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Proceedings of the London Mathematical Society Advance Access originally published online on December 10, 2007
Proceedings of the London Mathematical Society 2008 96(3):582-622; doi:10.1112/plms/pdm034
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© 2007 London Mathematical Society

Crystal structure on the set of Lakshmibai–Seshadri paths of an arbitrary level-zero shape

Satoshi Naito and Daisuke Sagaki

Institute of Mathematics
University of Tsukuba
Tsukuba
Ibaraki 305-8571
Japan
naito@math.tsukuba.ac.jp

Received 14 November 2006.

Let Formula , with miisinZ≥0 for iisinI0, be a level-zero dominant integral weight for an affine Lie algebra g over Q, where the {varpi}i, iisinI0, are the level-zero fundamental weights, and let B({lambda}) be the crystal of all Lakshmibai–Seshadri paths of shape {lambda}. First, we give an explicit description of the decomposition of the crystal B({lambda}) into connected components and show that all the connected components are pairwise ‘isomorphic’ (up to a shift of weights). Second, we ‘realize’ the connected component B0({lambda}) of B({lambda}) containing the straight line {pi}{lambda} as a specified subcrystal of the affinization Formula (with weight lattice P) of the crystal Formula (with weight lattice Formula , where {delta} is the null root of g ), which we studied in a previous paper (Int. Math. Res. Not. 2005 (2005) 815–840).

2000 Mathematics Subject Classification 17B37, 05E99 (primary); 81R10, 81R50 (secondary).


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