TITLE: Positivity of Iwahori–Hecke algebra characters on the Kazhdan-Lusztig basis
QUESTION [10 upvotes]: $\DeclareMathOperator\tr{tr}$I'm interested in the irreducible characters of a finite Iwahori–Hecke algebra evaluated at the Kazhdan–Lusztig basis. These are Laurent polynomials.
Are the coefficients of these polynomials always positive?
In type $A_n$, the irreducible representations are given by left cells. Denote by $(C_w)$ the Kazhdan-Lusztig basis and by $(C^w)$ its dual basis with respect to the standard trace $\tr$. Take $\Lambda$ such a left cell and denote by $\chi_\lambda$ the associated character. It is known that $\chi_\lambda(h) = \tr(\sum_{x \in \Lambda}C_xC^xh)$. Hence
$$\chi_\lambda(C_w) = \sum_{x\in\Lambda} \tr(C_xC^xC_w)$$
which is positive since the expression equals a sum of structure constants in the KL-basis.
In the literature, one finds character tables, but always on the standard basis, not the KL-basis. Is there a reference where the character tables on the KL basis are computed?
Finally, most of the positivity properties in the Hecke algebras can be proven using the categorification with Soergel bimodules. Is there a categorification of the irreducible characters?

REPLY [3 votes]: I checked using the Chevie package of Gap3. It seems that for all types apart $A_n$, there are negative coefficients in some values of characters on the KL-basis.
The smallest example is type $B_2$. Some character value of $C'_{rsr}$ has negative coefficients.
So the first question seems only to work in type $A_n$ for which everything is understood.