TITLE: What are the special parahoric subgroups in unitary groups?
QUESTION [8 upvotes]: Let $L$ be a $p$-adic field and let $L'/L$ be a quadratic extension.  Let $U_{L'/L}(n)$ be a quasi-split unitary group of $n\times n$ matrices with entries in $L'$.  I'm curious about what the special maximal subgroups of such a group are.
More specifically, let $\mathbf{K}$ be the subgroup consisting of unitary matrices whose entries lie in the ring of integers $\mathfrak{o}_{L'}$.  If the residue characteristic of $L'/L$ is not $2$, and $L'/L$ is unramified, then $\mathbf{K}$ is hyperspecial.  However, when the residue characteristic is $2$, then $\mathbf{K}$ is not even maximal when $n = 3$.
My question is:

When $L'/L$ is ramified and has residue characteristic not $2$, is $\mathbf{K}$ still maximal special (even though it is not hyperspecial)? If not, what are some of the special subgroups?
If the residue characteristic is $2$, when is $\mathbf{K}$ special? When it isn't, can we find a special subgroup containing it?

REPLY [2 votes]: If you can read french, explicit models of Bruhat-Tits buildings of classical groups over local fields (of any characteristic) are given in:
Bruhat, F.; Tits, J. Schémas en groupes et immeubles des groupes classiques sur un corps local. II : groupes unitaires. Bulletin de la Société Mathématique de France, 115 (1987), p. 141-195