This is a proposition in Lusztig's Intersection cohomology complexes on a reductive group (Invent. Math. 75, 205-272).
Proposition 6.3. The triple $(L,C_1, \mathcal{E}^{\cdot}_1)$ above is unqiuely determined (up to $G$-conjugacy) by $(C,\mathcal E^{\cdot})$. Moreover, $(C_1,\mathcal{E}^{\cdot}_1)$ is necessarily in $\mathcal N_L^{(0)}$.
For $(C,\mathcal E^{\cdot}) \in \mathcal N$, let $P \supseteq \tilde P$ be parabolic subgroups of $G$, and $L \supseteq \tilde L$ be Levi subgroups contained in $P$ and $\tilde P$ respectively. Let $C_1$ and $\tilde{C}_1$ be unipotent classes in $P/U_P$ and $\tilde P/U_{\tilde P}$ respectively. Let $\pi_P: P \rightarrow P/U_P$ and $\pi_{\tilde P}: \tilde P \rightarrow \tilde P/U_{\tilde P}$ be natural projections. Take $g$ and $\tilde g$ in $C_1$ and $\tilde{C}_1$ respectively. Let $\mathcal E^{\cdot}_1$ be an irreducible $P/U_P$-equivariant local system on $C_1$ and $\tilde{\mathcal E}^{\cdot}_1$ be an irreducible $\tilde P/U_{\tilde P}$-equivariant local system on $\tilde C_1$.
Let \begin{eqnarray} f_2 &:& \pi_P^{-1}(C_1) \cap C \rightarrow C_1, \nonumber \\ f_2' &:& \pi_{\tilde P}^{-1}(\tilde{C}_1) \cap C \rightarrow \tilde{C}_1, \nonumber \\ f_2'' &:& \pi_{\tilde P}^{-1}(\tilde{C}_1) \cap C_1 \rightarrow \tilde{C}_1\nonumber \end{eqnarray} where $f_2$ is the restriction of $\pi_P$, and $f_2',f_2''$ are the restrictions of $\pi_{\tilde P}$.
Let
(a) $H_c^{\dim C - \dim C_1}(\pi_P^{-1}(g) \cap C, \mathcal E^{\cdot}) \neq 0$ and $\mathcal E^{\cdot}_1$ is a direct summand of the local system $R^{\dim C - \dim C_1}(f_2)_!(\mathcal E^{\cdot})$,
(b) $H_c^{\dim C - \dim \tilde{C}_1}(\pi_{\tilde P}^{-1}(\tilde g) \cap C, \mathcal E^{\cdot}) \neq 0$ and $\tilde{\mathcal E}^{\cdot}_1$ is a direct summand of the local system $R^{\dim C - \dim \tilde{C}_1}(f_2')_!(\mathcal E^{\cdot})$,
(c) $H_c^{\dim C_1 - \dim \tilde{C}_1}(\pi_{\tilde P}^{-1}(\tilde g) \cap C_1, \mathcal E^{\cdot}_1) \neq 0$ and $\tilde{\mathcal E}^{\cdot}_1$ is a direct summand of the local system $R^{\dim C_1 - \dim \tilde{C}_1}(f_2'')_!(\mathcal E^{\cdot}_1)$.
Then it will follow from the proof of Proposition 6.3 that (a) and (c) imply the first part of (b). The second part of (b) also follows from the second parts of (a) and (b).
My problems is: do (a) and (b) imply (c)?
I think the answer is affirmative.
$R^{\dim C - \dim C_1}(f_2)_! (\mathcal E^{\cdot})$ is a $G$-equivariant local system on $C_1$.
If (a) holds, then the proof of Proposition 6.3 (for the uniqueness) shows that $R^{\dim C - \dim C_1}(f_2)_! (\mathcal E^{\cdot})$ is a direct sum of copies of $\mathcal E_1^{\cdot}$.
So the proof in Proposition 6.3 showing (a)+(c) implies (b) can also go conversely, showing that (a)+(b) implies (c).
But I am not sure about this reasoning.
Moreover, is it possible that (b) and (c) impy (a)?
I appreciate for any answers or comments. Thanks to everyone.
