I would like to count the size of a stabilizer of an arbitrary flag $\mathcal F$ in $Sp_n$ over a finite field $\mathbb F_q$. I am so far basing my attempt to count off of Paul Garrett's book on buildings, which has provided me with the guidance that counting the size of this is equivalent to counting the size of the Levi component and the unipotent radical, since $$P=M\rtimes R_uP$$ where $P$ is the parabolic subgroup stabilizing $\mathcal F$, $M$ is the Levi component, and $R_uP$ is the unipotent radical.
To begin with, I am trying to just count the size of $P$ for a flag with just one vector space, which, as outlined in his book, consists of matrices with entries like this. But then it says that there are further conditions which must be satisfied for the matrix to lie in $Sp_n$, which I am assuming are the relations described for any element of this group, which he gives earlier in the chapter, for a matrix $(a,b \backslash c, d)$ $$c^t a-a^t c = 0 \quad d^t b-b^t d = 0 \quad d^ta-b^tc=1$$ How do I count the number of matrices with entries that satisfy this criteria? If someone knows of a better way to do this with the Levi and unipotent radical components, then that would be appreciated as well. The standard Levi component seems to be easier to count, as $$|M|=|GL_l||SP_{n-l}|$$ but the unipotent radical gives me the same difficulty to count. Any help would be appreciated.
