Graded Characters of Modules Supported in the Closure of a Nilpotent Conjugacy Class

This is a combinatorial study of the Poincaré polynomials of isotypic components of a natural family of graded GL(n)-modules supported in the closure of a nilpotent conjugacy class. These polynomials generalize the Kostka–Foulkes polynomials and are q -analogues of Littlewood–Richardson coefficients. The coefficients of two-column Macdonald–Kostka polynomials also occur as a special case. It is conjectured that these q -analogues are the generating function of so-called catabolizable tableaux with the charge statistic of Lascoux and Schützenberger. A general approach for a proof is given, and is completed in certain special cases including the Kostka–Foulkes case. Catabolizable tableaux are used to prove a characterization of Lascoux and Schützenberger for the image of the tableaux of a given content under the standardization map that preserves the cyclage poset.