  We discuss an extension of the Weingarten formula, to the case of
noncommutative homogeneous spaces, under suitable "easiness" assumptions. The
spaces that we consider are noncommutative algebraic manifolds, generalizing
the spaces of type $X=G/H\subset\mathbb C^N$, with $H\subset G\subset U_N$
being subgroups of the unitary group, subject to certain uniformity conditions.
We discuss various axiomatization issues, then we establish the Weingarten
formula, and we derive some probabilistic consequences.
