see [@DiMi1 Prop. 6.4], [@L10a §3], [@first §4].

\[r12\] The “Fourier matrix” $\Upsilon:=
\bigl(\{x,y\} \bigr)_{x,y\in X(W)}$ is hermitian, and $\Upsilon^2$ is the identity matrix (see [@L1 4.14]). For each $x\in X(W)$, the corresponding unipotent “almost character” $R_x$ is defined by $$R_x:=\sum_{\rho\in{{\mathfrak{U}}}(G^F)} \delta_{\rho} \{x_\rho,x\}\rho;\qquad
\mbox{see \cite[4.24.1]{L1}}.$$ Note that $R_{x_\epsilon}=R_\epsilon$ for $\epsilon\in{{\operatorname{Irr}}}(W)$. For any $x,y \in X(W)$ we have $$\langle R_x,R_y\rangle=\left\{\begin{array}{cl} 1 & \qquad \mbox{if
$x=y$}, \\ 0 & \qquad \mbox{otherwise}.\end{array}\right.$$ Since $\Upsilon^2$ is the identity matrix, we obtain $$\rho=\delta_{\rho}\sum_{x\in X(W)} \{x,x_\rho\}R_x \qquad \mbox{for any
$\rho\in{{\mathfrak{U}}}(G^F)$}.$$ Thus, the problem of computing the values of $\rho\in{{\mathfrak{U}}}(G^F)$ is equivalent to the analogous problem for the unipotent almost characters $R_x$, $x\in X(W)$.

\[r13\] Let $\hat{G}$ be the set of character sheaves on $G$ (up to isomorphism). These are certain simple perverse sheaves in the bounded derived category $\mathcal{D}G$ of constructible $\overline{{{\mathbb{Q}}}}_\ell$-sheaves on $G$ (in the sense of Beilinson, Bernstein, Deligne [@bbd]), which are equivariant for the action of $G$ on itself by conjugation. For $w\in W$ let $K_w^{\mathcal{L}_0}\in\mathcal{D}G$ be defined as in [@L2a 2.4], where $\mathcal{L}_0=\overline{{{\mathbb{Q}}}}_\ell$ is the constant local system on the maximal torus $T$. Let $\hat{G}^{\text{un}}$ be the set of unipotent character sheaves, that is, those $A\in \hat{G}$ which are constituents of a 