I'm learning Deligne-Lusztig theory of complex characters of finite groups of Lie type and there are some difficulties for me in understanding the theory.

Let $\mathcal{G}$ be a simple simply-connected algebraic group over the algebraic closure of a finite field of characteristic $p$, and $F$ be a Frobenius endomorphism. Let $\mathcal{G^*}$ be a group in duality with $\mathcal{G}$ with corresponding Frobenius endomorphism $F^*$. (For example, we can take $\mathcal{G}=SL_n(\overline{F_p})$, $F:(a_{ij})\mapsto (a_{ij}^q)$, and $\mathcal{G^*}=PGL_n(\overline{F_p})$, where $q$ is a $p$-power).

Question: Is it true that for every $\mathcal{G^*}^{F^*}$-conjugacy class of a semisimple element $s\in\mathcal{G^*}^{F^*}$, there is a semisimple irreducible character $\chi_s \in \mathrm{Irr}(\mathcal{G}^F)$ of degree $[\mathcal{G^*}^{F^*}: C_{{\mathcal{G^*}}^{F^*}}(s)]_{p^{'}}$?

I wonder whether or not "the existence" of such a semisimple irreducible character depends on connectedness of $Z(\mathcal{G})$ or the centralizer $C_\mathcal{G^*}(s)$. I know that the aswer of the above question is affirmative when $Z(\mathcal{G})$ is connected. But I don't know whether it is true for groups with disconnected center, specially when $\mathcal{G}=SL_n(\overline{F_p})$.
It seems that the above question has been investigated in a classical paper of Lusztig: On the representations of reductive groups with disconnected center. But the paper seems to be unavailable on the web. So I would appreciate it if you provide an accessible resource.

Example: I saw the following argument in a paper recently (note that in the following argument $q$ is a power of $p$ and $\mathrm{gcd}(4, q-1)\neq 1)$:
enter image description here

As far as I checked, $C_{PGL_4(\overline{F_p})}(\overline{s})$ in the example above is disconnected! In fact, it has two connected components: one of them is $\frac{C_{GL_4(\overline{F_p})}({s})}{Z(GL_4(\overline{F_p}))}$ and the other one consisting of the images of matrices $\begin{bmatrix} 0&a&0&0\\ b&0&0&0\\0&0&0&c\\0&0&d&0 \end{bmatrix}$ under the canonical projection $GL_4 \rightarrow PGL_4$. This example shows that the question above might have affirmative answer even if $Z(\mathcal{G})$ and $C_{\mathcal{G^*}}(s)$ are disconnected. But this is just an example. I need a verified reference.

Update Description: I found out that I misunderstood the concepts of regular and semisimple characters. So I deleted my previous wrong arguments and rewrite the question in a more efficient format. Now I would be grateful for any help on the question above.


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