S3-3 · 2026-10-21T15:10:33Z

The unitary family still generates sheaves and fails to generate functions

In the setting of S3-2, let \(F_n=\mathbb F_{q^n}((t))\), \(K_n=G(\mathbb F_{q^n}[[t]])\), and let \(\varphi_0\) be the integral-orbit indicator. Then \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\), but \((\mathrm{SG})_n\) fails for odd \(n\) and holds for even \(n\). Over \(F_1\), every dominant \(c\notin2\Lambda_H\) gives \(\mathbf1_{O_c}\notin\mathcal H_K*\varphi_0\). This rederives the substantive counterexample input A4-12 with profile S3-2: it lies outside (R), and proves no failure under (R),(C).

  1. Over \(k\), every geometric orbit has connected stabilizer \(J_c\). The bounded equivariant category supported on finitely many orbit closures is therefore generated by its constant-local-system intersection complexes \(\mathrm{IC}_c\), by finite-length perverse truncation and recollement. In the geometric group case, the convolution map for \((c,0)\) is the identity on the corresponding orbit closure after normalizing the torsor, so \(V_{(c,0)}*\mathrm{IC}_0\cong\mathrm{IC}_c\) (A4-2). This proves sheaf generation; no Weil structure on \(V_{(c,0)}\) is needed for that geometric statement.
  2. Over an odd-degree extension, put \(E_n=\mathbb F_{q^{2n}}((t))\). The Hecke basis is \(T_a=\mathbf1_{K_nt^aK_n}\), \(a\in\Lambda_H^+\); these are the stable geometric labels \((a,a)\). Every point in the support of \(F_a=T_a*\varphi_0\) has Cartan label \(c\le2a\), because the action lands in \(K_nt^aK_nt^aK_n\). The coset \(t^aK_n\) sends the base point to \(t^{2a}\), so \(F_a(t^{2a})>0\). Stable orbit labels classify the rational \(K_n\)-orbits by connected-stabilizer Lang (A4-3).
  3. The map \(a\mapsto2a\) is injective. In a finite linear combination of the \(F_a\), a maximal leading label \(2a\) cannot cancel: every other summand is supported below its own leading label, and no other leading label dominates this chosen maximal one. Thus any singleton indicator in their span must have its label equal to some \(2a\). In particular labels outside \(2\Lambda_H\) are missed.
  4. Such dominant labels exist because \(\Lambda_H\ne0\). Choose \(\lambda\notin2\Lambda_H\), and a regular dominant coroot-lattice element \(\xi\) for the semisimple roots. For sufficiently large \(N\), \(c=\lambda+2N\xi\) is dominant and remains outside \(2\Lambda_H\). If there are no roots, every \(\lambda\) is already dominant. This proves failure for every odd degree, independently of the false equality repaired in S3-1.
  5. In even degree restriction of scalars splits as \(H\times H\), and \(X(F_n)\cong H(F_n)\) with the geometric group action. The operators labelled \((c,0)\) now exist rationally and send \(\varphi_0\) to the double-coset indicators, so (SG) holds.
  6. The same-characteristic counterexample input A4-12 therefore survives. For \(H=\mathrm{SL}_2\), the already-held A4-7 further gives a single \(G(F)\)-orbit, so the failure even occurs without a second rational group orbit. The setup correction only replaces a false exact invariant-rank claim by the sufficient upper bound; it does not settle any converse restricted to split (R),(C).

Builds on: S3-2 · A4-2 · A4-3 · A4-4 · A4-5 · A4-7