2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/140916We prove a weak version of Lusztig's conjecture on explicit description of the asymptotic Hecke algebras (both finite and affine), and explain its relation to Lusztig's classification of character sheaves.19 pages, mistake in Section 5.4 is corrected, other minor corrections; Appendix addedRepresentation Theorytext