  The totally nonnegative flag variety was introduced by Lusztig. It has
enriched combinatorial, geometric, and Lie-theoretic structures. In this paper,
we introduce a (new) $J$-total positivity on the full flag variety of an
arbitrary Kac-Moody group, generalizing the (ordinary) total positivity.
  We show that the $J$-totally nonnegative flag variety has a cellular
decomposition into totally positive $J$-Richardson varieties. Moreover, each
totally positive $J$-Richardson variety admits a favorable decomposition,
called a product structure. Combined with the generalized Poincare conjecture,
we prove that the closure of each totally positive $J$-Richardson variety is a
regular CW complex homeomorphic to a closed ball. Moreover, the $J$-total
positivity on the full flag provides a model for the (ordinary) totally
nonnegative partial flag variety. As a consequence, we prove that the closure
of each (ordinary) totally positive Richardson variety is a regular CW complex
homeomorphic to a closed ball, confirming conjectures of Galashin, Karp and
Lam.
