Pre-canonical bases on affine Hecke algebras

For any affine Weyl group, we introduce the pre-canonical bases. They are a set of bases $\{\mathbf{N}^i\}_{1\leq i \leq m+1}$ (where $m$ is the height of the highest root) of the spherical Hecke algebra that interpolates between the standard basis $\mathbf{N}^1$ and the canonical basis $\mathbf{N}^{m+1}$. The expansion of $\mathbf{N}^{i+1}$ in terms of the $\mathbf{N}^i$ is in many cases very simple and we conjecture that in type $A$ it is positive.