𝔸ⁿ under GLₙ · 1

\(\Aff^n\): geometry and orbits

\(\Aff^n\) under \(\GL_n\) is the simplest spherical variety that is affine but not homogeneous, so it sits just outside the Theorem. This page computes its invariants, says what the extra point \(0\) adds, draws \(F^n\) in valuation coordinates, and finds its \(K\)-orbits, its \(I\)-orbits and its \(I\)-stable lattices.

Sections · arrows: “is used by”

SetupWhy is \(\Aff^n\) neither in the Theorem nor far from it?

\(G=\GL_n\) acts on column vectors \(X=\Aff^n\), \(B\) upper triangular. Then

\[\Xo=B\cdot e_n=\{v_n\ne0\},\qquad\Stab_B(e_n)=\{b:\ b_{in}=0\ (i\lt n),\ b_{nn}=1\}\ \text{(connected)},\] \[\Xb:=\Aff^n\smallsetminus\{0\}=G/H,\qquad H=\Stab_G(e_n)\supseteq N^-\ \text{(lower unitriangular)}.\]

So \(X\) is affine, smooth and spherical with connected generic \(B\)-stabilizers, but not homogeneous. Its open \(G\)-orbit \(\Xb\) is homogeneous and but only quasi-affine. The Theorem needs both properties at once [Sak, §1.1, §8], so it applies to neither.

Over \(F\): \(X(F)=F^n\), \(X(\cO)=\cO^n\), \(\phiz=\one_{\cO^n}\), \(K=\GL_n(\cO)\), \(I=\{k\in K:\ k\bmod\vpi\in B(\Fq)\}\) and \(K_1=1+\vpi\Mat_n(\cO)\subseteq I\). In the examples and pictures \(F=\Fqt\), \(\cO=\Fq[[t]]\) and \(\vpi=t\).

The invariantsWhat are \(\AX\), \(\LXp\), \(\WX\), \(\dX\) here?

All invariants are those of the open orbit \(\Xb\), computed with the recipes of the Geometry tab.

Table 1. The invariants of \(\Xb=\Aff^n\smallsetminus\{0\}\).
objectvaluehow
\(\PX\), \(\LevX\)\(P_{n-1,1}\), \(\GL_{n-1}\times\GL_1\)last row \((0,\dots,0,*)\)
\(\Xc(X)\), \(\AX\)\(\Z\epsilon_n\) (eigenfunction \(v_n\)), \(\Gm\)\(\LevX/(\LevX\cap H)\), \(\LevX\cap H=\GL_{n-1}\)
\(\LT\to\LX\)\(\lambda\mapsto\lambda_n\)onto
\(\Vc\), \(\LXp\)\(\Q\), \(\Z\)horospherical
\(\WX\)\(1\)horospherical
\(\AXs\subseteq\As\)\(\{(1,\dots,1,y)\}\)dual of \(\lambda\mapsto\lambda_n\)
\(\dXh\AXs\)\(\{(q^{\frac{n-2}2},\dots,q^{-\frac{n-2}2},\,y)\}\)\(\rho\) of \(\GL_{n-1}\)

This agrees with Ben-Zvi–Sakellaridis–Venkatesh, who treat \(\Aff^n\) with the right action of \(\GL_n\): “the only \(B\)-stable divisor is defined by \(x_1=0\) and its stabilizer is the parabolic \(P(X)=P_{1,n-1}\)”, with dual group \(\GX=\Gm\) and \(\SL_2\) “principal for the Levi of type \((1,n-1)\)” [BZSV, Ex. 4.3.12]. The principal \(\SL_2\) of \(\GL_{n-1}\) at \(\diag(q^{1/2},q^{-1/2})\) is the string \(q^{\frac{n-2}2},\dots,q^{-\frac{n-2}2}\) in the table.

The colour and the point \(0\)What does the embedding \(\Xb\subseteq\Aff^n\) add?

\(\Aff^n\) has one colour, \(D=\{v_n=0\}\) (colours), and one extra \(G\)-orbit, \(\{0\}\), of codimension \(n\). In Luna–Vust terms \(\Aff^n\) is the simple embedding of \(\Xb\) whose closed orbit is \(\{0\}\), with coloured cone spanned by the valuation of \(D\): \(v_D(v_n)=1\), the positive direction of \(\LX=\Z\). In the \(K\)-orbit picture this is the direction of the shells \(X_m\), \(m\to+\infty\), which accumulate at \(0\) ():

\[\overline{X_m}=\vpi^m\cO^n=\{0\}\sqcup\bigsqcup_{m'\ge m}X_{m'} .\]

A function in \(\Sw(\Aff^n)\) may be nonzero along this direction, as long as it is eventually constant there. That is the difference between \(\Sw(\Xb)\) and \(\Sw(\Aff^n)\).

The valuation pictureHow do we draw \(F^n\)?

For \(n=2\), draw \(v=(v_1,v_2)\) at \((a,b)=(\val v_1,\val v_2)\), a zero coordinate at \(\infty\). One point of the plane stands for every vector with those two valuations; the orbits of \(K\) and \(I\) turn out to be unions of such points.

The valuation plane with axes a = val v1 and b = val v2 running from -2 to 3 plus a row and column for infinity. The quadrant a, b at least 0 is shaded and labelled X(O) = O^2, where the basic function is 1; the rest of the plane is X(F) = F^2. A star at (infinity, infinity) is the zero vector. Four example vectors are plotted with their Laurent series: (t^-1 + 1, t^2) at (-1, 2), (1 + t, t) at (0, 1), (t^3, t + t^2) at (3, 1), and (0, 1 + t^2) at (infinity, 0).
\(F^2\) in valuation coordinates: \(\cO^2\) is the quadrant \(a,b\ge0\), where \(\phiz=1\); the origin is the corner \((\infty,\infty)\).

(order and last coordinate) Definition

For \(v\in F^n\smallsetminus0\),

\[\ord v:=\min_i\val v_i\in\Z,\qquad\operatorname{last}v:=\max\{i:\ \val v_i=\ord v\}\in\{1,\dots,n\}.\]

Equivalently \(\ord v\) is the largest \(m\) with \(v\in\vpi^m\cO^n\), so \(v=\vpi^{\ord v}u\) with \(u\) primitive (below), and \(\operatorname{last}v\) is the last nonzero coordinate of \(u\bmod\vpi\in\Fq^n\). An upper triangular matrix cannot move that coordinate, which is why \(\operatorname{last}\) is an \(I\)-invariant.

(primitive vector) Definition

\(u\in\cO^n\) is primitive if the following equivalent conditions hold:

\[u\notin\vpi\cO^n\iff\ord u=0\iff u\bmod\vpi\ne0\iff\exists i:\ u_i\in\cO^\times .\]

For \(n=2\): \((1+t,t)\), \((t,1+t)\), \((1,1)\) are primitive; \((t,t^2)\) and \((0,0)\) are not.

(\(n=2\))

At \((a,b)\): \(\ord v=\min(a,b)\), so the vectors of order \(m\) fill the hook \(\{\min(a,b)=m\}\), the rays \(\{b=m\le a\}\) and \(\{a=m\le b\}\) with corner \((m,m)\); \(\operatorname{last}v=2\) if \(b\le a\), and \(1\) if \(a\lt b\).

Table 2. The two invariants on six vectors.
\(v\)\((a,b)\)\(v=t^{\ord v}u\)\(u\bmod t\)\((\ord v,\operatorname{last}v)\)
\((t^{-1}+1,\ t^2)\)\((-1,2)\)\(t^{-1}(1+t,\ t^3)\)\((1,0)\)\((-1,1)\)
\((1+t,\ t)\)\((0,1)\)\((1+t,\ t)\)\((1,0)\)\((0,1)\)
\((t,\ 1+t)\)\((1,0)\)\((t,\ 1+t)\)\((0,1)\)\((0,2)\)
\((0,\ 1+t^2)\)\((\infty,0)\)\((0,\ 1+t^2)\)\((0,1)\)\((0,2)\)
\((t^3,\ t+t^2)\)\((3,1)\)\(t\,(t^2,\ 1+t)\)\((0,1)\)\((1,2)\)
\((t^2,\ t^2+t^3)\)\((2,2)\)\(t^2(1,\ 1+t)\)\((1,1)\)\((2,2)\)
The same valuation plane split by the dashed diagonal a = b: the region a < b above it is tinted orange and labelled last v = 1, the region b at most a on and below it is tinted blue and labelled last v = 2. Three hooks, L-shaped lines with corners at (-1,-1), (0,0) and (1,1) opening up and to the right, are the level sets ord v = -1, 0, 1. The four example points are labelled by their pair (ord v, last v): (-1,1), (0,1), (1,2) and (0,2).
\(\ord v=\min(a,b)\) is the hook through \((a,b)\); \(\operatorname{last}v=2\) on and below the diagonal, \(1\) above it.

\(K\)- and \(I\)-orbitsWhat are the cells of \(F^n\)?

(orbits) Proved here

The \(K\)-orbits on \(F^n\) are \(\{0\}\) and the shells \(X_m=\{\ord v=m\}=\vpi^m(\cO^n\smallsetminus\vpi\cO^n)\), \(m\in\Z\). The \(I\)-orbits are \(\{0\}\) and \(O_{m,j}=\{\ord v=m,\ \operatorname{last}v=j\}\), \(1\le j\le n\), so \(X_m=\bigsqcup_{j=1}^nO_{m,j}\). Every orbit other than \(\{0\}\) is open and compact.

Proof
  1. The \(K\)-orbits.
    1. \(\ord\) is \(K\)-invariant. For \(n=2\), write \(k=\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}\in K\), so \(\alpha,\beta,\gamma,\delta\in\cO\). Then \[kv=\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}\begin{pmatrix}v_1\\v_2\end{pmatrix}=\begin{pmatrix}\alpha v_1+\beta v_2\\\gamma v_1+\delta v_2\end{pmatrix},\] \[\val(\alpha v_1+\beta v_2)\ge\min(\val\alpha+\val v_1,\ \val\beta+\val v_2)\ge\min(\val v_1,\val v_2)=\ord v,\] and the same holds for \(\gamma v_1+\delta v_2\). So \(\ord(kv)\ge\ord v\). Applying this to \(k^{-1}\in K\) and \(kv\) gives \(\ord v\ge\ord(kv)\). For example, with \(k=\begin{pmatrix}1&1\\0&1\end{pmatrix}\) and \(v=(t^2,t)\), \[kv=\begin{pmatrix}t^2+t\\t\end{pmatrix}:\qquad(a,b)=(2,1)\ \mapsto\ (1,1),\qquad\ord=1\ \text{before and after}.\] For general \(n\) the argument is the same, with \((kv)_i=\sum_lk_{il}v_l\). So each \(X_m\) is \(K\)-stable, and so is \(\{0\}\).
    2. \(K\) is transitive on primitive vectors. Let \(u\) be primitive with \(u_i\in\cO^\times\), and let \(k=(e_1|\cdots|\widehat{e_i}|\cdots|e_n|u)\) have columns \(e_l\) (\(l\ne i\), in order) and then \(u\).
      • \(ke_n\) is the last column of \(k\), so \(ke_n=u\).
      • The entries of \(k\) lie in \(\cO\), so \(k\in K=\GL_n(\cO)\) if and only if \(\det k\in\cO^\times\).
      • Row \(i\) of \(k\) is \((0,\dots,0,u_i)\): each \(e_l\) with \(l\ne i\) has a \(0\) in row \(i\).
      • Expand \(\det k\) along row \(i\). The only nonzero entry is \(u_i\), in position \((i,n)\), so \[\det k=(-1)^{i+n}\,u_i\,\det k',\] where \(k'\) is \(k\) with row \(i\) and column \(n\) deleted. Deleting row \(i\) from the columns \(e_1,\dots,\widehat{e_i},\dots,e_n\) leaves \(e_1,\dots,e_{n-1}\in\cO^{n-1}\), so \(k'=1_{n-1}\) and \(\det k=\pm u_i\).
      • \(u_i\in\cO^\times\), so \(\det k\in\cO^\times\) and \(k\in K\).
      For \(n=2\) and \(u=(x,y)\): \[y\in\cO^\times:\ k=\begin{pmatrix}1&x\\0&y\end{pmatrix},\ \det k=y;\qquad x\in\cO^\times:\ k=\begin{pmatrix}0&x\\1&y\end{pmatrix},\ \det k=-x;\qquad k\begin{pmatrix}0\\1\end{pmatrix}=\begin{pmatrix}x\\y\end{pmatrix}.\] For \(n=3\), \(u=(x,y,z)\) with \(y\in\cO^\times\) (\(i=2\)): \[k=(e_1|e_3|u)=\begin{pmatrix}1&0&x\\0&0&y\\0&1&z\end{pmatrix},\qquad\det k=(-1)^{2+3}\,y\det\begin{pmatrix}1&0\\0&1\end{pmatrix}=-y.\]
    3. The other shells. \(X_m=\vpi^mX_0=K\,\vpi^me_n\), because the scalar \(\vpi^m\) commutes with \(K\).
    The quadrant O^2 with the smaller quadrant tO^2 inside it; the hook ord v = 0 between them is drawn in black and labelled primitive. Green arrows k and k' carry the base point e_2 at (infinity, 0) to v = (t, 1 + t) at (1, 0) and to v' = (1 + t, t) at (0, 1); a grey arrow labelled times t carries the corner (0,0) to (1,1), the corner of the grey hook X_1 = t X_0.
    Step 1 for \(n=2\): \(k=\begin{pmatrix}1&t\\0&1+t\end{pmatrix}\) carries \(e_2\) to \((t,1+t)\), \(k'=\begin{pmatrix}0&1+t\\1&t\end{pmatrix}\) carries it to \((1+t,t)\), and \(t\cdot\) carries \(X_0\) to \(X_1\).
  2. Reduction to \(\Fq^n\). Let \(r_m:X_m\to\Fq^n\smallsetminus0\), \(r_m(v)=\vpi^{-m}v\bmod\vpi\).
    1. \(r_m\) is \(I\)-equivariant: \(r_m(kv)=\bar k\,r_m(v)\) with \(\bar k=k\bmod\vpi\in B(\Fq)\). So each \(I\)-orbit lies over one \(B(\Fq)\)-orbit.
    2. The fibres of \(r_m\) are \(K_1\)-orbits: \(K_1v=v+\vpi^{m+1}\cO^n\). Given \(w\in\cO^n\) and \(v=\vpi^mu\), \(u_i\in\cO^\times\), let \(A\) have column \(i\) equal to \(wu_i^{-1}\) and all other columns zero; then \((1+\vpi A)v=v+\vpi^{m+1}w\).
    3. Conclusion. If \(r_m(w)=\bar b\,r_m(v)\) with \(\bar b\in B(\Fq)\), lift \(\bar b\) to \(b\in B(\cO)\subseteq I\); then \(w\in K_1bv\subseteq Iv\). So the \(I\)-orbits in \(X_m\) are the preimages of the \(B(\Fq)\)-orbits.
    The hook ord v = 0 with its blue horizontal ray and orange vertical ray. A dashed blue box encloses the horizontal ray away from the corner, labelled K_1 v: all of a at least 1, and a dashed orange box encloses the vertical ray, labelled K_1 v: all of b at least 1; the corner is marked as having both coordinates frozen. An arrow labelled v maps to v mod t leads to an inset showing F_q^2 minus 0 for q = 3: six blue dots with w_2 nonzero, the orbit of e_2, and two orange squares with w_2 = 0, the orbit of e_1.
    Step 2 for \(n=2\): \(K_1v=v+t\cO^2\) moves \(v\) along its ray and cannot leave it; \(B(\Fq)\) permutes the fibres over \(\{w_2\ne0\}\), and separately those over \(\{w_2=0\}\).
  3. The \(B(\Fq)\)-orbits on \(\Fq^n\smallsetminus0\). For \(w\ne0\) with last nonzero coordinate \(j\), \((bw)_i=\sum_{l\ge i}b_{il}w_l\) is \(0\) for \(i\gt j\) and \(b_{jj}w_j\ne0\) for \(i=j\), so \(j\) is preserved. Conversely the matrix equal to the identity except for column \(j\), which is \(w\), is upper triangular, invertible, and sends \(e_j\) to \(w\). So the orbits are \(B(\Fq)e_j=\{w_j\ne0,\ w_i=0\ (i\gt j)\}\), and their preimages are the \(O_{m,j}\).
  4. Each \(O_{m,j}\) is a union of cosets of \(\vpi^{m+1}\cO^n\) inside the compact set \(\vpi^m\cO^n\), hence open and compact.

(a) \(K\)-orbits

One hook per order.

(b) \(I\)-orbits

Blue horizontal: last \(=2\). Orange vertical: last \(=1\).

Interactive, \(n=2\): pick valuations or click a grid point. \(K\cdot v=X_m\) is the whole hook, \(I\cdot v=O_{m,j}\) one of its two rays; the corner belongs to \(O_{m,2}\), and \(0\) (★) is a separate orbit.

The chain of \(I\)-stable latticesWhich lattices does \(I\) preserve, and what lies between them?

(the chain of lattices) Definition

For \(m\in\Z\) and \(0\le j\le n\),

\[\Lambda_{m,j}:=\{v:\ \val v_i\ge m\ (i\le j),\ \val v_i\ge m+1\ (i\gt j)\}=g_{m,j}\cO^n,\] \[g_{m,j}:=\vpi^m\begin{pmatrix}1_j&0\\0&\vpi\,1_{n-j}\end{pmatrix}=\vpi^m\begin{pmatrix}1&&&&&\\&\ddots&&&&\\&&1&&&\\&&&\vpi&&\\&&&&\ddots&\\&&&&&\vpi\end{pmatrix}\quad(j\text{ entries }1,\ n-j\text{ entries }\vpi,\ \text{zeros elsewhere}),\] \[n=2:\qquad g_{m,2}=\vpi^m\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad g_{m,1}=\vpi^m\begin{pmatrix}1&0\\0&\vpi\end{pmatrix},\qquad g_{m,0}=\vpi^m\begin{pmatrix}\vpi&0\\0&\vpi\end{pmatrix},\] \[n=3:\qquad g_{m,2}=\vpi^m\begin{pmatrix}1&0&0\\0&1&0\\0&0&\vpi\end{pmatrix},\qquad g_{m,1}=\vpi^m\begin{pmatrix}1&0&0\\0&\vpi&0\\0&0&\vpi\end{pmatrix},\qquad g_{0,1}\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}x\\\vpi y\\\vpi z\end{pmatrix},\] \[\cdots\supset\Lambda_{m,n}=\vpi^m\cO^n\supset\Lambda_{m,n-1}\supset\cdots\supset\Lambda_{m,0}=\vpi^{m+1}\cO^n=\Lambda_{m+1,n}\supset\cdots\]

Modulo \(\vpi^{m+1}\cO^n\), \(\Lambda_{m,j}\) is \(\langle e_1,\dots,e_j\rangle\subseteq\Fq^n\): the chain inserts the standard flag between \(\vpi^{m+1}\cO^n\) and \(\vpi^m\cO^n\). \(K\) acts on this quotient through \(\GL_n(\Fq)\), which moves the flag; \(I\) acts through \(B(\Fq)\), its stabilizer. Each step \(\Lambda_{m,j}\supset\Lambda_{m,j-1}\) imposes \(\val v_j\ge m+1\) and removes the vectors with \((\ord v,\operatorname{last}v)=(m,j)\). In the plane, \(\Lambda_{m,j}\) is a quadrant with corner \((m,m)\) for \(j=2\) and \((m,m+1)\) for \(j=1\); going down the chain moves the corner alternately up and right along the diagonal, one ray at a time.

(\(n=2\), around \(\cO^2\))

\[\Lambda_{-1,1}=t^{-1}\cO\oplus\cO\ \supset\ \Lambda_{0,2}=\cO^2\ \supset\ \Lambda_{0,1}=\cO\oplus t\cO\ \supset\ \Lambda_{1,2}=t\cO^2\ \supset\ \Lambda_{1,1}=t\cO\oplus t^2\cO,\]

with corners \((-1,0)\), \((0,0)\), \((0,1)\), \((1,1)\), \((1,2)\). The last lattice containing \(v\) is \(\Lambda_{\ord v,\operatorname{last}v}\):

Table 3. Membership in the chain (– = not in).
\(v\) at \((a,b)\)\(\Lambda_{-1,1}\)\(\cO^2\)\(\Lambda_{0,1}\)\(t\cO^2\)\(\Lambda_{1,1}\)
\((t^{-1}+1,\ t^2)\) at \((-1,2)\)in––––
\((1+t,\ t)\) at \((0,1)\)ininin––
\((t,\ 1+t)\) at \((1,0)\)inin–––
\((t^3,\ t+t^2)\) at \((3,1)\)inininin–

The permutation matrix swapping \((1+t,t)\) and \((t,1+t)\) lies in \(K\) but not in \(I\), and \(\Lambda_{0,1}\) contains the first but not the second.

\(j\)
The lattice Λ(m,j) in the valuation plane of F²

Interactive: choose \(n\le12\), \(m\) and \(j\), or step along the chain. Light shading is \(\Lambda_{m,j}\), dark shading the next lattice, and the difference is one \(I\)-orbit; for \(n\ge3\) each row bounds one coordinate.

(\(I\)-stable lattices) Proved here

  1. For \(g\in G(F)\) and \(L=g\cO^n\): \(L\) is \(I\)-stable \(\iff I\subseteq gKg^{-1}\iff IgK=gK\).
  2. Every \(\Lambda_{m,j}\) is \(I\)-stable, and \(\Lambda_{m,j}\smallsetminus\Lambda_{m,j-1}=O_{m,j}\).
  3. The \(I\)-stable lattices are exactly the \(\Lambda_{m,j}\); the \(K\)-stable lattices are exactly the \(\vpi^m\cO^n\).
Proof
  1. \(kL=L\iff g^{-1}kg\,\cO^n=\cO^n\iff g^{-1}kg\in K\), since \(K\) is the stabilizer of \(\cO^n\). So \(L\) is \(I\)-stable iff \(g^{-1}Ig\subseteq K\), i.e. \(kg\in gK\) for all \(k\in I\), i.e. \(IgK=gK\).
  2. For \(k\in I\) and \(g=g_{m,j}\), the entries of \(g^{-1}kg\) are \(\vpi^{-e_a}k_{ab}\vpi^{e_b}\) with \(e_a=0\) for \(a\le j\) and \(1\) for \(a\gt j\). The only negative exponent occurs for \(a\gt j\ge b\), below the diagonal, where \(k_{ab}\in\vpi\cO\). So \(g^{-1}kg\in K\) and \(\Lambda_{m,j}\) is \(I\)-stable by (a). The difference is \(\{v\in\Lambda_{m,j}:\ \val v_j=m\}=O_{m,j}\).
    Three nested quadrants: O^2 with corner (0,0), Lambda_{0,1} = g O^2 with corner (0,1), and tO^2 with corner (1,1). The blue horizontal ray at b = 0 is labelled O_{0,2} = O^2 minus Lambda_{0,1}, the set val v_2 = 0; the orange vertical ray at a = 0, b at least 1, is labelled O_{0,1} = Lambda_{0,1} minus tO^2, the set val v_1 = 0. A note reads g^{-1} k g is in K for k in I, since k_21 is in tO.
    \(g=\begin{pmatrix}1&0\\0&t\end{pmatrix}\):\(g^{-1}kg=\begin{pmatrix}k_{11}&tk_{12}\\t^{-1}k_{21}&k_{22}\end{pmatrix}\in K\), because \(k_{21}\in t\cO\) for \(k\in I\); a quadrant minus the next one is one ray.
  3. Let \(L\) be \(I\)-stable and \(m\) the largest integer with \(L\subseteq\vpi^m\cO^n\); some \(v\in L\) has \(\ord v=m\).
    1. \(K_1\subseteq I\) and \(K_1v=v+\vpi^{m+1}\cO^n\) (proof of , step 2b), so \(\vpi^{m+1}\cO^n\subseteq L\).
    2. So \(L\) is determined by its image \(W\subseteq\vpi^m\cO^n/\vpi^{m+1}\cO^n\cong\Fq^n\), a nonzero \(B(\Fq)\)-stable subspace.
    3. Let \(j\) be the largest last nonzero coordinate of a vector of \(W\), and \(w\in W\) one attaining it. Then \(W\subseteq\langle e_1,\dots,e_j\rangle\) and \(W\supseteq B(\Fq)w=B(\Fq)e_j\ni e_j,\ e_j+e_i\ (i\lt j)\), which span \(\langle e_1,\dots,e_j\rangle\). So \(W=\langle e_1,\dots,e_j\rangle\), the image of \(\Lambda_{m,j}\), and \(L=\Lambda_{m,j}\).
    With \(K\) in place of \(I\), \(W\) is a nonzero \(\GL_n(\Fq)\)-stable subspace, hence \(\Fq^n\), and \(L=\vpi^m\cO^n\).

In the building of \(\GL_n\), the classes of \(\Lambda_{0,j}\), \(0\le j\le n-1\), up to homothety are the vertices of the alcove fixed by \(I\). The Iwahori page turns each \(\Lambda_{m,j}\) into a Hecke translate of \(\phiz\).