§2 · Agent 2 · Summand criterion over levels · 2026-10-18T18:50:51Z

Proposition 2.10 · Other rows: the criterion for symmetric $X$; $\mathbb G_m$ decided; sign obstruction

proved

For the symmetric rows of Table 1, Chen-Yi supply (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 local-system variant (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.') and DT_I, so the Iwahori summand criterion reduces each row to one explicit summand statement; GL_2/Sp_2 = G_m is settled at both levels by the Omega-argument, a sign-obstruction lemma generalizing the group case gives a sufficient test for sheaf failure, and the remaining rows are left with exact obligations.

Proposition 2.10.

Over $k=\overline{\mathbb F}_q$, $p$ odd, let $X=G/G^\theta$ be a symmetric row of Table 1 (all rows except $\mathbb A^n$; the group is $(H\times H)/\Delta H$).

(a) (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).) holds for a level presentation by finite unions of $I$-orbit closures (Chen–Yi, Cor. 27(i), Thm. 36), with pieces equal to orbits at fine levels; (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.') holds with $\mathcal S_I=\{\mathrm{IC}(\overline{O_v},L_{\xi,v}):(\xi,v)\in\mathcal D\}$, $L_{\xi,v}$ the rank-one local systems attached to $\xi\in X(\pi_0(T_v))$ (Chen–Yi, Lemma 71(ii), Def. 72); (DT$I$) holds (Chen–Yi, Prop. 70(ii)). Hence, given (L1)–(L3) for this presentation, Theorem 2.4 gives $$\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\iff\text{every }\mathrm{IC}(\overline{O_v},L_{\xi,v}),\ (\xi,v)\in\mathcal D,\ \text{is a shifted summand of some }\mathrm{IC}_w\star\mathrm{IC}_0 .\qquad(\ast_X)$$ (b) $p^*\mathcal M\subseteq\mathcal M_I$ (Lemma 2.3(c)): where $\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K$ is known, the open part of $(\ast_X)$ is the IC's of non-$K$-stable $I$-orbit closures and the non-constant $\xi$. (c) $X=\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m$ (over $\overline{\mathbb F}_q$ or $\mathbb C$): with $Y_N=\bigsqcup{|m|\le N}t^m\mathcal O^\times$, $Y_{N,M}=\bigsqcup_{|m|\le N}t^m(\mathcal O/t^M)^\times$, $I_{N,M}=\det^{-1}(1+t^M\mathcal O)\cap I$, all of (L1)–(L3), (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).), (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.) hold, $\mathcal S_I=\{k_{t^m\mathcal O^\times}\}=\Omega\cdot\mathrm{IC}0$, hence $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$; and $T{\omega^m}\ast\varphi_0=\mathbf 1_{t^m\mathcal O^\times}$ span $C_I$, so (IG) holds. The row agrees (both hold). (d) For $\mathrm{GL}{2n}/\mathrm{Sp}{2n}$ ($n\ge2$), $\mathrm{GL}2/T$, $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$, $\mathrm{Spin}{2n}/\mathrm{Spin}{2n-1}$, $\mathrm{SL}{2n}/\mathrm{Sp}_{2n}$, $E_6/F_4$ the sheaf side is decided by $(\ast_X)$ and is open here; the $\Omega$-argument of Theorem 2.7 needs $\mathcal S_I=\Omega\cdot\mathrm{IC}_0$, which is not established (and heuristically false) for these rows.

(e) Sign-obstruction lemma ($\overline{\mathbb F}q$; generalizes Theorem 2.9). Assume every generator $G_w$ has a finite-type proper model (Lemma 2.5). Let $\lambda_r\colon C_I^{(r)}\to\overline{\mathbb Q}\ell$ ($r\ge1$) be linear with $\lambda_r(h\ast f)=\varepsilon_r(h)\lambda_r(f)$ for $h\in\mathcal H_r$. If for one Frobenius-stable simple object $S_0$ one has $\lambda_r(t^{(r)}S)=0$ for all simple $S\not\simeq S_0$ (at a level containing the generator considered) and $\lambda_r(t^{(r)}{S_0})=c\,\beta^r$ with $c\ne0$, then $S_0\notin\mathcal M_I^+$, hence $S_0\notin\mathcal C_I^{\mathrm{Iwa}}\ne\mathcal C_I$. (f) Rows where (IG) is known to fail (Table 1: $\mathrm{GL}{2n}/\mathrm{Sp}{2n}$ and $\mathrm{SL}{2n}/\mathrm{Sp}{2n}$ in characteristic $0$, $\mathrm{GL}2/T$, $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$ for $p=q$ and $(p,q)=(1,2)$, $\mathrm{Spin}{2n}/\mathrm{Spin}_{2n-1}$ for $n=2,3$): the row differs iff $(\ast_X)$ holds (given (L$_X$)); it agrees (both fail) as soon as (e) applies to one $S_0$. For the characteristic-$0$ entries the comparison with sheaves over $\overline{\mathbb F}_q$ needs the function statement over $\mathbb F_q(\!(t)\!)$ first.

  1. (a), inputs. Chen–Yi work with $X=G/G^\theta$ over $\overline{\mathbb F}q$, $p\ne2$ (chen-yi-v4.pdf §§5, 8). Cor. 27(i): each $I$-orbit closure contains finitely many $I$-orbits; Thm. 36: orbit closures are placid with finitely presented embeddings, giving $LX=\varinjlim_v\overline{O_v}$. At a level fine enough that the finitely many orbits of $\overline{O_v}$ are pullbacks, the pieces are these orbits, so (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).). Lemma 71(ii): an $I$-equivariant local system on $O_v$ is a sum of $L{\xi,v}$, $\xi$ a character of the finite abelian group $\pi_0(T_v)$, so (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.') with the displayed $\mathcal S_I$ (checklist item 4). Prop. 70(ii): semisimple $\star$ semisimple is semisimple; $\mathrm{IC}_w$ and $\mathrm{IC}_0$ are simple, so (DT$_I$). Then Theorem 2.4(c),(e).
  2. (a), what is not supplied. Chen–Yi's placid presentation is not checked here against (L1)–(L3): affine-space transition maps in $M$, Cartesian closed embeddings in $N$, and the full faithfulness of Setup 2.1. This is the same unproved reading as the pages' 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). (first part). Obligation (L$_X$).
  3. (c), levels. $\mathrm{GL}2$ acts on $X\simeq\mathbb G_m$ through $\det$; $\det(I)=\mathcal O^\times$, so the $I$-orbits on $LX=F^\times$ are the $t^m\mathcal O^\times$, open and closed. $(\mathcal O/t^{M+1})^\times\to(\mathcal O/t^M)^\times$ is a $\mathbb G_a$-torsor (L1); $Y{N,M}\subseteq Y_{N+1,M}$ is a union of components (L2); $I/I_{N,M}\simeq(\mathcal O/t^M)^\times$ is connected and $I_{N,M}/I_{N,M+1}\simeq\mathbb G_a$ (L3). $I/I_{N,M}$ acts simply transitively on each $t^m(\mathcal O/t^M)^\times$: $2N+1$ orbits (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).), trivial stabilizers (I2I2 · Connected stabilizers: every stabilizer of I/I_{N,M} on Y_{N,M} is connected, so equivariant local systems on pieces are constant.). Piece closures: the $t^m\mathcal O^\times$, smooth, so $\mathcal S_I=\{k_{t^m\mathcal O^\times}\}$.
  4. (c), $\Omega$. $\omega=P\,\mathrm{diag}(t,1)$ normalizes $I$ (Theorem 2.7(b) with $n=2$) and $\det\omega=-t$. By Theorem 2.7(c), $\Delta_{\omega^m}\ast_I\mathrm{IC}0=(\omega^m)*k_{\mathcal O^\times}=k_{t^m\mathcal O^\times}$. So every simple object is a generator, $\mathcal M_I=\mathcal S_I$, and Theorem 2.4(d) gives $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ (no weights; valid over $\mathbb C$ too). Functions: $I$-invariant compactly supported functions on $F^\times$ are finite combinations of $\mathbf 1_{t^m\mathcal O^\times}=T_{\omega^m}\ast\varphi_0$. This re-proves the pages' IV.3 by the route of Theorem 2.7.
  5. (d), heuristic. $L^+X$ is a union of several $I$-orbits (over the several $B$-orbits of $X(k)$); we do not expect the closures over non-open $B$-orbits to be $\Omega$-translates of $L^+X$, and non-constant $L_{\xi,v}$ never are. So $(\ast_X)$ needs genuine summand computations in Chen–Yi's affine Lusztig–Vogan module (Thm. 1, Prop. 74, Lemma 75): every $e_{\xi,v}$ must occur as a summand, not only as a coefficient. Purity excludes cancellation for constant coefficients (Theorem 2.9, Step 6), so a divergence "(IG) holds, sheaves fail" can only come from $\xi\ne1$.
  6. Proof of (e). $\lambda_r(\varphi_0)=0$: $\mathbf 1_K\ast\varphi_0=[K:I]\varphi_0$ and $\varepsilon_r(\mathbf 1_K)=0$ (Theorem 2.9, Step 4), so $[K:I]\lambda_r(\varphi_0)=0$. Hence $\lambda_r(t^{(r)}{G_w})=\lambda_r(t^{(r)}{\mathrm{IC}w}\ast\varphi_0)=\varepsilon_r(t^{(r)}{\mathrm{IC}w})\lambda_r(\varphi_0)=0$, using the trace of a convolution (Theorem 2.9, Step 3(i), with $\mathrm{vol}(I)=1$). As in Theorem 2.9, Step 1, $t^{(r)}{G_w}=\sum_{S,s}(-1)^s\operatorname{Tr}(F^r|V_{S,s})t^{(r)}S$ with $V{S,s}=\mathrm{Hom}(S,{}^pH^s(G_w))$ pure of weight $w_{G}+s-w_S$, where $w_S$ is the weight of the chosen Weil structure on $S$. Applying $\lambda_r$: $c\sum_s(-1)^s\operatorname{Tr}((\beta F)^r|V_{S_0,s})=0$ for all $r$, and $\beta F$ on $V_{S_0,s}$ has eigenvalues of absolute value $|\beta|q^{(w_G+s-w_{S_0})/2}$, distinct for distinct $s$; Theorem 2.9, Step 6 gives $V_{S_0,s}=0$. Then Theorem 2.4(b).

Builds on

  • Theorem 2.4 · Theorem (I$_I$): Iwahori summand criterion via $\mathcal M_I\subseteq\mathcal M_I^+$
  • Lemma 2.5 · Lemma: a finite-type proper model gives (DT$_I$) over $\overline{\mathbb F}_q$
  • Lemma 2.3 · Lemma: generators $\mathrm{IC}_w\ast_I\mathrm{IC}_0$; spherical summands persist
  • Theorem 2.7 · $\mathbb A^n$: $\mathcal S_I=\Omega\cdot\mathrm{IC}_0$, hence Iwahori generation

Built on by

  • Audit 2.11 · Audit: the nine failure modes checked against the criterion and the two cases
  • Summary 2.12 · Verdict: the Iwahori summand criterion and the two test cases
  • Audit 8.11 · Audit of $\mathbb G_m$ and the other rows (A2-10 (a)–(d), (f))
  • Objection 8.12 · Objection to the sign-obstruction lemma (A2-10(e))
  • Strategy 10.1 · Registry after round one: every target has a candidate proof; three adversarial audits are launched
  • Summary 10.2 · Verdict: the criterion and the bridge hold; the decided rows of Table 1 agree; GL₂/T fails at both levels
Original record, as the agent wrote it at 18:50 UTC

Other rows: the criterion for symmetric $X$; $\mathbb G_m$ decided; sign obstruction

  1. (a), inputs. Chen–Yi work with $X=G/G^\theta$ over $\overline{\mathbb F}q$, $p\ne2$ (chen-yi-v4.pdf §§5, 8). Cor. 27(i): each $I$-orbit closure contains finitely many $I$-orbits; Thm. 36: orbit closures are placid with finitely presented embeddings, giving $LX=\varinjlim_v\overline{O_v}$. At a level fine enough that the finitely many orbits of $\overline{O_v}$ are pullbacks, the pieces are these orbits, so (I1). Lemma 71(ii): an $I$-equivariant local system on $O_v$ is a sum of $L{\xi,v}$, $\xi$ a character of the finite abelian group $\pi_0(T_v)$, so (I2') with the displayed $\mathcal S_I$ (checklist item 4). Prop. 70(ii): semisimple $\star$ semisimple is semisimple; $\mathrm{IC}_w$ and $\mathrm{IC}_0$ are simple, so (DT$_I$). Then Theorem 2.4(c),(e).
  2. (a), what is not supplied. Chen–Yi's placid presentation is not checked here against (L1)–(L3): affine-space transition maps in $M$, Cartesian closed embeddings in $N$, and the full faithfulness of Setup 2.1. This is the same unproved reading as the pages' I1 (first part). Obligation (L$_X$).
  3. (c), levels. $\mathrm{GL}2$ acts on $X\simeq\mathbb G_m$ through $\det$; $\det(I)=\mathcal O^\times$, so the $I$-orbits on $LX=F^\times$ are the $t^m\mathcal O^\times$, open and closed. $(\mathcal O/t^{M+1})^\times\to(\mathcal O/t^M)^\times$ is a $\mathbb G_a$-torsor (L1); $Y{N,M}\subseteq Y_{N+1,M}$ is a union of components (L2); $I/I_{N,M}\simeq(\mathcal O/t^M)^\times$ is connected and $I_{N,M}/I_{N,M+1}\simeq\mathbb G_a$ (L3). $I/I_{N,M}$ acts simply transitively on each $t^m(\mathcal O/t^M)^\times$: $2N+1$ orbits (I1), trivial stabilizers (I2). Piece closures: the $t^m\mathcal O^\times$, smooth, so $\mathcal S_I=\{k_{t^m\mathcal O^\times}\}$.
  4. (c), $\Omega$. $\omega=P\,\mathrm{diag}(t,1)$ normalizes $I$ (Theorem 2.7(b) with $n=2$) and $\det\omega=-t$. By Theorem 2.7(c), $\Delta_{\omega^m}\ast_I\mathrm{IC}0=(\omega^m)*k_{\mathcal O^\times}=k_{t^m\mathcal O^\times}$. So every simple object is a generator, $\mathcal M_I=\mathcal S_I$, and Theorem 2.4(d) gives $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$ (no weights; valid over $\mathbb C$ too). Functions: $I$-invariant compactly supported functions on $F^\times$ are finite combinations of $\mathbf 1_{t^m\mathcal O^\times}=T_{\omega^m}\ast\varphi_0$. This re-proves the pages' IV.3 by the route of Theorem 2.7.
  5. (d), heuristic. $L^+X$ is a union of several $I$-orbits (over the several $B$-orbits of $X(k)$); we do not expect the closures over non-open $B$-orbits to be $\Omega$-translates of $L^+X$, and non-constant $L_{\xi,v}$ never are. So $(\ast_X)$ needs genuine summand computations in Chen–Yi's affine Lusztig–Vogan module (Thm. 1, Prop. 74, Lemma 75): every $e_{\xi,v}$ must occur as a summand, not only as a coefficient. Purity excludes cancellation for constant coefficients (Theorem 2.9, Step 6), so a divergence "(IG) holds, sheaves fail" can only come from $\xi\ne1$.
  6. Proof of (e). $\lambda_r(\varphi_0)=0$: $\mathbf 1_K\ast\varphi_0=[K:I]\varphi_0$ and $\varepsilon_r(\mathbf 1_K)=0$ (Theorem 2.9, Step 4), so $[K:I]\lambda_r(\varphi_0)=0$. Hence $\lambda_r(t^{(r)}{G_w})=\lambda_r(t^{(r)}{\mathrm{IC}w}\ast\varphi_0)=\varepsilon_r(t^{(r)}{\mathrm{IC}w})\lambda_r(\varphi_0)=0$, using the trace of a convolution (Theorem 2.9, Step 3(i), with $\mathrm{vol}(I)=1$). As in Theorem 2.9, Step 1, $t^{(r)}{G_w}=\sum_{S,s}(-1)^s\operatorname{Tr}(F^r|V_{S,s})t^{(r)}S$ with $V{S,s}=\mathrm{Hom}(S,{}^pH^s(G_w))$ pure of weight $w_{G}+s-w_S$, where $w_S$ is the weight of the chosen Weil structure on $S$. Applying $\lambda_r$: $c\sum_s(-1)^s\operatorname{Tr}((\beta F)^r|V_{S_0,s})=0$ for all $r$, and $\beta F$ on $V_{S_0,s}$ has eigenvalues of absolute value $|\beta|q^{(w_G+s-w_{S_0})/2}$, distinct for distinct $s$; Theorem 2.9, Step 6 gives $V_{S_0,s}=0$. Then Theorem 2.4(b).

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