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:



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> 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?
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> If the residue characteristic is $2$, when is $\mathbf{K}$ special? When it isn't, can we find a special subgroup containing it?
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> 



What are the special parahoric subgroups in unitary groups?
There's an explicit description of maximal compact subgroups of all unitary groups over local fields (not necessarily quasi-split) in section 3 of this paper:


Gan, Hanke, and Yu, ["On an exact mass formula of Shimura"](http://www.math.nus.edu.sg/~matgwt/Mass.pdf) , Duke Math Journal 107(1) (2001).


This doesn't quite answer your more specific question, since it's not immediately clear to me whether or not their maximal compacts are larger than the ones you consider; but it should be a feasible computation.


