Let $L/K$ be ramified quadratic extension of local fields, and let characteristic of the residue field of $K$ be $2$. Let $mathbb{G}=SU_3$, $G=mathbb{G}(K)$. Let $text{val}$ be a valuation on $K$ so that $text{val}(K^times) = mathbb{Z}$ (and $text{val}(L^times) = frac{1}{2}mathbb{Z}$).
Following Tits 1.15 and 3.11, I have been trying to work out the parahoric subgroups of $G$ attached to the special vertices $nu_0$ and $nu_1$ in the building of $G$.
Firstly, I’ll start with a description of the root subgroups of $G$. I’m using a slightly different notation from Tits’. Let $$u_+(c,d) = begin{pmatrix} 1 & -bar{c} & d \ 0 & 1 & c \ 0 & 0 & 1 end{pmatrix},$$
with $bar{c}c+d+bar{d}=0$.
Similarly, $$u_-(c,d) = begin{pmatrix} 1 & 0 & 0 \ c & 1 & 0 \ d & -bar{c} & 1 end{pmatrix},$$
with $bar{c}c+d+bar{d}=0$.
We have the root subgroups $U_{pm a}(K) = { u_pm(c,d) text{ : } c,d in L }$ and $U_{pm 2a} = { u_pm(0,d) text{ : } d in L}$.
Tits later defines $delta = sup{text{val}(d) text{ : } d in L, , bar{d}+d+1=0}$. $delta=0$ in the unramified case and in the ramified, residue characteristic $pneq 2$ case. However, when $L/K$ is ramified with residue characteristic $2$, $delta$ is strictly negative.
From here, Tits finds the set of affine roots of $G$ as $$Big{pm a + frac{1}{2}mathbb{Z} +frac{delta}{2}Big} cup Big{pm 2a +mathbb{Z}+ frac{1}{2} + delta Big}.$$
Affine root subgroups are given by $$U_{pm a + gamma/2} = { u_pm(c,d) text{ : } text{val}(d) geq gamma},$$
$$U_{pm 2a+ gamma} = { u_pm(0,d) text{ : } text{val}(d) geq gamma}.$$
The special points $nu_0$ and $nu_1$ i the standard apartment are defined by $$a(nu_1)=frac{delta}{2}, , a(nu_0) = frac{delta}{2} + frac{1}{4}.$$
From here, one can find that $$G_{nu_1} = langle T_0, U_{a-frac{delta}{2}}, U_{-a+frac{delta}{2}}, U_{2a+frac{1}{2}-delta}, U_{-2a+frac{1}{2}+delta} rangle,$$
$$G_{nu_0} = langle T_0, U_{a-frac{delta}{2}}, U_{-a+frac{1}{2}+frac{delta}{2}}, U_{2a-frac{1}{2}-delta}, U_{-2a+frac{1}{2}+delta} rangle.$$
In 3.11, Tits takes a $lambda in L$ with $text{val}(lambda) = delta$, satisfying $lambda+bar{lambda}+1=0$ in a way such that $lambda varpi_L + overline{(lambda varpi_L)}=0$ for some uniformizer $varpi_L$ of the ring of integers $mathcal{O}_L$ of $L$.
In 3.11, Tits defines the lattices $$Lambda_{nu_1} = mathcal{O}_L oplus mathcal{O}_L oplus lambdamathcal{O}_L,$$
$$Lambda_{nu_0} = varpi_L^{-1}mathcal{O}_L oplus mathcal{O}_L oplus lambdamathcal{O}_L.$$ Let $P_{nu_1}$ and $P_{nu_0}$ be their respective stabilizers.
Tits then states that $G_{nu_i} = P_{nu_i} cap G_{nu_i}$ for $i=0,1$.
Here’s where my problem comes in.
Consider $G_{nu_1} = langle T_0, U_{a-frac{delta}{2}}, U_{-a+frac{delta}{2}}, U_{2a+frac{1}{2}-delta}, U_{-2a+frac{1}{2}+delta} rangle.$ The stabilizer of the lattice $Lambda_{nu_1}$ in $GL_3(L)$ has the form
$$begin{pmatrix} mathcal{O}_L & mathcal{O}_L & mathfrak{p}_L^{-2delta} \ mathcal{O}_L & mathcal{O}_L & mathfrak{p}_L^{-2delta} \ mathfrak{p}_L^{2delta} & mathfrak{p}_L^{2delta} & mathcal{O}_L end{pmatrix}.$$
Since $text{val}(delta) < 0$, intersecting this stabilizer with $G$ would give us a matrix roughly looking like
$$begin{pmatrix} mathcal{O}_L & mathfrak{p}_L^{-2delta} & mathfrak{p}_L^{-2delta} \ mathcal{O}_L & mathcal{O}_L & mathfrak{p}_L^{-2delta} \ mathfrak{p}_L^{2delta} & mathcal{O}_L & mathcal{O}_L end{pmatrix},$$
Presumeably, this would tell us that $$U_{a-frac{delta}{2}} = { u_+(c,d) text{ : } c,d in L, , text{val}(d) geq -delta textbf{ and } text{val}(c) geq -delta },$$
$$U_{-a+frac{delta}{2}} = {u_{-}(c,d) text{ : } c,d in L, , text{val}(d) geq delta textbf{ and } text{val}(c) geq 0 }.$$
Normally, one would expect that if $text{val}(d) = gamma$, then $text{val}(d) = frac{gamma}{2}$ or $frac{gamma}{2}+frac{1}{4}$, as whether $gamma in mathbb{Z}$ or just $frac{1}{2}mathbb{Z}$.
I cannot work out algebraically why we have these improved bounds on the valuation of $c$ for these affine root subgroups. I assume it involves some manipulation with $lambda$, but I am not making any progress.
Thank you
