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 $y\leq w$, $P\_{w,w}=1$. 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



Kazhdan-Lusztig C-basis and categorification
If you identify the Hecke algebra with the Grothendieck principal block of a graded lift of category $\mathcal{O}$ for the corresponding semi-simple Lie algebra (oddly, enough which of the two Langlands dual options you pick doesn't matter) such that $T\_y$ is the class of a Verma module with highest weight $-y\rho-\rho$ (so $T\_1$ corresponds to the anti-dominant Verma), then the $C'$-basis will match with the classes of tilting modules, and the $C$ basis with the simple modules (I tend to prefer matching the $C'$-basis with projectives, but then the $C$-basis is something horrible). 


I'm not really sure where this is written down properly; philosophically, the point is that everything will be fixed in place once you decide what functor to send the bar involution to. If you send it the contragredient dual, then simples and tiltings are obvious sets of modules fixed by this duality, and they satisfy the triangularities you've written above.


