The 0-Hecke algebra is an interesting deformation of the group algebra of the symmetric group. It is known that these two algebras are analogous in many aspects. We find more analogies by studying their actions on the usual polynomial ring and the Stanley-Reisner ring of the Boolean algebra. For example, we show that the coinvariant algebra carries the regular representation of the 0-Hecke algebra. We also give an interpretation for the noncommutative Hall-Littlewood symmetric function introduced by Bergeron and Zabrocki, and recover a result of Garsia and Gessel on generating functions of the joint distribution of five permutation statistics.

Some of our results can be generalized to the setting of the Hecke algebra acting on the Stanley-Reisner ring of the Coxeter complex.