Quantum Schur-Weyl duality and projected canonical bases

Let \H_r be the generic type A Hecke algebra defined over \ZZ[u, u^{-1}]. The Kazhdan-Lusztig bases {C_w}{w \in \S_r} and {C’w}{w \in \S_r} of \H_r give rise to two different bases of the Specht module M\lambda, \lambda \vdash r, of \H_r. These bases are not equivalent and we show that the transition matrix S(\lambda) between the two is the identity at u = 0 and u = \infty. To prove this, we first prove a similar property for the transition matrices \tilde{T}, \tilde{T}’ between the Kazhdan-Lusztig bases and their projected counterparts {\tilde{C}w}{w \in \S_r}, {\tilde{C}'w}{w \in \S_r}, where \tilde{C}w := C_w p\lambda, \tilde{C}‘w := C’w p\lambda and p\lambda is the minimal central idempotent corresponding to the two-sided cell containing w. We prove this property of \tilde{T},\tilde{T}’ using quantum Schur-Weyl duality and results about the upper and lower canonical basis of V^{\tsr r} (V the natural representation of U_q(\gl_n)) from \cite{GL, FKK, Brundan}. We also conjecture that the entries of S(\lambda) have a certain positivity property.