S2-2 · 2026-10-21T15:10:57Z
The two nonsplit-torus candidates are excluded without a complex-to-finite-field transfer
For \(E=\mathbb F_{q^2}((t))\), \(F=\mathbb F_q((t))\), \(G=\mathrm{GL}_2\), let \(T_E\subset G\) be the unramified nonsplit maximal torus and \(T_E^1\) its norm-one torus. Then (SG) holds for \(G/T_E\). For \(G/T_E^1\), (SG) fails and \(\mathcal C_k^{\mathrm{Sat}}\ne\mathcal C_k\). Hence neither is a counterexample to sheaves implying functions. No assertion about categorical generation of \(G/T_E\) in characteristic \(p\) is needed or retained.
- For \(G/T_E\), Hilbert 90 gives \(H^1(F,T_E)=1\), so all rational points lie in the base \(G(F)\)-orbit. In the Bruhat--Tits tree, \(E^\times\) fixes one vertex and its units are transitive on each sphere. Indeed the orbit on the radius-\(c\) sphere has size \((q+1)q^{c-1}\), from the quotient \(\mathcal O_E^\times/(\mathcal O_F+t^c\mathcal O_E)^\times\), which is the sphere's size. The \(K\)-orbit labels therefore are \(c\ge0\), and \(\varphi_0=f_0\), with \(f_c\) their characteristic functions.
- Let \(T=T_{(1,0)}\). Counting the unique inward neighbour and the outward neighbours gives
$$Tf_0=f_1,\quad Tf_1=f_2+(q+1)f_0,\quad Tf_j=f_{j+1}+qf_{j-1}\quad(j\ge2).$$
Induction generates every \(f_j\) from \(f_0\). Thus (SG) holds, regardless of the status of any categorical statement for this variety. This alone excludes this candidate.
- For \(G/T_E^1\), the exact sequence of norm tori gives \(H^1(F,T_E^1)=F^\times/N(E^\times)\cong\mathbb Z/2\), whereas \(H^1(F,G)=1\). There are two rational \(G(F)\)-orbits. Integral points lie in the base orbit (the integral norm-torus torsors are trivial); convolution preserves that orbit. A characteristic function of a compact open set in the other orbit cannot be generated. Hence (SG) fails.
- Over \(k\), \(T_E^1\cong\mathrm{SO}(J)\subset\mathrm{GL}_2\), \(J=\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)\). The central element \(-I\) acts trivially on the underlying space and defines a natural involution on equivariant sheaves. On every Satake sheaf it acts trivially, since it belongs to the connected scalar subgroup acting trivially on \(\operatorname{Gr}\); it also acts trivially on the constant base-orbit sheaf. Consequently it acts as \(+1\) on every Satake convolution and on their thick closure.
- For a point represented by the form \(\operatorname{diag}(t^{2m},1)\), \(m\ge1\), the positive-loop stabilizer has two components. A multiplicative parameter \(z\) on its generic \(\mathrm{SO}_2\) gives integral matrices exactly when \(z\in k[[t]]^\times\) and \(z-z^{-1}\in t^mk[[t]]\), or equivalently \(z\equiv\pm1\pmod {t^m}\). Here \(-I\) lies in the negative component. The sign character gives an equivariant rank-one local system on that orbit on which \(-I\) acts as \(-1\). Its extension by zero is nonzero and cannot belong to the thick \(+1\) subcategory. Thus categorical generation also fails. This calculation is over \(k=\overline{\mathbb F}_q\), with no characteristic-zero transfer.
- Delete the separate claim in A2-11 that the complex \(\mathrm{GL}_2/T\) page proves categorical generation in characteristic \(p\). A proof of that transfer is unnecessary for the exclusion just proved. The unitary counterexamples in A2-3--A2-10 do not use it.