  A description of the fundamental degrees of freedom underlying a generalized
K\"ahler manifold, which separates its holomorphic moduli from the space of
compatible metrics in a similar way to the K\"ahler case, has been sought since
its discovery in 1984. In this paper, we describe a full solution to this
problem for arbitrary generalized K\"ahler manifolds, which involves the new
concept of a holomorphic symplectic Morita 2-equivalence between double
symplectic groupoids, equipped with a Lagrangian bisection of its real
symplectic core. Essentially, any generalized K\"ahler manifold has an
associated holomorphic symplectic manifold of quadruple dimension and equipped
with an anti-holomorphic involution; the metric is determined by a Lagrangian
submanifold of its fixed point locus. This finally resolves affirmatively a
long-standing conjecture by physicists concerning the existence of a
generalized K\"ahler potential.
  We demonstrate the theory by constructing explicitly the above Morita
2-equivalence and Lagrangian bisection for the well-known generalized K\"ahler
structures on compact even-dimensional semisimple Lie groups, which have until
now escaped such analysis. We construct the required holomorphic symplectic
manifolds by expressing them as moduli spaces of flat connections on surfaces
with decorated boundary, through a quasi-Hamiltonian reduction.
