semisimple support of character sheaves

Question

So the essential question is:

How should we think about, or if possible compute, the semisimple support of a cuspidal character sheaf?

For example, let $G=SL_2$. We have the cuspidal character sheaf $IC(\mathcal{O},\mathcal{F})[2]=\mathcal{F}[2]$ where $\mathcal{O}$ is the regular unipotent orbit and $\mathcal{F}$ is the non-trivial $G$-equivariant local system on it. It has semisimple support $s=\left(\begin{matrix}1&0\\0&-1\end{matrix}\right)\in G^{\vee}=PGL_2$. In other words $\mathcal{F}[2]\in\hat{G}_{\mathcal{L}}$ in Lusztig's original language where $\mathcal{L}$ is the unique order 2 local system on the maximal torus of $G$. Do we have any method/algorithm to compute this semisimple support in general without a case-by-case study?

The most elementary way to realize the $SL_2$ case is probably that the Deligne-Lusztig induction of $\theta_{\mathcal{L}}$ (which corresponds to $s\in G^{\vee}$ as in Deligne-Lusztig) contains two irreducible representations. The perverse sheaf $\mathcal{F}[2]$ then appears in the difference of their characters, again via function-sheaf correspondence. But this seems ad hoc as this decomposition relies implicitly on $G$ having disconnected center.

My apology for my ignorance if the answer is somewhere in the obvious literatures (e.g. [L1]-[L5]) that I fail to fully read. And thanks a lot for any answer and comment.

Answer 1

I will try to provide an answer to my own question, which really comes from $\S2$ of Characteristic varieties of character sheaves by Mirkovic and Vilonen. (I think Dongkwan Kim might have hinted me about this before; regrettably I did not understand.)

In the above example, consider $\pi:G\rightarrow G/N$ the natural projection. Let $\mathcal{F}[2]$ be as above. Consider $\mathcal{G}:=\pi^*\pi_*\mathcal{F}[2]$. On the closed Bruhat cell $\mathcal{G}$ is trivial, while on the open Bruhat cell it will be bi-$N$-invariant and thus comes from a (shifted) local system on $T$. That local system will then be $\mathcal{L}$. I don't know if one always get an irreducible local system (could this be an interesting question?) on a unique Bruhat cell, but at least some constituent on some Bruhat cell will give what is sought.