Crystal bases and two-sided cells of quantum affine algebras

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Let be an affine Kac-Moody Lie algebra. Let + be the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated to. We construct a basis of + 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 qₛ⁻¹ [qₛ⁻¹]. Using this construction we study the global crystal basis () of the modified quantum enveloping algebra defined by Lusztig. We obtain a Peter-Weyl like decomposition of the crystal () (Theorem 4.18), as well as an explicit description of two-sided cells of () and the limit algebra of at q=0 (Theorem 6.45).