Cyclage, Catabolism, and the Affine Hecke Algebra

We identify a subalgebra Ĥ + n of the extended affine Hecke algebra Ĥn of type A. The subalgebra Ĥ + n is a u-analogue of the monoid algebra of Sn⋉Z n ≥0 and inherits a canonical basis from that of Ĥn. 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 R1n of Ĥ + n that is a u-analogue of the ring of coinvariants C[y1, . . . , yn]/(e1, . . . , en) with left cells labeled by PAT that are essentially standard Young tableaux with cocharge labels. Multiplying canonical basis elements by a certain element π ∈ Ĥ + n corresponds to rotations of words, and on cells corresponds to cocyclage. We further show that R1n has cellular quotients Rλ that are u-analogues of the Garsia-Procesi modules Rλ with left cells labeled by (a PAT version of) the λ-catabolizable tableaux. We give a conjectural description of a cellular filtration of Ĥ + n , the subquotients of which are isomorphic to dual versions of Rλ under the perfect pairing on R1n . This turns out to be closely related to the combinatorics of the cells of Ĥn worked out by Shi, Lusztig, and Xi, and we state explicit conjectures along these lines. We also conjecture that the k-atoms of Lascoux, Lapointe, and Morse [9] and the Rcatabolizable tableaux of Shimozono and Weyman [20] have cellular counterparts in Ĥ + n . We extend the idea of atom copies from [9] to positive affine tableaux and give descriptions, mostly conjectural, of some of these copies in terms of catabolizability.