Positivity conjectures for the canonical basis of the pseudo-centralizer

Let $\mathcal{H}$ be the Hecke algebra with standard basis $\{H_x\mid x\in W_{\mathrm{aff}}\}$ and Kazhdan–Lusztig basis $\{\underline{H}_x\mid x\in W_{\mathrm{aff}}\}$, and let $\mathcal{B}$ be its pseudo-centralizer with standard basis elements $B_w$ indexed by $w\in W_{\mathrm{aff}}^+$. Assume that the standard basis exists and that

$$B_w\in\mathcal{H}^{\leq\ell(w)}$$

for every $w\in W_{\mathrm{aff}}^+$. For the resulting canonical basis $\{\underline{B}_w\mid w\in W_{\mathrm{aff}}^+\}$, write

$$\underline{B}_w=\sum_{x\in W_{\mathrm{aff}}}Q_{x,w}\cdot\underline{H}_x,$$

and write its products as

$$\underline{B}_x\cdot\underline{B}_y=\sum_{z\in W_{\mathrm{aff}}^+}\mu_{x,y}^z\cdot\underline{B}_z.$$

Positivity conjectures. The Laurent polynomials $Q_{x,w}\in\mathbb{Z}[v,v^{-1}]$ have non-negative coefficients for all $x\in W_{\mathrm{aff}}$ and $w\in W_{\mathrm{aff}}^+$, and the Laurent polynomials $\mu_{x,y}^z\in\mathbb{Z}[v,v^{-1}]$ have non-negative coefficients for all $x,y,z\in W_{\mathrm{aff}}^+$. These conjectures extend Kazhdan–Lusztig positivity to the canonical basis of $\mathcal{B}$; their status is open, as the paper presents them as proposed positivity conjectures and leaves the underlying categorification as an open problem.