Generalizing Lusztig's total positivity II : geometric properties

Positive structures in Lie groups with respect to a subset $\Theta$ of the set of positive roots provide a generalization of Lusztig's total positivity in split real Lie groups to the setting of general real semisimple Lie groups. In [GW25] Lie groups G admitting a positive structure were classified and many key properties of the unipotent positive semigroups were established. In this article we focus on the positive semigroup in G. We establish key geometric properties of elements in the positive semigroup. Further we introduce corresponding positive and non-negative parts of flag varieties and determine the topology of the non-negative parts of flag varieties in many cases. For symplectic flag varieties we provide explicit descriptions of the positive and non-negative flag varieties.