Pre-canonical bases on affine Hecke algebras

For any affine Weyl group, we introduce the pre-canonical bases. They are a set of bases $\{{\mathbf{N}}^i{\}}_{1\le i\le m+1}$ (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.