I am going through a book (Roberts and Schmidt, Local Newforms for GSp(4)), which states the following group decomposition. Let $F$ be a non-archimedean field, $\mathfrak{o}$ its ring of integers and $\mathfrak{p}$ its maximal ideal. Then, in terms of subgroups of $\mathrm{GSp}(4)$,
$$\left(
\begin{array}{cccc}
\mathfrak{o}&\mathfrak{o}&\mathfrak{o}&\mathfrak{o}\\
\mathfrak{p}^n&\mathfrak{o}&\mathfrak{o}&\mathfrak{o}\\
\mathfrak{p}^n&\mathfrak{o}&\mathfrak{o}&\mathfrak{o}\\
\mathfrak{p}^n&\mathfrak{p}^n&\mathfrak{p}^n&\mathfrak{o}
\end{array}
\right)
=
\left(
\begin{array}{cccc}
1& & & \\
\mathfrak{p}^n&1& & \\
\mathfrak{p}^n& &1& \\
\mathfrak{p}^n&\mathfrak{p}^n&\mathfrak{p}^n&1
\end{array}
\right)
\left(
\begin{array}{cccc}
\mathfrak{o}^\times&&&\\
&\mathfrak{o}&\mathfrak{o}&\\
&\mathfrak{o}&\mathfrak{o}& \\
& & &\mathfrak{o}^\times
\end{array}
\right)
\left(
\begin{array}{cccc}
\mathfrak{1}&\mathfrak{o}&\mathfrak{o}&\mathfrak{o}\\
&\mathfrak{1}& &\mathfrak{o}\\
& &\mathfrak{1}&\mathfrak{o}\\
& & &\mathfrak{1}
\end{array}
\right)
$$
where missing entries are zeros. I would like to understand this decomposition more generally, for I would like to apply that for other groups. Is there any general setting for this Iwahori factorization? (it seems similar to an $LU$ factorization, however is it always true and what are exactly the three groups appearing on the right?)
I got only partial answers on MSE hence I post the question here.
Best Answer
A good general setting can be found in "Bruhat-Tits 1", p.152, Propositions 6.4.9 and 6.4.48.
Bruhat, François; Tits, Jacques, Reductive groups over a local field, Publ. Math., Inst. Hautes Étud. Sci. 41, 5-251 (1972). ZBL0254.14017.
The special case of Moy-Prasad subgroups is in "Moy-Prasad 2", Theorem 4.2 (which refers to Bruhat-Tits above).
Moy, Allen; Prasad, Gopal, Jacquet functors and unrefined minimal $K$-types, Comment. Math. Helv. 71, No. 1, 98-121 (1996). ZBL0860.22006.
These are all more general than you need, but basically what you've got is this: You have a group $GSp_4$ defined over ${\mathbb Z}$ with a "standard" maximal torus. Let $\Phi$ be the set of roots. There's a concave function $f \colon \Phi \rightarrow {\mathbb R}$ (see Bruhat-Tits) which takes the value $n$ on the roots which locate your $p^n$'s and value $0$ on all other roots. This concave function gives your compact-open subgroup and the Iwahori decomposition you've given.
To be more precise, this concave function gives you a group $K$ generated by certain lattices in the root subgroups and the integer-points of the torus. Bruhat-Tits 1 tells you that $K$ has the Iwahori factorization property that you desire. But it doesn't tell you that $K$ equals the left side. For that last step, you can probably appeal to a fancier argument with group schemes (Bruhat-Tits 2) or you can check it directly. One containment is easy. The other requires a bit more work I think.
Oh - also see J.K. Yu's paper on smooth models for more on concave functions, compact opens, and group schemes.
Yu, Jiu-Kang, Smooth models associated to concave functions in Bruhat-Tits theory, Edixhoven, Bas (ed.) et al., Autour des schémas en groupes. École d’Été “Schémas en groupes”. Volume III. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-820-6/pbk). Panoramas et Synthèses 47, 227-258 (2015). ZBL1356.20018.