Crystal bases and two-sided cells of quantum affine algebras

Abstract

Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra. Let $\mathbf{U}^+$ be the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated to $\mathfrak{g}$. We construct a basis of $\mathbf{U}^+$ 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 $q_s^{-1} \mathbf{Z}[q_s^{-1}]$. Using this construction, we study the global crystal basis $\mathscr{B}(\widetilde{\mathbf{U}})$ of the modified quantum enveloping algebra defined by Lusztig. We obtain a Peter-Weyl-like decomposition of the crystal $\mathscr{B}(\widetilde{\mathbf{U}})$ (Th. 4.18), as well as an explicit description of two-sided cells of $\mathscr{B}(\widetilde{\mathbf{U}})$ and the limit algebra of $\widetilde{\mathbf{U}}$ at $q=0$ (Th. 6.44).