  We identify a subalgebra \pH_n of the extended affine Hecke algebra \eH_n of
type A. The subalgebra \pH_n is a \u-analogue of the monoid algebra of \S_n
\ltimes \ZZ_{\geq 0}^n and inherits a canonical basis from that of \eH_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 \emph{positive
affine tableaux} (PAT).
  We then exhibit a cellular subquotient \R_{1^n} of \pH_n that is a
\u-analogue of the ring of coinvariants \CC[y_1,...,y_n]/(e_1,...,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 \pi
\in \pH_n corresponds to rotations of words, and on cells corresponds to
cocyclage. We further show that \R_{1^n} has cellular quotients \R_\lambda that
are \u-analogues of the Garsia-Procesi modules R_\lambda with left cells
labeled by (a PAT version of) the \lambda-catabolizable tableaux.
  We give a conjectural description of a cellular filtration of \pH_n, the
subquotients of which are isomorphic to dual versions of \R_\lambda under the
perfect pairing on \R_{1^n}. We conjecture how this filtration relates to the
combinatorics of the cells of \eH_n worked out by Shi, Lusztig, and Xi. We also
conjecture that the k-atoms of Lascoux, Lapointe, and Morse and the
R-catabolizable tableaux of Shimozono and Weyman have cellular counterparts in
\pH_n. We extend the idea of atom copies of Lascoux, Lapoint, and Morse to
positive affine tableaux and give descriptions, mostly conjectural, of some of
these copies in terms of catabolizability.
