perverse cohomology sheaf ${^p\!H}^i
(K_w^{\mathcal{L}_0})$ for some $w\in W$ and some $i\in{{\mathbb{Z}}}$ (see [@L2a Def. 2.10]). Let $\epsilon \in{{\operatorname{Irr}}}(W)$. In analogy to the above definition of $R_\epsilon$, we formally define $$K_\epsilon^{\mathcal{L}_0}:=\frac{1}{|W|} \sum_{w\in W} \epsilon(w)
\sum_{i\in{{\mathbb{Z}}}} (-1)^{i+\dim G}\, {^p\!H}^i(K_w^{\mathcal{L}_0});$$ see [@L2c 14.10.3]. (We write $K_\epsilon^{\mathcal{L}_0}$ in order to avoid confusion with $R_\epsilon$ in \[r11\].) As in [@L2c 14.10.4], we also denote by $(A:K_\epsilon^{\mathcal{L}_0})$ the multiplicity of $A\in\hat{G}^{\text{un}}$ in $K_\epsilon^{\mathcal{L}_0}$ (in the appropriate Grothendieck group).

\[r14\] Now [@L2e Theorem 23.1] (see also the comments in [@L10 3.10]) provides a classification of $\hat{G}^{\text{un}}$ in terms of similar ingredients as in \[r11\]. Indeed, let the parameter set $X(W)$, the pairing $\{\;,\;\} \colon X(W)\times X(W)\rightarrow 
\overline{{{\mathbb{Q}}}}_\ell$ and the embedding ${{\operatorname{Irr}}}(W) \hookrightarrow X(W)$ be as above. Then there is a bijection $$\hat{G}^{\text{un}}\leftrightarrow X(W),\qquad A\leftrightarrow x_A,$$ such that $(A:K_\epsilon^{\mathcal{L}_0}) =\hat{\varepsilon}_{A}\{x_A,
x_\epsilon\}$ for any $A\in \hat{G}^{\text{un}}$ and $\epsilon\in{{\operatorname{Irr}}}(W)$. Here, we set $$\hat{\varepsilon}_K:=(-1)^{\dim G-\dim\text{supp}(K)}\qquad \mbox{for any
$K\in\mathcal{D}G$},$$ where $\mbox{supp}(K)$ is the Zariski closure of the set $\{g\in G\mid {{\mathcal{H}}}_g^i(K)\neq \{0\} \mbox{ for some $i$}\}$. (Cf.  [@L2c 15.11].) Here, ${{\mathcal{H}}}_g^i(K)$ are the stalks at $g\in G$ of the cohomology sheaves of $K$, for any $i\in {{\mathbb{Z}}}$.

Assume that $G$ is simple modulo its centre. Then, again, the bijection $\hat{G}^{\text{un}}\leftrightarrow X(W)$ 