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We identify a subalgebra \pH_n of the extended affine Hecke algebra \eH_n oftype 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. Weshow that its left cells are naturally labeled by tableaux filled with positiveinteger entries having distinct residues mod n, which we term \emph{positiveaffine 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) withleft cells labeled by PAT that are essentially standard Young tableaux withcocharge labels. Multiplying canonical basis elements by a certain element \pi\in \pH_n corresponds to rotations of words, and on cells corresponds tococyclage. We further show that \R_{1^n} has cellular quotients \R_\lambda thatare \u-analogues of the Garsia-Procesi modules R_\lambda with left cellslabeled by (a PAT version of) the \lambda-catabolizable tableaux.  We give a conjectural description of a cellular filtration of \pH_n, thesubquotients of which are isomorphic to dual versions of \R_\lambda under theperfect pairing on \R_{1^n}. We conjecture how this filtration relates to thecombinatorics of the cells of \eH_n worked out by Shi, Lusztig, and Xi. We alsoconjecture that the k-atoms of Lascoux, Lapointe, and Morse and theR-catabolizable tableaux of Shimozono and Weyman have cellular counterparts in\pH_n. We extend the idea of atom copies of Lascoux, Lapoint, and Morse topositive affine tableaux and give descriptions, mostly conjectural, of some ofthese copies in terms of catabolizability.

Cyclage, catabolism, and the affine Hecke algebra