A lattice chain to manipulate

· The standard GLₙ model Definition

Let \(\Loc\) be a non-Archimedean local field, \(\Ring\) its valuation ring, \(\unif\) a uniformizer, and \(\resq\) the residue cardinality; fix \(\nDim\ge2\).

$$\Grp=\mathrm{GL}_{\nDim},\qquad X(\Loc)=\Loc^{\nDim},\qquad \Iw=\{g\in\mathrm{GL}_{\nDim}(\Ring):\bar g\text{ upper triangular}\}.$$

For the standard basis \(e_1,\ldots,e_{\nDim}\), define the linear automorphism \(\shift:\Loc^{\nDim}\to\Loc^{\nDim}\) by

$$\shift e_1=\unif e_{\nDim},\qquad \shift e_i=e_{i-1}\quad(2\le i\le\nDim).$$

For every \(\idx\in\mathbb Z\), write \(\idx=\nDim\period+\resid\), with \(\period\in\mathbb Z\) and \(0\le\resid<\nDim\), and set

$$\Lat_{\idx}=\shift^{\idx}\Ring^{\nDim}=\unif^{\period}\bigl(\Ring^{\nDim-\resid}\oplus(\unif\Ring)^{\resid}\bigr).$$

The index counts individual steps; \(\period\) counts complete periods.

· Two coordinates Example

$$\nDim=2,\qquad \shift=\begin{pmatrix}0&1\\\unif&0\end{pmatrix},\qquad \shift^2=\unif 1_2.$$$$\Lat_0=\Ring^2\supset\Lat_1=\Ring\oplus\unif\Ring\supset\Lat_2=\unif\Ring^2.$$

· Shift and stabilizer Proved here

$$\shift\Lat_{\idx}=\Lat_{\idx+1},\qquad \Lat_{\idx+\nDim}=\unif\Lat_{\idx}.$$ $$\Iw=\bigcap_{\idx\in\mathbb Z}\operatorname{Stab}(\Lat_{\idx}),\qquad\shift\Iw\shift^{-1}=\Iw.$$

Proof

1. On the basis vectors, a full cycle contributes one factor of the uniformizer: $$\shift^{\nDim}=\unif 1_{\nDim}.$$
2. Stabilizing \(\Lat_0\) means being integral and invertible; stabilizing its sublattices modulo \(\unif\) means preserving the standard flag: $$g\in\bigcap_{\idx}\operatorname{Stab}(\Lat_{\idx})\iff g\in\mathrm{GL}_{\nDim}(\Ring),\quad \bar g\text{ upper triangular}.$$
3. Shifting the index permutes the family of stabilizers: $$\shift\left(\bigcap_{\idx}\operatorname{Stab}(\Lat_{\idx})\right)\shift^{-1}=\bigcap_{\idx}\operatorname{Stab}(\Lat_{\idx+1}).$$