In course of my research I met the following situation :
1) I have a bunch of open subgroup (so of finite index) in $\mathbb{GL}_{n}(\mathbb{Z}_p)$.
2) My groups arises naturally as stabilizers of sort of flags, namely :as stabilizers of a sequence of sublattices $L_i$ where $L_i/L_{i+1}$ is an $\mathbb{F}_p$ vector space.
3)I have a nice description of the orbits of vectors under each subgroup.
My question are:
What is the relation of such a setting with the theory of Bruhat-Tits? Does this theory unables me to have 3) once 1) and 2) are ensured (or more generally even)? Does this theory unable to control the orbits of any congruence subgroup(I mean finite index) of $\mathbb{GL}_n(\mathbb{Z}_p)$ in terms of some (nicely intelligible) combinatorial data? Is there a general theory of such finite index subgroups?
I have been pointed to this theory while exposing my research to other people, so I have started learning it, but it would be a great motivational guide to have answers to the above question, possibly with reference to the relevant literature (regarding the question not general books about Bruhat Tits theory I have already found them) or an explanation.
I thank you in advance!
Let $G=GL_n(\Bbb{Q}_p)$ and $K=GL_n(\Bbb{Z}_p)$. The groups you describe are (examples of) parahoric subgroups of $G$. The group $K$ contains a normal subgroup $K^1=K\cap(1+Mat_n(p\Bbb{Z}_p))$, and the quotient is $\bar{K}=GL_n(\Bbb{F}_p)$. You're describing, essentially, the inverse images under the projection $K/K^1\rightarrow\bar{K}$ of the stabilizers in $\bar{K}$ of flags of subspaces of $\Bbb{F}_p^n$, i.e. the inverse images in $K$ of parabolic subgroups of $\bar{K}$. This is the "naive" definition of a parahoric subgroup -- it works in this case, but in general it's hard to rigorously get going, because the point is that this construction only works once you already have $K$ -- itself a maximal parahoric subgroup of $G$.
Now letting $G$ be an arbitrary connected reductive group over some finite extension $F$ of $\Bbb{Q}_p$ (or even over a field of Laurent series over $\Bbb{F}_p$), Bruhat and Tits associate a certain Euclidean simplicial complex $\mathscr{B}$, which is the enlarged building of $G$. There is a canonical action of $G$ on $\mathscr{B}$ by similicial automorphisms. Given a simplex in $\mathscr{B}$, its isotropy group under this action is a parahoric subgroup. (Well, at least up to being careful about connectedness: the isotropy group is naturally the rational points of an integral group scheme; the rational points of the connected component of this scheme is the parahoric subgroup).
Returning to the case of $GL_n(\Bbb{Q}_p)$, a very convenient alternative definition of parahoric subgroups is via hereditary orders and lattice chains, which is precisely what you describe. This crops up a lot in the Bushnell--Kutzko theory for constructing representations of this group (but is nowhere near as deep). There is, for example, a short introduction to stabilizers of lattice chains in one of the early sections of the Bushnell--Henniart book on local Langlands for GL(2). Alternatively, there is an introductory set of notes by Bushnell and Kutzko entitled Supercuspidals of $GL_n$ which gives a more detailed look at such things for any $n$; I'm not sure how easy it will be to find online though.
As for your question on the orbits, how do you mean? Orbits of the parahoric subgroups under the action of which group? If it's the action of $GL_n(\Bbb{Q}_p)$, then this is done by looking at the actions on lattice chains, which essentially reduces things to consider actions of $GL_n(\Bbb{F}_p)$ on parabolic subgroups, which is pretty well understood.