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Consider $\mathrm{SO}(5)$, or maybe $\mathrm{SO}(n)$ over your favorite locally compact non-Archimedian field of characteristic $0$. There are two interesting families of compact open subgroups. The first is those coming from the Moy-Prasad filtration. The second are the stabilizers of lattices. My question is as follows: is it the case that a lattice stabilizer is a subgroup of an arbitrary step in the Moy-Prasad filtration?

I know the top of the Moy-Prasad filtration is a maximial parahoric, which is known (by Gan-Yu) to be a subgroup of the stabilizer of a lattice. But I don't know much more than that.

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Let $F$ be a non-archimedean locallly compact field (of odd residue characteristic). Let $G$ be a classical group attached to a non-degenerate (symmetric, symplectic, or hermitian) form on an $F$-space $V$. Let $G'={\rm Aut}_F(V)$, a general linear group. Then Bruhat and Tits proved (Bull. SMF 1987) that the building $X$ of $G$ embeds in the building $X'$ of $G'$ in a $G$-equivariant way, the image being the fixed point set of a certain involution. A lattice $L$ in $V$ corresponds to a point $x\in X\subset X'$. Therefore its stabilizer $G_L$ is the stabilizer of $x\in X$ in $G$. It follows that $G_L$ is non necessarily a parahoric subgroup of $G$, but that its connected component $G_L^0$ is a parahoric subgroup.

More generally we have the notion of a self dual lattice sequence in $V$. There are decreasing sequences of lattices satisfying a certain symmetry condition relative to the form defining $G$. To such a sequence $\Lambda$ one attaches :

-- a point $x=x_\Lambda$ in $X$,

-- a filtration of the Lie algebra of G and a filtration of the parahoric subgroup attached to $x$.

It turns out that these filtrations are nothing other than the filtrations defined by Moy and Prasad. The proof is due to Bertrand Lemaire:

Lemaire, Bertrand Comparison of lattice filtrations and Moy-Prasad filtrations for classical groups. J. Lie Theory 19 (2009), no. 1, 29--54.

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  • $\begingroup$ Hmm. It seems like the answer is "not really, but there is this other gadget that works". I'll see if I can get the other gadget to fit into some sort of global construction akin to the genus of a lattice, but I'm not optimistic. $\endgroup$ – Watson Ladd Feb 20 '18 at 18:57

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