Lemma 5.1 · Orbits of $I$ on $L\mathbb A^n$: closures $\tau^k\mathcal O^n$, $\tau\in\Omega$

Orbits of $I$ on $L\mathbb A^n$: closures $\tau^k\mathcal O^n$, $\tau\in\Omega$

Proof.

1. (a) $\Lambda_{m,j}$ is spanned over $\mathcal O$ by $t^me_1,\dots,t^me_j,t^{m+1}e_{j+1},\dots,t^{m+1}e_n$. Applying $\tau$: $t^me_1\mapsto t^{m+1}e_n$, $t^me_i\mapsto t^me_{i-1}$ ($2\le i\le j$), $t^{m+1}e_i\mapsto t^{m+1}e_{i-1}$ ($i>j$). The images span $t^me_1,\dots,t^me_{j-1},t^{m+1}e_j,\dots,t^{m+1}e_n$, i.e. $\Lambda_{m,j-1}$. Since $\mathcal O^n=\Lambda_{0,n}$ and $k(m,j)$ increases by one when $j$ drops by one, with $k(m,0)=n(m+1)=k(m+1,n)$, induction gives $\tau^{k(m,j)}\mathcal O^n=\Lambda_{m,j}$; writing $k=nm+(n-j)$ with $0\le n-j\le n-1$ inverts $k$.

2. (b) $g\in G(F)$ stabilizing $\mathcal O^n$ lies in $K$; $g\in K$ stabilizes $\Lambda_{0,j}$ iff $g\bmod t$ preserves $\Lambda_{0,j}/t\mathcal O^n=\langle e_1,\dots,e_j\rangle$. So the stabilizer of the whole chain $\{\Lambda_{m,j}\}=\{t^m\Lambda_{0,j}\}$ is $\{g\in K: g\bmod t\in B\}=I$. By step 1, $\tau$ permutes the chain, hence $\tau I\tau^{-1}=I$, $I\tau I=\tau I$, and the $I$-orbit of $\tau I$ in $\mathrm{Fl}$ is a point: $\ell(\tau)=0$. Writing $\tau^k=t^\mu\sigma^k$ (diagonal times permutation), $t^\mu\mathcal O^n=\tau^k\mathcal O^n=t^{\lambda_{m,j}}\mathcal O^n$ forces $\mu=\lambda_{m,j}$, so $\tau^k\in t^{\lambda_{m,j}}W$.

3. (c) $K_1=1+tM_n(\mathcal O)\subseteq I$ and $I/K_1=B(k)$. For $v=t^mv_0$, $v_0$ primitive, $(1+tA)v=v+t^{m+1}Av_0$ and $A\mapsto Av_0$ is onto $\mathcal O^n$; so $K_1v=v+t^{m+1}\mathcal O^n$, and the $I$-orbits in $\{\operatorname{ord}=m\}$ are the preimages of the $B(k)$-orbits on $k^n\setminus0$ under $v\mapsto t^{-m}v\bmod t$. These are $B(k)e_j=\{w:w_j\ne0,\ w_i=0\ (i>j)\}$, with preimage $O_{m,j}$. Finally $v\in\Lambda_{m,j}$ lies outside $\Lambda_{m,j-1}$ iff $\operatorname{val}v_j=m$, i.e. iff $(\operatorname{ord}v,\operatorname{last}v)=(m,j)$. (This is Lemma 1 of classical-affine.html, reproved over an arbitrary field.)

4. (d), orbits and closures. $(1+t^{N+M}A)v-v\in t^{N+M}t^{-N}\mathcal O^n=t^M\mathcal O^n$, so $I_\alpha$ acts trivially. For $m<M$, $O_{m,j}$ is a union of cosets of $t^{m+1}\mathcal O^n\supseteq t^M\mathcal O^n$, so its image $o_k$ is an orbit with preimage $O_{m,j}$; every $v$ with $\operatorname{ord}v\ge M$, and $0$, maps to $0$. The quotient $\tau^k\mathcal O^n/\tau^{k+1}\mathcal O^n$ is one-dimensional (coordinate: the $t^m$-coefficient of $v_j$), so $\dim V_{\alpha,k}=nM-k$ and $o_k=V_{\alpha,k}\setminus V_{\alpha,k+1}$ is open dense in the irreducible $V_{\alpha,k}$.

5. (d), stabilizers. $H_\alpha=\{g\in M_n(\mathcal O/t^{N+M}):g_{ab}\in t\mathcal O\ (a>b),\ g_{aa}(0)\ne0\}$ is an open subset of an affine space cut out by $n$ non-vanishing conditions, hence irreducible and connected; it is the stabilizer of $0$. A point of $o_{k(m,j)}$ is $t^me_j\bmod t^M$; its stabilizer is given by the linear conditions $g_{aj}\equiv\delta_{aj}\bmod t^{M-m}$ on column $j$ (compatible with the Iwahori conditions since $M-m\ge1$), which force $g_{jj}(0)=1$, plus the $n-1$ open conditions $g_{aa}(0)\ne0$, $a\ne j$: a nonempty open subset of a linear space, hence connected.

Steps 1–3 hold over any field; the connectedness statements are for $k$ algebraically closed. Over $\mathbb F_q$ the same sets of rational points give the function-level orbits.