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We prove that a Weyl module for the current Lie algebra associated with asimple Lie algebra of type $ADE$ is rigid, that is, it has a unique Loewyseries. Further we use this result to prove that the grading on a Weyl moduledefined by the degree of currents coincides with another grading which comesfrom the degree of the homology group of the quiver variety. As a corollary weobtain a formula for the Poincar\'{e} polynomials of quiver varieties of type$ADE$ in terms of the energy functions defined on the crystals for tensorproducts of level-zero fundamental representations of the corresponding quantumaffine algebras.

Loewy series of Weyl modules and the Poincar\'{e} polynomials of quiver varieties