  A normalizable complex distribution $P(x)$ on a manifold $\mathcal{M}$ can be
regarded as a complex weight, thereby allowing to define expectation values of
observables $A(x)$ defined on $\mathcal{M}$. Straightforward importance
sampling, $x\sim P$, is not available for non positive $P$, leading to the
well-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 analytically
continued function on $\mathcal{M}^c$. Such representations allow to carry out
Monte Carlo calculations to obtain estimates of $\langle A(x) \rangle_P$,
through the sampling $z \sim \rho$. In the present work we tackle the problem
of constructing positive representations for complex weights defined on
manifolds of compact Lie groups, both Abelian and non Abelian, as required in
lattice gauge field theories. Since the variance of the estimates increase for
broad representations, special attention is put on the question of localization
of the support of the representations.
