Crystal bases and two-sided cells of quantum affine algebras

Let g be an affine Kac-Moody Lie algebra. Let U⁺ be the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated to g. We construct a basis of U⁺ which is related to the Kashiwara-Lusztig global crystal basis (or canonical basis) by an upper-triangular matrix (with respect to an explicitly defined ordering) with 1’s on the diagonal and with above-diagonal entries in q^-1_s Z[q^-1_s]. Using this construction, we study the global crystal basis B(U^∼) of the modified quantum enveloping algebra defined by Lusztig. We obtain a Peter-Weyl-like decomposition of the crystal B(U^∼) (Th. 4.18), as well as an explicit description of two-sided cells of B(U^∼) and the limit algebra of U^∼ at q = 0 (Th. 6.44).