§2 · Agent 2 · Summand criterion over levels · 2026-10-18T18:46:02Z

Proposition 2.6 · $\mathbb A^n$: all hypotheses hold; simple objects are lattice sheaves $k_\Lambda$

proved

For X = A^n every hypothesis of the Iwahori summand criterion is verified directly, stabilizers are connected because I/I_{N,M} is the unit group of a finite-dimensional algebra, and the simple objects of C_I are exactly the constant sheaves on the lattices t^m L_j.

Proposition 2.6.

$G=\mathrm{GL}n$, $X=\mathbb A^n$, $LX=F^n$, $L^+X=\mathcal O^n$, $I=\mathfrak A^\times$ with $\mathfrak A=\{x\in M_n(\mathcal O):x\bmod t\ \text{upper triangular}\}$. Put $$L_j=\mathcal Oe_1+\dots+\mathcal Oe_j+t\mathcal Oe_{j+1}+\dots+t\mathcal Oe_n\ (0\le j\le n),\qquad \mathcal L=\{t^mL_j:m\in\mathbb Z,\ 1\le j\le n\},$$ ($t^mL_0=t^{m+1}L_n$), $O{m,j}=\{v:\min_i\operatorname{val}v_i=m,\ j=\max\{i:\operatorname{val}v_i=m\}\}$, and the level presentation $Y_N=t^{-N}\mathcal O^n$, $Y_{N,M}=t^{-N}\mathcal O^n/t^M\mathcal O^n$, $I_{N,M}=1+t^{N+M}M_n(\mathcal O)$ ($N\ge0$, $M\ge1$). Over $k=\overline{\mathbb F}_q$ or $\mathbb C$:

(i) (L1)–(L3) hold, except that the full-faithfulness clause of (L3) is the standard consequence recorded in Setup 2.1, step 1. (ii) (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).): the $I$-orbits on $F^n\setminus0$ are the $O_{m,j}=I\cdot t^me_j$; the $I/I_{N,M}$-orbits on $Y_{N,M}$ are $0$ and the images of $O_{m,j}$, $-N\le m<M$: $|\mathcal P_{N,M}|=(N+M)n+1$. (iii) (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.): every stabilizer of $I/I_{N,M}$ on $Y_{N,M}$ is connected. (iv) The piece closures are exactly the lattices in $\mathcal L$ ($\overline{O_{m,j}}=t^mL_j$, $\overline{T_M}=T_M=t^ML_n$), so $\mathcal S_I=\{k_\Lambda:\Lambda\in\mathcal L\}$ up to shift. (v) (FM$_w$) holds for every $w\in\widetilde W$, hence (DT$_I$) holds over $\overline{\mathbb F}_q$ (Lemma 2.5), and over $\mathbb C$ by the cited equivariant decomposition theorem.

  1. (L1)–(L3). $\pi_{N,M',M}$ is the linear surjection with kernel $t^M\mathcal O^n/t^{M'}\mathcal O^n$, a trivial vector bundle, equivariant for $I/I_{N,M'}\to I/I_{N,M}$. $Y_{N,M}\subseteq Y_{N+1,M}$ is a linear subspace and $\pi_{N+1,M}^{-1}(Y_{N,M})=Y_N$ because $t^M\mathcal O^n\subseteq t^{-N}\mathcal O^n$ (L2). $I_{N,M}$ decreases, $I_{N,M}/I_{N',M'}=(1+t^aM_n(\mathcal O))/(1+t^bM_n(\mathcal O))$, $1\le a\le b$, is unipotent, and $I/I_{N,M}=(\mathfrak A/\mathfrak J)^\times$ is connected (Step 3).
  2. (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).). Invariants. For $g\in I$ and $v\in O_{m,j}$: if $i>j$, $(gv)i=\sum_kg{ik}v_k$ with $g_{ik}\in t\mathcal O$ for $k<i$ and $\operatorname{val}v_k\ge m+1$ for $k>j$, so $\operatorname{val}(gv)i\ge m+1$; $(gv)_j=g{jj}v_j+(\text{terms of valuation}\ge m+1)$ with $g_{jj}\in\mathcal O^\times$; all coordinates have valuation $\ge m$. So $(m,j)$ is $I$-invariant. Transitivity. $g=1-\sum_{i\ne j}(v_i/v_j)E_{ij}$ has $v_i/v_j\in\mathcal O$ for $i<j$ and $\in t\mathcal O$ for $i>j$, and is unipotent ($E_{ij}E_{i'j}=0$ for $i,i'\ne j$), so $g\in I$ and $gv=v_je_j$; a diagonal unit takes this to $t^me_j$. Levels. For $m<M$, $m$ and $j$ are read off $v\bmod t^M$, so $O_{m,j}$ is a union of cosets of $t^M\mathcal O^n$ (a piece) with image one orbit; every $v$ with all valuations $\ge M$ maps to $0$, whose preimage is the piece $T_M=t^M\mathcal O^n$, containing infinitely many orbits and $\{0\}$. (Finite check: for $n=2$ over $\mathbb F_2$ and $(N,M)\in\{(0,1),(0,2),(1,1),(1,2)\}$ the orbit counts are $3,5,5,7=(N+M)n+1$.)
  3. (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.), units lemma. $\mathfrak J=t^{N+M}M_n(\mathcal O)$ is a two-sided ideal of $\mathfrak A$ contained in $t\mathfrak A$, so $\mathfrak A^\times=I\to(\mathfrak A/\mathfrak J)^\times$ is surjective (a lift of a unit is a unit: $ab\in1+\mathfrak J$ is invertible by the $t$-adic geometric series) with kernel $1+\mathfrak J=I_{N,M}$. $V=Y_{N,M}$ is a module over the finite-dimensional $k$-algebra $\mathfrak A/\mathfrak J$ (since $\mathfrak J\,t^{-N}\mathcal O^n\subseteq t^M\mathcal O^n$). The stabilizer of $\bar v$ is $(1+\mathrm{Ann}(\bar v))\cap(\mathfrak A/\mathfrak J)^\times$: a nonempty (it contains $1$) open subset ($\det$ of left multiplication $\ne0$) of the affine space $1+\mathrm{Ann}(\bar v)$, hence irreducible, hence connected; it is even smooth as a group scheme, being open in an affine space. The same argument shows $(\mathfrak A/\mathfrak J)^\times$ is connected.
  4. (iv). $O_{m,j}=\{v\in t^mL_j:\operatorname{val}v_j=m\}$ is open and dense in the linear space $t^mL_j/t^M\mathcal O^n$, so $\overline{O_{m,j}}=t^mL_j$; $T_M=t^ML_n$ is closed. Every element of $\mathcal L$ occurs (as $\overline{O_{m,j}}$ at any level with $-N\le m<M$). The closures are linear at each level, so the IC is the shifted constant sheaf, and by Lemma 2.2(c) $\mathcal S_I=\{k_\Lambda:\Lambda\in\mathcal L\}$. Distinct $(m,j)$, $1\le j\le n$, give distinct lattices.
  5. (FM$_w$). $\mathrm{Fl}_w$ is quasi-compact, so there are $a,b$ with $t^a\mathcal O^n\subseteq g\mathcal O^n\subseteq t^{-b}\mathcal O^n$ for all $g\in\overline{IwI}$ ($I$ preserves both lattices and $\dot v\mathcal O^n=t^\mu\mathcal O^n$ for each of the finitely many cells $IvI\subseteq\overline{IwI}$). Take $N\ge b$, $M\ge\max(a,1)$ and
$$Z=\{(gI,\bar x)\in\mathrm{Fl}_w\times Y_{N,M}:\ \bar x\in g\mathcal O^n/t^M\mathcal O^n\},$$

the pullback of the tautological subbundle of a Grassmannian of $Y_{N,M}$ along $gI\mapsto g\mathcal O^n/t^M\mathcal O^n$: a vector bundle over $\mathrm{Fl}w$ of rank $nM-\operatorname{val}\det g$, constant because $\operatorname{val}\circ\det$ is locally constant on $LG$ and $\overline{IwI}$ is connected. $r=\mathrm{pr}_1$ is smooth; $p=\mathrm{pr}_2$ is proper ($Z$ closed, $\mathrm{Fl}_w$ projective). Since $t^M\mathcal O^n\subseteq g\mathcal O^n$, $\overline{IwI}\times^I\mathcal O^n\simeq\{(gI,x):x\in g\mathcal O^n\}=Z\times{Y_{N,M}}Y_N$, and $\mathrm{IC}w\tilde\boxtimes k{\mathcal O^n}$ corresponds to $r^*\mathrm{IC}w$. $I$ acts on $Z$ through $I/(I{N,M}\cap I_{M''})$ for $M''$ with $I_{M''}$ acting trivially on $\mathrm{Fl}_w$; everything is defined over $\mathbb F_q$. Lemma 2.5 applies.

Builds on

  • Setup 2.1 · Set-up: level hypotheses, simple objects, $\mathcal M_I\subseteq\mathcal M_I^+$
  • Lemma 2.2 · Lemma (PH$_I$): perverse hearts glue; $\mathcal C_I=\mathrm{Thick}(\mathcal S_I)$
  • Lemma 2.5 · Lemma: a finite-type proper model gives (DT$_I$) over $\overline{\mathbb F}_q$

Built on by

  • Theorem 2.7 · $\mathbb A^n$: $\mathcal S_I=\Omega\cdot\mathrm{IC}_0$, hence Iwahori generation
  • Audit 8.8 · Audit of $\mathbb A^n$ orbits, levels and simple objects (A5-1, A5-2, A2-6)
Original record, as the agent wrote it at 18:46 UTC

$\mathbb A^n$: all hypotheses hold; simple objects are lattice sheaves $k_\Lambda$

  1. (L1)–(L3). $\pi_{N,M',M}$ is the linear surjection with kernel $t^M\mathcal O^n/t^{M'}\mathcal O^n$, a trivial vector bundle, equivariant for $I/I_{N,M'}\to I/I_{N,M}$. $Y_{N,M}\subseteq Y_{N+1,M}$ is a linear subspace and $\pi_{N+1,M}^{-1}(Y_{N,M})=Y_N$ because $t^M\mathcal O^n\subseteq t^{-N}\mathcal O^n$ (L2). $I_{N,M}$ decreases, $I_{N,M}/I_{N',M'}=(1+t^aM_n(\mathcal O))/(1+t^bM_n(\mathcal O))$, $1\le a\le b$, is unipotent, and $I/I_{N,M}=(\mathfrak A/\mathfrak J)^\times$ is connected (Step 3).
  2. (I1). Invariants. For $g\in I$ and $v\in O_{m,j}$: if $i>j$, $(gv)i=\sum_kg{ik}v_k$ with $g_{ik}\in t\mathcal O$ for $k<i$ and $\operatorname{val}v_k\ge m+1$ for $k>j$, so $\operatorname{val}(gv)i\ge m+1$; $(gv)_j=g{jj}v_j+(\text{terms of valuation}\ge m+1)$ with $g_{jj}\in\mathcal O^\times$; all coordinates have valuation $\ge m$. So $(m,j)$ is $I$-invariant. Transitivity. $g=1-\sum_{i\ne j}(v_i/v_j)E_{ij}$ has $v_i/v_j\in\mathcal O$ for $i<j$ and $\in t\mathcal O$ for $i>j$, and is unipotent ($E_{ij}E_{i'j}=0$ for $i,i'\ne j$), so $g\in I$ and $gv=v_je_j$; a diagonal unit takes this to $t^me_j$. Levels. For $m<M$, $m$ and $j$ are read off $v\bmod t^M$, so $O_{m,j}$ is a union of cosets of $t^M\mathcal O^n$ (a piece) with image one orbit; every $v$ with all valuations $\ge M$ maps to $0$, whose preimage is the piece $T_M=t^M\mathcal O^n$, containing infinitely many orbits and $\{0\}$. (Finite check: for $n=2$ over $\mathbb F_2$ and $(N,M)\in\{(0,1),(0,2),(1,1),(1,2)\}$ the orbit counts are $3,5,5,7=(N+M)n+1$.)
  3. (I2), units lemma. $\mathfrak J=t^{N+M}M_n(\mathcal O)$ is a two-sided ideal of $\mathfrak A$ contained in $t\mathfrak A$, so $\mathfrak A^\times=I\to(\mathfrak A/\mathfrak J)^\times$ is surjective (a lift of a unit is a unit: $ab\in1+\mathfrak J$ is invertible by the $t$-adic geometric series) with kernel $1+\mathfrak J=I_{N,M}$. $V=Y_{N,M}$ is a module over the finite-dimensional $k$-algebra $\mathfrak A/\mathfrak J$ (since $\mathfrak J\,t^{-N}\mathcal O^n\subseteq t^M\mathcal O^n$). The stabilizer of $\bar v$ is $(1+\mathrm{Ann}(\bar v))\cap(\mathfrak A/\mathfrak J)^\times$: a nonempty (it contains $1$) open subset ($\det$ of left multiplication $\ne0$) of the affine space $1+\mathrm{Ann}(\bar v)$, hence irreducible, hence connected; it is even smooth as a group scheme, being open in an affine space. The same argument shows $(\mathfrak A/\mathfrak J)^\times$ is connected.
  4. (iv). $O_{m,j}=\{v\in t^mL_j:\operatorname{val}v_j=m\}$ is open and dense in the linear space $t^mL_j/t^M\mathcal O^n$, so $\overline{O_{m,j}}=t^mL_j$; $T_M=t^ML_n$ is closed. Every element of $\mathcal L$ occurs (as $\overline{O_{m,j}}$ at any level with $-N\le m<M$). The closures are linear at each level, so the IC is the shifted constant sheaf, and by Lemma 2.2(c) $\mathcal S_I=\{k_\Lambda:\Lambda\in\mathcal L\}$. Distinct $(m,j)$, $1\le j\le n$, give distinct lattices.
  5. (FM$_w$). $\mathrm{Fl}_w$ is quasi-compact, so there are $a,b$ with $t^a\mathcal O^n\subseteq g\mathcal O^n\subseteq t^{-b}\mathcal O^n$ for all $g\in\overline{IwI}$ ($I$ preserves both lattices and $\dot v\mathcal O^n=t^\mu\mathcal O^n$ for each of the finitely many cells $IvI\subseteq\overline{IwI}$). Take $N\ge b$, $M\ge\max(a,1)$ and
$$Z=\{(gI,\bar x)\in\mathrm{Fl}_w\times Y_{N,M}:\ \bar x\in g\mathcal O^n/t^M\mathcal O^n\},$$

the pullback of the tautological subbundle of a Grassmannian of $Y_{N,M}$ along $gI\mapsto g\mathcal O^n/t^M\mathcal O^n$: a vector bundle over $\mathrm{Fl}w$ of rank $nM-\operatorname{val}\det g$, constant because $\operatorname{val}\circ\det$ is locally constant on $LG$ and $\overline{IwI}$ is connected. $r=\mathrm{pr}_1$ is smooth; $p=\mathrm{pr}_2$ is proper ($Z$ closed, $\mathrm{Fl}_w$ projective). Since $t^M\mathcal O^n\subseteq g\mathcal O^n$, $\overline{IwI}\times^I\mathcal O^n\simeq\{(gI,x):x\in g\mathcal O^n\}=Z\times{Y_{N,M}}Y_N$, and $\mathrm{IC}w\tilde\boxtimes k{\mathcal O^n}$ corresponds to $r^*\mathrm{IC}w$. $I$ acts on $Z$ through $I/(I{N,M}\cap I_{M''})$ for $M''$ with $I_{M''}$ acting trivially on $\mathrm{Fl}_w$; everything is defined over $\mathbb F_q$. Lemma 2.5 applies.

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