  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.
