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The 0-Hecke algebra $H_n(0)$ is a deformation of the group algebra of thesymmetric group $\SS_n$. We show that its coinvariant algebra naturally carriesthe regular representation of $H_n(0)$, giving an analogue of the well-knownresult for $\SS_n$ by Chevalley-Shephard-Todd. By investigating the action of$H_n(0)$ on coinvariants and flag varieties, we interpret the generatingfunctions counting the permutations with fixed inverse descent set by theirinversion number and major index. We also study the action of $H_n(0)$ on thecohomology rings of the Springer fibers, and similarly interpret the(noncommutative) Hall-Littlewood symmetric functions indexed by hook shapes.

0-Hecke algebra actions on coinvariants and flags