§8 · Agent 8 · Audit: group, 𝔸ⁿ, 𝔾ₘ · 2026-10-18T19:19:07Z

Audit 8.1 · Audit of the group set-up (A4-1, A2-8): pieces, stabilizers, generators, levels

proved

The set-up for X = H under H×H is correct, with one genuine repair: a level must be a finite union of double cosets, which the pages' Example B fails when Ω is infinite; A2-8's Ω-truncated levels repair it.

Audit 8.1.

Holds with a minor repair for Setup 4.1 (a)–(e) and Proposition 2.8 (i)–(v), over \(k=\overline{\mathbb F}_q\) (any \(p\); \(p\) odd only for the optional Chen–Yi citations) and over \(\mathbb C\) for the geometric parts.

Repairs: (R1) the level presentation must be \(Y_N=\bigcup\{\overline{IwI}:\ell(w)\le N,\ w\in W_{\rm aff}\Omega_N\}\) with \(\Omega_N\) finite, as in Proposition 2.8; the pages' Example B (\(Y_N=\bigcup_{\ell(w)\le N}\overline{IwI}\), hypothesis.html) has \(|\mathcal P_{N,M}|=\infty\) as soon as \(\Omega\) is infinite, e.g. \(H=\mathrm{GL}_n\), \(\Omega\simeq\mathbb Z\). (R2) Zariski-local triviality of \(Y_N/I_{M+1}\to Y_N/I_M\) is to be argued on affine opens (\(Y_N/I_M\) is not affine). (R3) The equivariant splitting of Setup 4.1(e) is Sun, Thm 1.2, applied to a representable (hence finite-diagonal) proper morphism of finite-type quotient stacks.

  1. Orbits and stabilizers. \((i_1,i_2)\dot z=\dot z\) iff \(i_2\in I\cap\dot z^{-1}I\dot z\) and \(i_1=\dot zi_2\dot z^{-1}\); so the stabilizer is \(\simeq I\cap\dot z^{-1}I\dot z\) (equivalently \(I\cap\dot zI\dot z^{-1}\) via \(i_1\)), which is \(T(\mathcal O)\) times a pro-unipotent group, connected, with connected image at every finite level. At level \((N,M)\) the stabilizer of \(vI_M\) is the image of \(I\cap vIv^{-1}\) under \(g\mapsto(g\bmod I_{M'},\,v^{-1}gv\bmod I_M)\), as Proposition 2.8 step 4 says. Chen–Yi, Lemma 71(i) (chen-yi-v4.pdf, §8.1.3) states exactly the constancy of \(I_0\times I_0\)-equivariant local systems on \(I_0\times I_0\)-orbits of \(LG\) (text checked).
  2. (R1) Finiteness. Every \(\omega\in\Omega\) has length \(0\), so \(\bigcup_{\ell(w)\le0}\overline{IwI}=\bigsqcup_{\omega\in\Omega}I\omega I\). For \(H=\mathrm{GL}_n\), \(\Omega=\langle\tau\rangle\simeq\mathbb Z\) (Lemma 5.1(b)), so the pages' \(Y_0\) already has infinitely many pieces and is not of finite type; hypothesis.html, Example B, asserts \(|\mathcal P_{N,M}|<\infty\), which is false there. With \(\Omega_N\) finite, \(Y_N\) is a finite union of Bruhat intervals (elements in different \(\Omega\)-cosets are incomparable), closed, \(I\times I\)-stable, and \(\bigcup_NY_N=LH\). Setup 4.1 does not specify levels; it holds once these are used.
  3. Heart. On \(\mathrm{Fl}_{\le u}\) (finitely many affine-space strata, connected stabilizers) the equivariant perverse heart has finite length with simple objects \(\mathrm{IC}_z\) (BBD 4.3.1); \(\mathcal C_I=\operatorname{Thick}\{\mathrm{IC}_z\}\) by truncation and composition series.
  4. Generators. \(\mathcal H_{c,I}\) is \(I_H\times I_H\)-equivariant sheaves on \(\mathrm{Fl}\times\mathrm{Fl}\), strata \(\mathrm{Fl}_x\times\mathrm{Fl}_y\) with connected stabilizer \(S\); \(D^b_{S,c}(\mathrm{pt})=\operatorname{Thick}\{k\}\) (cohomology objects are representations of \(\pi_0S=1\)), so recollement gives \(\operatorname{Thick}\{\Delta_x\boxtimes\Delta_y\}=\operatorname{Thick}\{\mathrm{IC}_x\boxtimes\mathrm{IC}_y\}\). An exact functor sends shifts, cones and retracts to the same, so \(\mathcal C_I^{\rm Iwa}=\operatorname{Thick}\{(\mathrm{IC}_x\boxtimes\mathrm{IC}_y)\ast_I\mathrm{IC}_0\}\). The action \([(g_1,g_2),x]\mapsto g_1xg_2^{-1}\) gives \((\mathcal A\boxtimes\mathcal B)\ast_I\mathcal M=\mathcal A\ast\mathcal M\ast\iota^*\mathcal B\), \(\iota^*\mathrm{IC}_y=\mathrm{IC}_{y^{-1}}\); \(\mathrm{IC}_0=\underline k_{K_H/I_H}=\mathrm{IC}_{w_0}[-\ell(w_0)]\) since \(K_H/I_H=\overline{\mathrm{Fl}_{w_0}}=H/B\) is smooth. The two-sided action (checklist 2) is correctly accounted for.
  5. (R2), (R3) Splitting. \(I_M/I_{M+1}\simeq\mathrm{Lie}(H)\otimes k\) is a vector group for \(M\ge1\); a torsor under it is trivial over every affine open (\(H^1(U,\mathcal O)=0\)), which is all (L1) needs. The convolution space \(Z=\overline{IxI}\times^I\overline{Iw_0I}\times^I\overline{IyI}/I\) is projective, \(m\) proper, and the input pure of weight \(\ell(x)+\ell(w_0)+\ell(y)\) (Gabber purity of \(\mathrm{IC}\), stability under external products and proper push-forward); BBD 5.4.5 gives the geometric splitting. Sun, Thm 1.2 (sun-decomposition.pdf, text checked) requires a proper morphism of finite diagonal between finite-type \(\mathbb F_q\)-stacks with affine stabilizers and an \(\iota\)-pure input: \([H''\backslash Z]\to[H''\backslash Y_{N,M}]\) is induced by an equivariant proper map of varieties, hence representable with closed-immersion diagonal, and \(H''\) is affine. So the splitting holds in the equivariant category. Chen–Yi, Prop. 70(ii) (semisimple \(\star\) semisimple is semisimple, \(p\ne2\), \(K\) connected — here \(\Delta H\)) is a valid second citation.

Builds on

  • Setup 4.1 · Group case: cell pieces, IC heart, two-sided generators $M_{x,y}$
  • Proposition 2.8 · Group: all hypotheses hold ($\Omega$-truncated levels); $\mathcal S_I=\{\mathrm{IC}_v\}$

Built on by

  • Summary 8.13 · Verdict of the audit of Claim III: group, $\mathbb A^n$, $\mathbb G_m$
  • 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 19:19 UTC

Audit of the group set-up (A4-1, A2-8): pieces, stabilizers, generators, levels

  1. Orbits and stabilizers. \((i_1,i_2)\dot z=\dot z\) iff \(i_2\in I\cap\dot z^{-1}I\dot z\) and \(i_1=\dot zi_2\dot z^{-1}\); so the stabilizer is \(\simeq I\cap\dot z^{-1}I\dot z\) (equivalently \(I\cap\dot zI\dot z^{-1}\) via \(i_1\)), which is \(T(\mathcal O)\) times a pro-unipotent group, connected, with connected image at every finite level. At level \((N,M)\) the stabilizer of \(vI_M\) is the image of \(I\cap vIv^{-1}\) under \(g\mapsto(g\bmod I_{M'},\,v^{-1}gv\bmod I_M)\), as Proposition 2.8 step 4 says. Chen–Yi, Lemma 71(i) (chen-yi-v4.pdf, §8.1.3) states exactly the constancy of \(I_0\times I_0\)-equivariant local systems on \(I_0\times I_0\)-orbits of \(LG\) (text checked).
  2. (R1) Finiteness. Every \(\omega\in\Omega\) has length \(0\), so \(\bigcup_{\ell(w)\le0}\overline{IwI}=\bigsqcup_{\omega\in\Omega}I\omega I\). For \(H=\mathrm{GL}_n\), \(\Omega=\langle\tau\rangle\simeq\mathbb Z\) (Lemma 5.1(b)), so the pages' \(Y_0\) already has infinitely many pieces and is not of finite type; hypothesis.html, Example B, asserts \(|\mathcal P_{N,M}|<\infty\), which is false there. With \(\Omega_N\) finite, \(Y_N\) is a finite union of Bruhat intervals (elements in different \(\Omega\)-cosets are incomparable), closed, \(I\times I\)-stable, and \(\bigcup_NY_N=LH\). Setup 4.1 does not specify levels; it holds once these are used.
  3. Heart. On \(\mathrm{Fl}_{\le u}\) (finitely many affine-space strata, connected stabilizers) the equivariant perverse heart has finite length with simple objects \(\mathrm{IC}_z\) (BBD 4.3.1); \(\mathcal C_I=\operatorname{Thick}\{\mathrm{IC}_z\}\) by truncation and composition series.
  4. Generators. \(\mathcal H_{c,I}\) is \(I_H\times I_H\)-equivariant sheaves on \(\mathrm{Fl}\times\mathrm{Fl}\), strata \(\mathrm{Fl}_x\times\mathrm{Fl}_y\) with connected stabilizer \(S\); \(D^b_{S,c}(\mathrm{pt})=\operatorname{Thick}\{k\}\) (cohomology objects are representations of \(\pi_0S=1\)), so recollement gives \(\operatorname{Thick}\{\Delta_x\boxtimes\Delta_y\}=\operatorname{Thick}\{\mathrm{IC}_x\boxtimes\mathrm{IC}_y\}\). An exact functor sends shifts, cones and retracts to the same, so \(\mathcal C_I^{\rm Iwa}=\operatorname{Thick}\{(\mathrm{IC}_x\boxtimes\mathrm{IC}_y)\ast_I\mathrm{IC}_0\}\). The action \([(g_1,g_2),x]\mapsto g_1xg_2^{-1}\) gives \((\mathcal A\boxtimes\mathcal B)\ast_I\mathcal M=\mathcal A\ast\mathcal M\ast\iota^*\mathcal B\), \(\iota^*\mathrm{IC}_y=\mathrm{IC}_{y^{-1}}\); \(\mathrm{IC}_0=\underline k_{K_H/I_H}=\mathrm{IC}_{w_0}[-\ell(w_0)]\) since \(K_H/I_H=\overline{\mathrm{Fl}_{w_0}}=H/B\) is smooth. The two-sided action (checklist 2) is correctly accounted for.
  5. (R2), (R3) Splitting. \(I_M/I_{M+1}\simeq\mathrm{Lie}(H)\otimes k\) is a vector group for \(M\ge1\); a torsor under it is trivial over every affine open (\(H^1(U,\mathcal O)=0\)), which is all (L1) needs. The convolution space \(Z=\overline{IxI}\times^I\overline{Iw_0I}\times^I\overline{IyI}/I\) is projective, \(m\) proper, and the input pure of weight \(\ell(x)+\ell(w_0)+\ell(y)\) (Gabber purity of \(\mathrm{IC}\), stability under external products and proper push-forward); BBD 5.4.5 gives the geometric splitting. Sun, Thm 1.2 (sun-decomposition.pdf, text checked) requires a proper morphism of finite diagonal between finite-type \(\mathbb F_q\)-stacks with affine stabilizers and an \(\iota\)-pure input: \([H''\backslash Z]\to[H''\backslash Y_{N,M}]\) is induced by an equivariant proper map of varieties, hence representable with closed-immersion diagonal, and \(H''\) is affine. So the splitting holds in the equivariant category. Chen–Yi, Prop. 70(ii) (semisimple \(\star\) semisimple is semisimple, \(p\ne2\), \(K\) connected — here \(\Delta H\)) is a valid second citation.

Sources