Geometric pieces and rational points

Setup and action

Let \(n\ge1\), \(k=\overline{\mathbb F}_q\), \(\mathcal O=k[[t]]\), \(F=k((t))\), \(G=\mathrm{GL}_n\), \(T\) diagonal and \(B\) upper triangular. The action is \(G\times\mathbb A^n\to\mathbb A^n\), \((g,v)\mapsto gv\), on column vectors; \(e_i\in F^n\) is the standard column basis and \(\varepsilon_i\in\mathbb Z^n=X_*(T)\) the standard cocharacter basis.

$$I(R)=\{g\in\mathrm{GL}_n(R[[t]]):g\bmod t\in B(R)\},\qquad LX(R)=R((t))^n,$$ $$t^\lambda=\operatorname{diag}(t^{\lambda_1},\ldots,t^{\lambda_n}),\quad \tau e_1=te_n,\quad \tau e_i=e_{i-1}\ (i>1),\quad L_a(R)=\tau^aR[[t]]^n.$$

Here \(R\) is any \(\mathbb F_q\)-algebra, \(LX(k)=F^n\), and \(L_a(k)=\tau^a\mathcal O^n\). Geometric objects are base-changed to \(k\) for sheaves; all formulas have models over \(\mathbb F_q\).

Are we working with points?

Geometric statements use schemes; counting statements use rational points.

$$Y_{N,M}(R)=t^{-N}R[[t]]^n/t^MR[[t]]^n,\qquad Y_{N,M}\cong\mathbb A^{n(N+M)}_{\mathbb F_q},\qquad N,M\ge0,\ N+M\ge1.$$ $$v_i=\sum_{a=-N}^{M-1}x_{a,i}t^a,\qquad H_{N,M}(R)=\{g\in\mathrm{GL}_n(R[t]/t^{N+M}):g\bmod t\in B(R)\}.$$

For \(k=nm+n-j\), \(1\le j\le n\), define the closed linear subspace \(V_k\) by \(x_{a,i}=0\) for \(a<m\), and \(x_{m,i}=0\) for \(i>j\). Its open subscheme \(o_k\) is defined by \(x_{m,j}\ne0\), meaning this coefficient is invertible on \(R\)-points.

$$o_k=V_k\setminus V_{k+1}\cong\mathbb G_m\times\mathbb A^{nM-k-1},\quad o_{nM}=V_{nM}=\{0\},\quad P_k:=Y_N\times_{Y_{N,M}}o_k.$$

Thus \(o_k(k)\) means geometric points and \(o_k(\mathbb F_q)\) means rational points; \(P_k\) is the scheme-theoretic inverse image, whose \(k\)-points are the lattice shell in \(F^n\).

Why the vector calculation proves the geometry

  1. Equations. The displayed vanishing equations and one inverted coefficient define the entire stratum over \(\mathbb F_q\), and give its product decomposition, dimension and closure.
  2. An algebraic orbit map. For \(v\in o_k(R)\), take its unique truncated polynomial representative and let \(g(v)\) have column \(j\) equal to \(t^{-m}v\), and all other columns equal to the identity columns, in \(R[t]/t^{N+M}\). $$g(v)\in H_{N,M}(R),\qquad g(v)t^me_j=v\pmod{t^M}.$$ The constant diagonal entry \(g_{jj}(0)=x_{m,j}\) is a unit, and entries below it have zero constant coefficient, so this is a regular section \(o_k\to H_{N,M}\) of the orbit map. The action preserves \(V_k\) and its hyperplane complement. Hence \(o_k\cong H_{N,M}/\operatorname{Stab}(t^me_j)\) as a homogeneous space, not just as a set of geometric points.
  3. Stabilizer schemes. The equations \(g_{aj}\equiv\delta_{aj}\pmod{t^{M-m}}\) are linear coefficient equations; the remaining invertibility conditions give \(\mathbb G_m^{n-1}\times\mathbb A^b\). This is a smooth connected variety; the product description is as varieties, not necessarily as groups.
  4. Pull back. \(P_k=\pi^{-1}_{N,M}(o_k)\) and \(L_k=\pi^{-1}_{N,M}(V_k)\) are identities of spaces from the same coefficient equations. The last piece \(L_{nM}\) contains zero and infinitely many loop orbits.

Sufficiency for generation. These geometric strata, smooth linear closures, and connected stabilizers are precisely the inputs to Proposition 4a: they classify equivariant simple sheaves; Proposition 4b then proves generation by finite length and perverse truncation. Counting \(\mathbb F_q\)-points alone supplies none of those geometric assertions.

Étale sheaf maps can be tested on all geometric stalks, which also requires the scheme and its étale neighborhoods: Stacks Project, §59.29, Definition 59.29.1 (checked 22 October 2026).

The finite-level stratification · \(\mathcal P_{N,M}\)

Why orbits. Both proofs run on the pieces \(P=\pi_{N,M}^{-1}(o)\), \(o\) an \(H_{N,M}\)-orbit on \(Y_{N,M}\):

  1. the simple objects of \(\mathcal C_I\) are the \(\mathrm{IC}(P)\) (Proposition 4a, used in §6);
  2. the indicators \(\mathbf 1_{P(\mathbb F_q)}\) are a basis of the functions \(C_I^{(N,M)}\) (used in §7).

I2 (smooth connected stabilizers) gives one simple object per piece, since every equivariant local system on \(o\) is constant, and, by Lang, one indicator per piece.

Definition.

$$\mathcal P_{N,M}=\{\pi_{N,M}^{-1}(o):\ o\ \text{an }H_{N,M}\text{-orbit on }Y_{N,M}\},\qquad P'\le P\iff P'\subseteq\overline P .$$

Proposition 3a · Geometric pieces of level \((N,M)\)

Let \(N,M\ge0\), \(N+M\ge1\). For \(-nN\le k\lt nM\) write \(k=nm+n-j\) with \(1\le j\le n\). Then:

  1. Pieces. \(\mathcal P_{N,M}=\{P_{-nN},\dots,P_{nM}\}\), \(|\mathcal P_{N,M}|=n(N+M)+1\), with $$P_k=L_k\setminus L_{k+1}\ \ (k\lt nM),\qquad P_{nM}=T_M=L_{nM}=\pi_{N,M}^{-1}(0).$$
  2. Closures. \(\overline{P_k}=L_k\), so the order is total: \(P_{-nN}>P_{-nN+1}>\cdots>P_{nM}\).
  3. Finite level. The orbits are \(o_k=\pi_{N,M}(P_k)\ni t^me_j\) and \(o_{nM}=\{0\}\), with $$\overline{o_k}=V_k:=L_k/L_{nM},\qquad\dim V_k=nM-k,\qquad o_k=V_k\setminus V_{k+1}\quad(k\lt nM).$$
  4. Smooth connected stabilizers. For every \(R\), \(\operatorname{Stab}(t^me_j)(R)=\{g\in H_{N,M}(R):g_{aj}\equiv\delta_{aj}\bmod t^{M-m}\ \text{for all }a\}\). As varieties, \(\operatorname{Stab}(t^me_j)\cong\mathbb G_m^{n-1}\times\mathbb A^b\); \(\operatorname{Stab}(0)=H_{N,M}\) is also smooth connected. This gives I2.
  5. Refinement. Raising \(M\) splits the closed piece; raising \(N\) adds \(n\) open pieces: $$\begin{aligned}M\to M+1:&\quad T_M=P_{nM}\sqcup\cdots\sqcup P_{nM+n-1}\sqcup T_{M+1},\\ N\to N+1:&\quad Y_{N+1}\setminus Y_N=P_{-nN-n}\sqcup\cdots\sqcup P_{-nN-1}.\end{aligned}$$
Table 2. The pieces of level \((N,M)\).
\(n\)pieces on \(k\)-points\(|\mathcal P_{N,M}|\)
1\(P_k(k)=t^k\mathcal O^\times\) \((-N\le k\lt M)\), \(P_M(k)=t^M\mathcal O\)\(N+M+1\)
2\(P_{2m}(k)=\{\operatorname{val}v_2=m\le\operatorname{val}v_1\}\), \(P_{2m+1}(k)=\{\operatorname{val}v_1=m\lt\operatorname{val}v_2\}\) \((-N\le m\lt M)\), \(P_{2M}(k)=t^M\mathcal O^2\)\(2(N+M)+1\)
\(n\)\(O_{m,j}\) \((-N\le m\lt M,\ 1\le j\le n)\), \(t^M\mathcal O^n\)\(n(N+M)+1\)
Proof
  • (a) Orbits. For \(\operatorname{ord}v=m\), \(K_1=1+tM_n(\mathcal O)\subseteq I\) gives \(K_1v=v+t^{m+1}\mathcal O^n\), so the \(I\)-orbits in \(\{\operatorname{ord}=m\}\) are the preimages of the \(B\)-orbits \(Be_j=\{w_j\ne0,\ w_i=0\ (i\gt j)\}\) on \(\overline{\mathbb F}_q^{\,n}\setminus0\): these are the \(O_{m,j}\). For \(m\lt M\), \(O_{m,j}\) is a union of cosets of \(t^{m+1}\mathcal O^n\supseteq t^M\mathcal O^n\), hence the preimage of its image; every \(v\) with \(\operatorname{ord}v\ge M\), and \(0\), maps to \(0\) (A5-1(c)(d)).
  • (b), (c) Closures. \(\tau^k\mathcal O^n/\tau^{k+1}\mathcal O^n\) is one-dimensional (coordinate: the \(t^m\)-coefficient of \(v_j\)), so \(V_k\) is linear of dimension \(nM-k\) and \(o_k=V_k\setminus V_{k+1}\) is the complement of a hyperplane, open and dense. Hence \(\overline{o_k}=V_k=\bigsqcup_{k'\ge k}o_{k'}\), \(\overline{P_k}=\pi_{N,M}^{-1}(V_k)=\tau^k\mathcal O^n\), and the order is total.
  • (d) Stabilizers. Fixing \(t^me_j\) modulo \(t^M\) means \(g_{aj}\equiv\delta_{aj}\bmod t^{M-m}\): linear conditions on column \(j\), compatible with the Iwahori conditions because \(M-m\ge1\), and forcing \(g_{jj}(0)=1\). The remaining conditions are \(g_{aa}(0)\ne0\), \(a\ne j\). So the stabilizer is \(\mathbb G_m^{\,n-1}\times\mathbb A^b\), connected, with \(b=n^2(N+M)-\binom n2-n(M-m)+(n-j)-(n-1)\); \(\operatorname{Stab}(0)=I/I_{N,M}\) (A5-1, step 5).
  • (e) Refinement. \(T_M=t^M\mathcal O^n=\tau^{nM}\mathcal O^n\) and \(T_{M+1}=\tau^{nM+n}\mathcal O^n\), so the shells \(P_k\), \(nM\le k\lt nM+n\), fill \(T_M\setminus T_{M+1}\); likewise \(Y_N=\tau^{-nN}\mathcal O^n\).
  • Brute force. levels_check.py enumerates \((I/I_{N,M})(\mathbb F_q)\) and its orbits on \(Y_{N,M}(\mathbb F_q)\) (Table 3). In every case each orbit is one class \(O_{m,j}\) (or \(0\)) of the predicted size, the unions \(V_k(\mathbb F_q)\) are subspaces with \(q^{nM-k}\) points, and the stabilizer of \(t^me_j\) is the predicted subgroup with \((q-1)^{n-1}q^b\) points. Example \(n=2\), \((1,1)\), \(q=3\): orbit sizes \(54,18,6,2,1\) for \(k=-2,\dots,2\).
    Table 3. Orbits of \((I/I_{N,M})(\mathbb F_q)\) on \(Y_{N,M}(\mathbb F_q)\); \(q=2\) lies outside Proposition II but the count is the same.
    \(n\)levels \((N,M)\)\(q\)orbits \(=n(N+M)+1\)
    1(0,1) (0,2) (1,1) (1,2) (2,2) (0,4)2, 32, 3, 3, 4, 5, 5 ✓
    2(0,1) (1,1) (0,2)2, 33, 5, 5 ✓
    2(1,2) (0,3) (2,1)27, 7, 7 ✓
    3(0,1) (1,1)24, 7 ✓
Pieces of level (1,1): three nested discs for n=1, a chain of five pieces for n=2 P−1​ = t−1​𝒪×​ P0​ = 𝒪×​ P1​ = t𝒪 0 ∞ many orbits n = 1: Y1,1​ = t−1​𝒪 / t𝒪 ≅ 𝔸2​ piece representative dim of closure in Y1,1​ P−2​ = O−1,2​ t−1​e2​ 4 (open) P−1​ = O−1,1​ t−1​e1​ 3 P0​ = O0,2​ e2​ 2 P1​ = O0,1​ e1​ 1 P2​ = T1​ = t𝒪2​ 0 0 (closed) n = 2: Y1,1​ ≅ 𝔸4​; each piece lies in the closure of the one above
Figure 3. Level \((1,1)\): the pieces are nested lattice shells, and the closed piece \(T_1=t\mathcal O^n\) is one piece although it contains infinitely many \(I\)-orbits.

Proposition 3b · Rational orbits and counts

For \(-nN\le k<nM\), \(k=nm+n-j\), and every \(r\ge1\),

$$o_k(\mathbb F_{q^r})=H_{N,M}(\mathbb F_{q^r})\cdot t^me_j,\qquad |o_k(\mathbb F_{q^r})|=(q^r-1)(q^r)^{nM-k-1},\qquad |o_{nM}(\mathbb F_{q^r})|=1.$$
Proof

The section \(g(v)\) above is defined over \(\mathbb F_q\), hence gives transitivity over every extension field. The product \(\mathbb G_m\times\mathbb A^{nM-k-1}\) gives the count.

Alternatively, the fibre of the orbit map over a rational point is a torsor under the smooth connected stabilizer; Lang’s theorem gives a rational lift. See B. Conrad, Lang’s theorem and dynamic methods, Theorem 1.1 and Example 1.2, p. 1 (checked 22 October 2026).

These finite rational sets give the orbit indicators used for traces and functions; the categories remain categories of sheaves on the geometric spaces.