  We prove that a Weyl module for the current Lie algebra associated with a
simple Lie algebra of type $ADE$ is rigid, that is, it has a unique Loewy
series. Further we use this result to prove that the grading on a Weyl module
defined 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 Poincar\'{e} polynomials of quiver varieties of type
$ADE$ in terms of the energy functions defined on the crystals for tensor
products of level-zero fundamental representations of the corresponding quantum
affine algebras.
