We identify a subalgebra ˆH+n of the extended affine Hecke algebra ˆHn of type A. The subalgebra ˆH+n is a u-analogue of the monoid algebra of \mathcal{S}_n ⋉ℤ_≥0^n and inherits a canonical basis from that of \widehat{\mathscr{H}}_n. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod n, which we term positive affine tableaux (PAT). We then exhibit a cellular subquotient \mathscr{R}_1^n of \widehat{\mathscr{H}}^+_n that is a u-analogue of the ring of coinvariants ℂ[y_1,\ldots,y_n]/(e_1, \ldots,e_n) with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element *π ∈ \widehat{\mathscr{H}}^+_n corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that \mathscr{R}_1^n has cellular quotients \mathscr{R}_λ that are u-analogues of the Garsia-Procesi modules R_λ with left cells labeled by (a PAT version of) the λ -catabolizable tableaux.