Does anyone know if there exists some natural way to interpret the Kazhdan-Lusztig C-basis in a categorification of the Hecke algebra ? The category of Soergel bimodules categorifies the C'-basis but it doesn't seem clear to me whether this or some other related structure (Rouquier complexes, sheaves...) could be adapted to categorify the C-basis instead ? I am currently working on some elements of the Hecke algebra which seem to have remarkable positivity properties when written in the C-basis (---> (-1) signs occur when writing them in the C'-basis).

The $C$ and $C'$-basis are defined in Kazhdan-Lusztig 79 :

For each element $w\in W$ there is a unique selfdual $C_w\in H$ such that

$$C_w=\sum_{y\leq w} \epsilon_y \epsilon_w q_w^{1/2} q_y^{-1} \bar{P_{y,w}} T_y$$

with $P_{y, w}\in\mathbb{Z}[q^{1/2}, q^{-1/2}]$ a polynomial in $q$ having degree at most $\frac{1}{2}(\ell(w)-\ell(y)-1)$ for $y\leq w$, $P_{w,w}=1$. The set $(C_w)_{w\in W}$ forms a basis, the $C$-basis of the Hecke algebra. 

For each element $w\in W$, there is a unique selfdual $C'_w\in H$ such that 

$$C'_w=q_w^{-1/2} \sum_{y\leq w} P_{y,w} T_y$$

with $P_{y, w}\in\mathbb{Z}[q^{1/2}, q^{-1/2}]$ a polynomial in $q$ having degree 
at most $\frac{1}{2}(\ell(w)-\ell(y)-1)$ for $y\leq w$ and $P_{w,w}=1$. The set $(C'_w)_{w\in W}$ forms a basis, the $C'$-basis of the Hecke algebra. 



the problem of applying the involution $j$ from 7.9 of Humphreys to my elements is that one gets (-1) signs before some coefficients of $C'$-elements