A Birkhoff–Bruhat atlas for partial flag varieties

Abstract

A partial flag variety PK 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 PK was constructed in He et al. (2013): PK 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 PK of a Kac–Moody group. We also construct a combinatorial atlas for the index set QK of the projected Richardson varieties in PK. As a consequence, we show that QK 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.