The Coxeter Flag Variety

For a Coxeter element $c$ in a Weyl group $W$, we define the $c$-Coxeter flag variety ${\mathrm{CFl}}_c\subset G\mathrm{/}B$ as the union of left-translated Richardson varieties $w^{-1}{X}_w^{wc}$. This is a complex of toric varieties whose geometry is governed by the lattice $\mathrm{NC}(W,c)$ of $c$-noncrossing partitions. We show that ${\mathrm{CFl}}_c$ is the common vanishing locus of the generalized Plücker coordinates indexed by $W\setminus \mathrm{NC}(W,c)$. We also construct an explicit affine paving of ${\mathrm{CFl}}_c$ and identify the $T$-weights of each cell in terms of $c$-clusters. This paving gives a GKM description of $H^{\bullet }({\mathrm{CFl}}_c)$ and $H_{T_{ad}}^{\bullet }({\mathrm{CFl}}_c)$ in terms of the induced Cayley subgraph on $\mathrm{NC}(W,c)$, and we show these rings are naturally isomorphic for different choices of $c$. In type $\mathrm{A}$, this recovers the quasisymmetric flag variety for a special $c$, and for general $c$ we show the cohomology ring has a presentation as permuted quasisymmetric coinvariants.