I need some information about centralizer of involutions in finite simple groups of lie type. Actually I want to know if $G$ is a simple group of lie type over a finite field,
How many conjugacy classes of involution does it have?
If there are more than one, what is their sizes?
I would be grateful, if someone answers these question or introduce a good reference.
There is a table in GLS vol 3, page 172, but I haven't too much luck getting the information out of it. In the low rank cases that I've tried, its been hard to match things up nicely. Also I don't think the table is enough to compute the exact size.
Example of using the GLS table
For example: $\operatorname{Aut}(\operatorname{PSL}(3,5)) = A_2(5)$ has involutions $t_1$ and $\gamma_1$ (every other row in the table does not apply). The Out-class of $t_1$ is determined by the 2-adic valuation of $q-\epsilon$ where $\epsilon=1$ indicates PSL instead of PSU. The 2-adic valuation of $q-\epsilon=5-1=4$ is 2 which is larger than the 2-adic valuation of $n=3$ (which is 0). Hence the Out-class of $t_1$ is "1", inner, as in, $t_1$ is in PSL itself, not just Aut(PSL). The Out-class of $\gamma_1$ is determined by the residue of $m$ and $q$ mod 4: now $m\not\equiv -1 \mod 4$ does not apply, but $q\equiv \epsilon \mod 4$ does ($5\equiv 1 \mod 4$), so we are in the first case "g", and $\gamma_1$ is the graph automorphism $x \mapsto x^{-T}$. In particular, it is not inner, so we probably don't care about it.
Now the centralizer of $t_1$ has $5'$-residual $A^\epsilon_{m-1}(q) = A_1(5)$ -- the version is "u" for universal, which I believe means $\operatorname{SL}(2,5)$. It may also have extra factors of $2$ associated with it. In this example, it has one such factor, the centralizer is $\operatorname{GL}(2,5)$. (In unrelated examples I've done, these extra factors can be predicted, but are complicated; if one chooses the "right" group instead of the simple one, then everything is fine.) I believe no extra complications arise in this particular example. At any rate, $\operatorname{PSL}(3,5)$ contains exactly one class of involutions, and the centralizer of an involution is isomorphic to $\operatorname{GL}(2,5)$.
Machine calculations
Here are some explicit results for low rank groups over small fields of odd characteristic:
Some of the groups have a single class of involution, but typically the number of classes of involutions is about half the Lie rank for the large rank groups. I list each class on its own line. Not all classes exist in all fields. The table in the book also includes classes in the automorphism group. I've been able to accurately count how many classes are in the simple group from the table, but the centralizer structures have not matched up for me.