0-Hecke algebra actions on coinvariants and flags

The 0-Hecke algebra $H_n(0)$ is a deformation of the group algebra of the symmetric group ${\text{SS}}_n$. We show that its coinvariant algebra naturally carries the regular representation of $H_n(0)$, giving an analogue of the well-known result for ${\text{SS}}_n$ by Chevalley-Shephard-Todd. By investigating the action of $H_n(0)$ on coinvariants and flag varieties, we interpret the generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. We also study the action of $H_n(0)$ on the cohomology rings of the Springer fibers, and similarly interpret the (noncommutative) Hall-Littlewood symmetric functions indexed by hook shapes.