# Iwahori decomposition of general groups

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.

• You can find some statements by looking up "Iwahori factorization." – Not a grad student Sep 10 '18 at 9:41
• @CoffeeBliss That is exactly what does not work for me, maybe I do not search properly but all statement are either not understandable for me because far too general, or in specific settings that do not seem to apply here. In particular, I already found precise statements of Iwahori's factorization with p instead of p^n, however I do not know whether or not this is critical or just for convenience. – TheStudent Sep 10 '18 at 9:54

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.

It seems that such a decomposition holds over any local ring $R$. Indeed, the group on the left is the preimage of a parabolic subgroup $P(R/I)$ under the reduction map $\rho\colon G(R)\to G(R/I)$. Now by Levi decomposition $P(R/I)=L_P(R/I)\cdot U_P(R/I)$, where $L_P(R/I)$ is the (semisimple) Levi factor of $P(R/I)$ and $U_P(R/I)$ its unipotent radical. Denote $G(R,I)=\rho^{-1}(G(R/I))$, the principal congruence subgroup of level $I$, then $$\rho^{-1}(P(R/I))=\rho^{-1}(L_P(R/I))\cdot\rho^{-1}(U_P(R/I)) = L_P(R)\cdot U_P(R)\cdot G(R,I).$$ Now since $R$ is local, $G(R,I)$ can be decomposed into a finite product of factors of the form $U_P(I)$, $U_P^-(I)$ and $L_P(R,I)\leqslant L_P(R)$ (for example, see this paper or arXiv version), and collecting similar terms with the help of some of the factors being normalized by the others, the result follows.

UPD On second thought, this is even more simple if the parabolic subgroup has quasisimple Levi factor (its root system is irreducible), as in the case considered in the question. Then the entry in the upper left corner is invertible, and the factorization in question is a special case of Chevalley-Matsumoto decomposition (see Theorem 1.3 of [1]).