We prove that a Weyl module for the current Lie algebra associated with a simple Lie algebra of type ADE<\i> is rigid, that is, it has a unique Loewy series. Further we use this result to prove that the grading on a Weyl module dened by the degree of currents coincides with another grading which comes from the degree of the homology group of the quiver variety. As a corollary we obtain a formula for the Poincare polynomials of quiver varieties of type ADE<\i> in terms of the energy functions dened on the crystals for tensor products of level-zero fundamental representations of the corresponding quantum ane algebras.
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Ryosuke Kodera, Katsuyuki Naoi, Loewy Series of Weyl Modules and the Poincaré Polynomials of Quiver Varieties. Publ. Res. Inst. Math. Sci. 48 (2012), no. 3, pp. 477–500DOI 10.2977/PRIMS/77