A Birkhoff-Bruhat Atlas for partial flag varieties

A partial flag variety ${\mathcal {P}}_K$ of a Kac-Moody group $G$ has a natural stratification into projected Richardson varieties. When $G$ is a connected reductive group, a Bruhat atlas for ${\mathcal {P}}_K$ was constructed by He, Knutson and Lu: ${\mathcal {P}}_K$ is locally modeled with Schubert varieties in some Kac-Moody flag variety as stratified spaces. The existence of Bruaht atlases implies some nice combinatorial and geometric properties on the partial flag varieties and the decomposition into projected Richardson varieties. A Bruhat atlas does not exist for partial flag varieties of an arbitrary Kac-Moody group due to combinatorial and geometric reasons. To overcome obstructions, we introduce the notion of Birkhoff-Bruhat atlas. Instead of the Schubert varieties used in a Bruhat atlas, we use the $J$-Schubert varieties for a Birkhoff-Bruhat atlas. The notion of the $J$-Schubert varieties interpolates Birkhoff decomposition and Bruhat decomposition of the full flag variety (of a larger Kac-Moody group). The main result of this paper is the construction of a Birkhoff-Bruhat atlas for any partial flag variety ${\mathcal {P}}_K$ of a Kac-Moody group. We also construct a combinatorial atlas for the index set $Q_K$ of the projected Richardson varieties in ${\mathcal {P}}_K$. As a consequence, we show that $Q_K$ has some nice combinatorial properties. This gives a new proof and generalizes the work of Williams in the case where the group $G$ is a connected reductive group.