§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T22:11:02Z

Audit 4.2 · Audit: \(V_{(c,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_c\); sheaves generate for all forms

proved

The convolution space over \(\operatorname{Gr}_{\le(a,b)}\) is recomputed: it is the convolution space of the Hecke category of \(H\), so \(V_{(a,b)}\ast\mathrm{IC}_0\cong\mathrm{IC}_a\star\mathrm{IC}_{b'}\) and \(V_{(c,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_c\). With connected stabilizers this gives \(\mathcal C_k=\operatorname{Thick}\{\mathrm{IC}_c\}=\mathcal C_k^{\mathrm{Sat}}\) for the split, Galois and unitary forms alike.

Audit 4.2.

Let \(H\) be connected reductive over \(k\), \(\vartheta'\) an automorphism of \(H\) with \(\vartheta'(T_H)=T_H\), and let \(G_k=H\times H\) act on \(LH\) by \((g_1,g_2)\cdot x=g_1x\,\vartheta'(g_2)^{-1}\), \(x_0=1\).

  1. Over \(\operatorname{Gr}_{\le(a,b)}=\operatorname{Gr}_{\le a}\times\operatorname{Gr}_{\le b}\) the convolution space is \(LH_{\le a}\times^{L^+H}LH_{\le b'}\) with \(m(u,v)=uv\), where \(b'\) is the dominant representative of \(-\vartheta'(b)\). Hence \(V_{(a,b)}\ast\mathrm{IC}_0\cong\mathrm{IC}_a\star\mathrm{IC}_{b'}\) (pulled back to \(LH\)); \(b'=b\) for a Chevalley involution, \(b'=-w_0b\) for \(\vartheta'=1\). For \(b=0\), \(m\) is an isomorphism onto \(\overline{\mathcal O_a}\) and \(V_{(a,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_a\).
  2. \(\mathcal C_k=\operatorname{Thick}\{\mathrm{IC}_c:c\in\Lambda_H^+\}\) (indeed the triangulated hull), hence \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\).
  3. For a torus \(T\) over \(\mathbb F_q\) and a subtorus \(H\subseteq T\): \(\operatorname{Tri}\{V_\lambda\ast\mathrm{IC}_0\}=\mathcal C_k\).

Outcome: holds for Counterexample 2.3 steps 1–4 and Counterexample 1.5(a); holds with a minor defect for Lemma 2.5: "pure" in Lemma 2.5(2) needs a Frobenius-stable \(\lambda\) (Audit 4.1, item 5).

  1. Normalizing a point of the convolution space. A point over \(\operatorname{Gr}_{\le(a,b)}\) is a class \([(g_1,g_2),y]\), \(y\in L^+H\), modulo \(((g_1,g_2)k,y)\sim((g_1,g_2),k\cdot y)\), \(k\in L^+G\). Taking \(k=(y,1)\) gives \(y=1\). The stabilizer of \(1\) is \(\{(\vartheta'(h),h):h\in L^+H\}\). Put \(u=g_1\), \(v=\vartheta'(g_2)^{-1}\); the residual relation is \((u,v)\sim(u\vartheta'(h),\vartheta'(h)^{-1}v)\), and \(m=g_1\vartheta'(g_2)^{-1}=uv\).
  2. The second factor. \(v\) runs over \(\iota(LH_{\le b})\) with \(\iota(g)=\vartheta'(g)^{-1}\), an automorphism of the ind-scheme \(LH\) exchanging the two \(L^+H\)-actions; \(\iota(t^b)=t^{-\vartheta'(b)}\), so \(\iota(LH_{\le b})=LH_{\le b'}\) and \(\iota_*\mathrm{IC}_b\cong\mathrm{IC}_{b'}\). The twisted product \(V_{(a,b)}\mathbin{\tilde\boxtimes}\mathrm{IC}_0\) becomes \(\mathrm{IC}_a\mathbin{\tilde\boxtimes}\mathrm{IC}_{b'}\), and \(m_!\) is the convolution of the Hecke category \(L^+H\backslash LH/L^+H\). For a Chevalley involution \(-\vartheta'(b)=b\); for \(\vartheta'=1\), \(\operatorname{dom}(-b)=-w_0b\).
  3. The case \(b=0\). Then \(v\in L^+H\) is absorbed by \(h\), the representative \([(g,1),1]\) with \(g=g_1\vartheta'(g_2)^{-1}\in LH_{\le a}\) is unique, and \(m([(g,1),1])=g\). The sheaf is the pullback of \(\mathrm{IC}(\operatorname{Gr}_{\le a})\) along the \(L^+H\)-torsor \(LH_{\le a}\to\operatorname{Gr}_{\le a}\), \(g\mapsto gL^+H\), and smooth pullback of an IC sheaf is the IC sheaf (Zhu, Lemma A.1.4(1)). So \(V_{(a,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_a\).
  4. Generation. The orbits are \(\mathcal O_c=L^+H\,t^c\,L^+H\), \(c\in\Lambda_H^+\), each closure a finite union of orbits (Zhu, Prop. 2.1.5), and \(\operatorname{Stab}(t^c)\cong J_c=L^+H\cap t^cL^+Ht^{-c}\), connected (Audit 4.3). For \(M\in\mathcal C_k\) supported on a finite union \(Z\) of orbit closures, pick an open orbit \(j:\mathcal O\hookrightarrow Z\) and use \(j_!j^*M\to M\to i_*i^*M\). Since \(\pi_0J_c=1\), the cohomology objects of \(j^*M\in D^b_{L^+G}(\mathcal O)\simeq D^b_{J_c}(\mathrm{pt})\) are constant, so \(j^*M\) is an iterated extension of shifted constant sheaves; and \(j_!\overline{\mathbb Q}_\ell\) lies in \(\operatorname{Thick}\{\mathrm{IC}_{c'}:c'\le c\}\) by induction, because the cone of \(j_!\overline{\mathbb Q}_\ell[\delta]\to\mathrm{IC}_c\) is supported on smaller orbits. So \(\mathcal C_k=\operatorname{Thick}\{\mathrm{IC}_c\}\), and step 3 puts every \(\mathrm{IC}_c\) among the Satake objects: \(\mathcal C_k\subseteq\mathcal C_k^{\mathrm{Sat}}\subseteq\mathcal C_k\).
  5. Use of all \(\lambda\). The generating objects are \(V_{(c,0)}\ast\mathrm{IC}_0\), and \(\sigma(c,0)=(0,c)\ne(c,0)\) in both twists: these Satake objects have no Weil structure over \(\mathbb F_q\). This is legitimate, because \(\mathcal C_k^{\mathrm{Sat}}\) is defined with all of \(\Lambda_T^+\); it is exactly the gap exploited at function level (Audit 4.8).
  6. Independence of the form. Steps 1–4 use only \(G_k\), \(X_k\) and the action, with \(\vartheta'=\vartheta\) (unitary) or \(1\) (Galois, split).
  7. Tori. \((LX)_{\mathrm{red}}=\bigsqcup_{\mu\in X_*(X)}t^\mu L^+X\), each piece open and closed with stabilizer \(L^+H\), connected for a torus \(H\). The sequence \(0\to X_*(H)\to X_*(T)\to X_*(X)\to0\) is exact (dual of the exact sequence of character groups, whose cokernel \(X^*(H)\) is free), so each piece is \(t^\lambda L^+X\), on which \(V_\lambda\ast\mathrm{IC}_0\) is the constant sheaf (\(m\) is \(y\mapsto t^\lambda y\)). Step 4 without the induction gives \(\operatorname{Tri}=\mathcal C_k\).
  8. Central elements. The elements of \(G_k\) acting trivially are \((z,\vartheta'z)\), \(z\in Z(H)\) (for an involution \(\vartheta'\)); they lie in every connected \(J_c\), so they act trivially on every equivariant local system. There is no analogue of the \(\mathrm{SL}_2/\mathrm{SO}_2\) obstruction.

Builds on

  • Lemma 2.5 · Lemma: \(V_{(c,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_c\), so sheaves generate for all forms
  • Counterexample 2.3 · Counterexample: \(\operatorname{Res}\mathbb G_m\) on \(\mathbb G_m\) through the norm
  • Counterexample 1.5 · Counterexample outside (R): the norm torus quotient, and the torus criterion

Built on by

  • Summary 4.12 · Verdict of the audit: the unitary twist is a valid counterexample; the repairs are minor

Sources