the group of rational points $G^F=G({{\mathbb{F}}}_q)$ is a finite group of Lie type of “split type”.

\[r11\] For each $w\in W$, let $R_w$ be the virtual character of $G^F$ defined by Deligne–Lusztig [@DeLu §1]. Let ${{\mathfrak{U}}}(G^F)$ be the set of unipotent characters of $G^F$, that is, $${{\mathfrak{U}}}(G^F)=\{\rho\in{{\operatorname{Irr}}}(G^F)\mid \langle \rho,R_w\rangle\neq 0\mbox{ for 
some $w\in W$}\}.$$ Now [@L1 Main Theorem 4.23] provides a classification of ${{\mathfrak{U}}}(G^F)$ in terms of

-   a parameter set $X(W)$ and a pairing $\{\;,\;\} \colon X(W)\times 
    X(W)\rightarrow \overline{{{\mathbb{Q}}}}_\ell$ (which only depend on $W$),

-   an embedding ${{\operatorname{Irr}}}(W) \hookrightarrow X(W)$, $\epsilon\mapsto 
    x_\epsilon$.

(See [@L1 4.21] for precise definitions; recall that we assume that $F$ acts trivially on $W$). Indeed, there is a bijection $${{\mathfrak{U}}}(G^F)\leftrightarrow X(W),\qquad \rho\leftrightarrow x_\rho,$$ such that for any $\rho\in{{\mathfrak{U}}}(G^F)$ and any $\epsilon\in{{\operatorname{Irr}}}(W)$, we have $$\langle \rho,R_\epsilon\rangle=\delta_\rho\{x_\rho,x_\epsilon\}\qquad
\mbox{where} \qquad R_\epsilon:=\frac{1}{|W|} \sum_{w\in W} \epsilon(w)R_w.$$ Here, for $\rho\in{{\mathfrak{U}}}(G^F)$, we define a sign $\delta_\rho=\pm 1$ by the condition that $\delta_\rho D_G(\rho)\in{{\mathfrak{U}}}(G^F)$, where $D_G$ denotes the duality operator on the character ring of $G$; see [@L1 6.8]. Note that [@L1 6.20] identifies $\delta_\rho$ with the sign $\Delta(x_\rho)$ appearing in the formulation of [@L1 4.23].

\[r11a\] Assume that $G$ is simple modulo its centre. The bijection ${{\mathfrak{U}}}(G^F)\leftrightarrow X(W)$ in \[r11\] is not 