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

Abstract

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.