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

Let λ = ∑ i ∈ I 0 ​m i ϖ i, with m i ∈ℤ ⩾0 for i ∈ I 0 , be a level‐zero dominant integral weight for an affine Lie algebra ℊ over ℚ, where the ϖ i , i ∈ I 0 , are the level‐zero fundamental weights, and let B (λ) be the crystal of all Lakshmibai–Seshadri paths of shape λ. First, we give an explicit description of the decomposition of the crystal B (λ) 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 B 0 (λ) of B (λ) containing the straight line π λ as a specified subcrystal of the affinizationB( λ )cl^(with weight lattice P ) of the crystal B( λ )cl ≃ ⊗ i ∈ I 0( B( ϖ i )cl)⊗ m i(with weight latticeP cl ≃ P / ( Q δ ∩ P ) , where δ is the null root of ℊ ), which we studied in a previous paper ( Int. Math. Res. Not. 2005 (2005) 815–840).