forms. The formula obtained is related to one found by Reshetikhin-Yakimov. Using our integration formula, we reduce the Mathieu conjecture to two stronger conjectures about Laurent polynomials in several complex variables with polynomial coefficients in several real variable polynomials. In these stronger conjectures there is no reference to Lie group theory.