Positivity of Iwahori–Hecke algebra characters on the Kazhdan-Lusztig basis $\DeclareMathOperator\tr{tr}$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?
 A: This is a really good question. What is known of an answer is somewhat
complicated, and perhaps others have more useful things to say.
As you mention, the Hecke algebra $H$ is categorified by the Hecke category $\mathcal{H}$, which has one incarnation as Soergel bimodules. (See our recent book for much too much detail!) One might hope that all irreducible $H$-modules $M$ admit categorifications in the form of abelian (or additive, or triangulated, ...) $\mathcal{H}$-module categories $\mathcal{M}$.
One example where this happens is in (finite) type $A$, when $H$ is the Hecke algebra of the symmetric group. In this case the left cells already yield all irreducible $H$-modules (as was noticed in Kazhdan and Lusztig's first paper on their basis!), and one can obtain categorifications as a quotient (of additive categories):
$$\mathcal{M}_c := \mathcal{H}_{\le_L c}/\mathcal{H}_{<_L c}$$
for a fixed left cell $c$. A good explanation of this is provided by Lusztig's paper "On tensor categories associated to cells in affine Weyl groups". Here the literature is substantial, and encompasses papers on cells, Lusztig's $J$ ring a.k.a. the assymptotic Hecke algebra, etc.
As a quotient of additive Krull-Schmidt categories, the indecomposable objects in $\mathcal{M}_c$ are given by the images of the indecomposable objects in $\mathcal{H}_{\le_L c}$ which stay non-zero in the quotient. In particular, the structures constants are in $\mathbb{Z}_{\ge 0}[v,v^{-1}]$.
In general the same procedure produces categorifications $\mathcal{M}_c$ associated to any left cell. However in all other types these cell modules aren't (quite) irreducible. However, if one is willing to replace "irreducible module" with "cell module" then the answer to your question is positive (no pun intended!).
A more recent initiative asks for a classification of all $\mathcal{H}$-modules satisfying the appropriate adjectives. There has been progress on this recently, and contacting one of the authors of the beautiful paper is probably a good idea.
To finish, let me mention a fascinating link to unipotent character sheaves which probably deserves further study. There are many things parametrized by irreducible characters $\chi$ of $W$, a Weyl group. Two of which are irreducible characters $\chi_q$ of the Hecke algebra $H$, and irreducible unipotent characters sheaves in the principal series $\mathcal{V}_\chi$. There is is an amazing formula of Lusztig which reads something like:
$$
HC(\mathcal{IC}_w) = \bigoplus_{\chi, \chi'} \{ \chi, \chi' \} \chi'_q(C_w') \cdot \mathcal{V}_\chi 
$$
Where $HC$ denotes the Harish-Chandra transform, and $\{ \chi, \chi' \}$ is Lusztig's non-abelian Fourier transform. (This formula is somewhere in Lusztig's series on character sheaves. It was invaluable here.)
Anyway the moral is that the value $\chi_q'(C_w)$ almost (i.e. up to Lusztig's Fourier transform) measures the multiplicity of a unipotent character sheaf in the Harish-Chandra transform of an IC. This gives another proof that these polynomials are positive in type $A$ (as here Lusztig's Fourier transform is trivial).
A: Following the hint of Geordie Williamson, I contacted Daniel Tubbenhauer, one of the authors of the paper on categorification. With his agreement, I put here his answer:
"These are actually very interesting questions, in particular from the viewpoint of categorification.
Let me try to summarize what I know about your questions. First things first, we need to distinguish between "positive" and "positive integral" - the later is the one being reachable from the viewpoint of categorification. I can't say much about the "positive" case, but we will get there. (Basically, pi is positive, but I do not think it comes up "very natural" from a categorification point of view.)
Ok, let us start with your questions.
The second question is easy to answer: I certainly have never seen any reference discussing/calculating how the simple characters of KL basis elements look like - left aside positivity or positive integral properties. Obviously this doesn't imply that there are no references, but I can only help you with the calculations I did myself, some of which I explain below. However, there is one family of exceptions. As Geordie mentions, for anything related to cell representations there is a vast literature on the subject and positive integral properties, most notably via categorification. The recent adorable book on Soergel bimdoles in Geordie's answer covers a good part of this flavor of the theory.
Let me now try to answer your other two questions.
In any computation I ever made the simple characters of the KL basis acted positively. But this comes with a huge catch: I stopped doing these calculations very quickly since in everything except type A you will immediately hit a wall, namely non-integral coefficients turning up in the simple representations.
Here is an explicit example for the dihedral group of order 2n. This group (and its Hecke algebra, of course) has its natural, a.k.a. rotation, representation which is of dimension 2. In matrices, the two simple reflections $s,t$ act as
$$s \mapsto \begin{pmatrix} \cos(2\pi /n) & \sin(2\pi /n) \\
\sin(2\pi /n) & -\cos(2\pi /n) \end{pmatrix}, \; t \mapsto \begin{pmatrix} \cos(4\pi /n) & \sin(4\pi /n) \\
\sin(4\pi /n) & -\cos(4\pi /n) \end{pmatrix} .$$
There is basically "no way" that compositions of these have integral traces. And indeed, if you would feed these matrices into the corresponding sums defining the KL basis elements in these types (summarized in Elias' dihedral paper https://arxiv.org/abs/1308.6611), then you will see that these have non-integral characters. They have non-negative characters, you get something like $\frac{1}{2}(1+\sqrt{5})$ for $n=5$, for example. I do not know whether that is a "law of small numbers coincidence" whether this holds in general.
For the same reasons as above (non-integral coefficients), I do not have any answer to the question of how to categorify simple representations. With the current technology (which is mostly about natural or integral decategorifications) this seems to be out of reach. Sadly, my paper Geordie mentioned also doesn't help because decategorifications of simple 2-representations are not simple, again because I do not know how to categorify expressions such as $\frac{1}{2}(1+\sqrt{5})$. The decategorifications of the simple 2-representations are indecomposable representations over the integers, which, depending on your field, might decompose further into non-integral representation. The above is an example of such, it turns up if you decompose the cell representations for the middle cell in dihedral type.
"
A: 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.
