Quiver varieties and $t$-analogs of $q$-characters of quantum affine algebras

Abstract

We consider a specialization of an untwisted quantum affine algebra of type ${\rm {\rm ADE}}$ at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of “computable” polynomials, similar to Kazhdan-Lusztig polynomials. At the same time we “compute” $q$-characters for all simple modules. The result is based on “computations” of Betti numbers of graded/cyclic quiver varieties. (The reason why we use “ ” will be explained at the end of the introduction.)