2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61506Let us consider a specialization of an untwisted quantum affine algebra of type $ADE$ 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 put `` '' will be explained in the end of the introduction.)32 pages, The definition of the multiplication of the $t$--analog of the representation ring is corrected. Several ref's are addedQuantum AlgebraAlgebraic Geometry17B37; 14D21, 14L30, 16G20text