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Answering a question of Kuniba, Misra, Okado, Takagi, and Uchiyama, it isshown that certain Demazure characters of affine type A, coincide with thegraded characters of coordinate rings of closures of conjugacy classes ofnilpotent matrices. This entails a translation of the affine type A crystaltheory into the language of tableaux following Nakayashiki and Yamada, for thecase of tensor products of the classical crystals indexed by rectangularpartitions. In particular the explicit action of the zero-th crystal raisingoperator on the above crystals is given, and its direct connection with thegeneralized cocyclage on Littlewood-Richardson tableaux is explained.

Affine Type A Crystal Structure on Tensor Products of Rectangles, Demazure Characters, and Nilpotent Varieties