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 in He et al. (2013): ${\mathcal{P}}_K$ is locally modelled with Schubert varieties in some Kac–Moody flag variety as stratified spaces. The existence of Bruhat 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 (2007) in the case where the group $G$ is a connected reductive group.

部分旗帜品种 ${\mathcal{磷}}_{钾}$ Kac-Moody 群 $G$具有自然分层为预计的理查森品种。什么时候$G$ 是一个连通的还原群，一个 Bruhat 图集 ${\mathcal{磷}}_{钾}$是在 He 等人中构建的。(2013):${\mathcal{磷}}_{钾}$在一些 Kac-Moody flag 变体中使用 Schubert 变体局部建模作为分层空间。Bruhat 图集的存在意味着部分标志变体和分解为投影的理查森变体的一些很好的组合和几何特性。

由于组合和几何原因，对于任意 Kac-Moody 群的部分标志变体，不存在 Bruhat 图集。为了克服障碍，我们引入了 Birkhoff-Bruhat 图集的概念。我们不使用 Bruhat 地图集中使用的舒伯特变体，而是使用$J$- Birkhoff-Bruhat 图集的舒伯特变体。的概念$J$-Schubert 变体对完整标志变体（更大的 Kac-Moody 群）进行 Birkhoff 分解和 Bruhat 分解。本文的主要结果是为任何部分旗帜品种构建了一个 Birkhoff-Bruhat 图集${\mathcal{磷}}_{钾}$属于 Kac-Moody 小组。我们还为索引集构建了一个组合图集${问}_{钾}$ 预计的理查森品种 ${\mathcal{磷}}_{钾}$. 因此，我们证明${问}_{钾}$有一些很好的组合属性。这给出了一个新的证明并概括了威廉姆斯（2007）在小组$G$ 是连通的还原群。