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A normalizable complex distribution $P(x)$ on a manifold $\mathcal{M}$ can beregarded as a complex weight, thereby allowing to define expectation values ofobservables $A(x)$ defined on $\mathcal{M}$. Straightforward importancesampling, $x\sim P$, is not available for non positive $P$, leading to thewell-known sign (or phase) problem. A positive representation $\rho(z)$ of$P(x)$ is any normalizable positive distribution on the complexified manifold$\mathcal{M}^c$, such that, $\langle A(x)\rangle_P = \langle A(z) \rangle_\rho$for a dense set of observables, where $A(z)$ stands for the analyticallycontinued function on $\mathcal{M}^c$. Such representations allow to carry outMonte Carlo calculations to obtain estimates of $\langle A(x) \rangle_P$,through the sampling $z \sim \rho$. In the present work we tackle the problemof constructing positive representations for complex weights defined onmanifolds of compact Lie groups, both Abelian and non Abelian, as required inlattice gauge field theories. Since the variance of the estimates increase forbroad representations, special attention is put on the question of localizationof the support of the representations.

Positive representations of complex distributions on groups