# Loewy Series of Weyl Modules and the Poincaré Polynomials of Quiver Varieties

### Ryosuke Kodera

Kyoto University, Japan### Katsuyuki Naoi

Kavli Institute for the Physics and Mathematics of the Universe, Kashiwa, Japan

## 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 Poincaré 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.

## Cite this article

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. 477–500

DOI 10.2977/PRIMS/77