2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70622We study the representations of the W-algebra W(g) associated to an arbitrary finite-dimensional simple Lie algebra g via the quantized Drinfeld-Sokolov reductions. The characters of irreducible representations with non-degenerate highest weights are expressed by Kazhdan-Lusztig polynomials. The irreduciblity conjecture of Frenkel, Kac and Wakimoto is proved completely for the "-" reduction and partially for the "+" reduction. In particular, the existence of the minimal series representations (= the modular invariant representations) of W(g) is proved.22 pages, fixed font problemsQuantum Algebra17B69, 17b56 (Primary) 81R10, 81T40 (Secondary)text