2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/126812We investigate the affine canonical basis and the monomial basis constructed in [LXZ] in Lusztig's geometric setting. We show that the transition matrix between the two bases is upper triangular with 1's in the diagonal and coefficients in the upper diagonal entries in N[v, v^{-1}]. As a consequence, we show that part of the monomial basis elements give rise to resolutions of support varieties of the affine canonical basis elements as simple perverse sheaves.24 pages. Title changed. The relation with Nakajima's conjecture clarified.Representation TheoryQuantum Algebra17B37, 16G20text