The character and specialness conjecture for Deligne–Lusztig cohomology

Let ${\bf w}$ be an $F$-root of ${\boldsymbol\pi}$, let ${\mathcal H}_q(w)$ be the specialization at $x^{1/a}\mapsto q^{1/a}$ of the cyclotomic Hecke algebra ${\mathcal H}(w)$, and let $\rho_w$ be the resulting virtual representation on $\sum_i(-1)^iH^i_c({\bf X}({\bf w}))$. Writing

$$\sum_i(-1)^iH^i_c({\bf X}({\bf w}))=\sum_{\lambda\in\operatorname{Irr}({{\bf G}^F})}a_\lambda\lambda,$$

let $\chi_\lambda$ be the corresponding virtual character of ${\mathcal H}_q(w)$, and call a representation special when its trace is, up to a scalar, the canonical symmetrizing trace form. Character and specialness conjecture. (i) The $\chi_\lambda$ generate the Grothendieck group of ${\mathcal H}_q(w)$ and are irreducible up to sign. (ii) The representation $\rho_w$ is special. The paper proves this only in selected cases, including some cases with ${\bf w}={\boldsymbol\pi}$, so the general conjecture remains open.