Hall-Littlewood polynomials and a Hecke action on ordered set partitions

We construct an action of the Hecke algebra $H_n(q)$ on a quotient of the polynomial ring $F[x_1,\dots ,x_n]$, where $F=\mathbb{Q}(q)$. The dimension of our quotient ring is the number of $k$-block ordered set partitions of ${1, 2, \dots, n }$. This gives a quantum analog of a construction of Haglund-Rhoades-Shimozono and interpolates between their result at $q=1$ and work of Huang-Rhoades at $q=0$.