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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?

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  • $\begingroup$ I suppose your hermitian form is given by, e.g. $\langle x,y\rangle=x_1\bar{y}_n+...+x_n\bar{y}_1$. In this case the answer is yes. It is special because the reductive quotient is (split) $\mathrm{SO}(n)$, which has the same Weyl group (or same set of "roots up to homothety") as a quasi-split unitary group, at least in residue characteristic not $2$. p.s. There is a list somewhere in Tits' Corvallis article, which marks all (hyper)special parahorics. $\endgroup$ Mar 29, 2016 at 4:16
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    $\begingroup$ In Bruhat, Tits, Groupes réductifs sur un corps local I. Données radicielles valuées IHES (1972), they compute the explicit down-to-earth description of stabilizers of point of the apartment (for all classical groups). The result you want is the Corollaire (10.1.33). Translating that result into something really explicit is surely quicker than going through the description of the integral model. $\endgroup$ Mar 31, 2016 at 12:27

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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" , 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.

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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

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