TITLE: Positivity of Schur elements in Iwahori-Hecke algebras
QUESTION [6 upvotes]: I'm interested in finite Iwahori-Hecke algebras.
If $\mathcal{H}$ is such a Hecke algebra, defined over $\mathbb{Z}[q^{\pm 1/2}]$, and $\Lambda$ an irreductible representation, there is the notion of a Schur element $S_\Lambda$. Roughly speaking, to $\Lambda$ you can associate a central element whose matrix in $\Lambda$ is given by $S_\Lambda$ times the identity.
I computed several of these Schur elements in type $A_n$ and I noticed a remarkable positivity property: all the coefficients in $q$ seem to be positive (see below)! I looked up in the literature, but I don't find explicitely this property.
Is there a concrete reference for this positivity? Does it hold in other types?
Here a small list of Schur elements:

For $\mathfrak{sl}_2$:     $1+q$, $1+q^{-1}$
For $\mathfrak{sl}_3$:     $q+1+q^{-1}$, $1+2q+2q^2+q^3$, $1+2q^{-1}+2q^{-2}+q^{-3}$
For $\mathfrak{sl}_4$:     $[4]!$ (quantum factorial), $q^2+3q+4+3q^{-1}+q^{-2}$,  $q^{-1}+2+2q+2q^2+q^3$, ...

REPLY [2 votes]: Together with Maria Chlouveraki, we determined the answer:
The Schur elements are positive in all classical types $A_n, B_n$ and $D_n$ (use the formula of Theorem 4.3. in https://hal.archives-ouvertes.fr/tel-01411063/document, which uses generalized hook lengths). For dihedral groups $I_2(m)$, Theorem 8.3.4. in the book of Geck-Pfeiffer on characters of finite Coxeter groups and Hecke algebras shows that the Schur elements have negative coefficients for $m>4$. For the exceptional cases, by brute force, there are Schur elements with negative coefficients only in type $H_4, F_4$ and $E_8$.