§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T22:14:42Z

Audit 4.10 · Audit: split \(\mathrm{GL}_2\) over a non-split torus gives no counterexample

proved

Dead end 2.11 holds as a dead end. For \(\mathrm{GL}_2/T_E\), (SG) holds by the tree recursion. For \(\mathrm{GL}_2/T_E^1\), the central element \(-I\) lies in a non-identity stabilizer component, which forces \(\mathcal C_k^{\mathrm{Sat}}\ne\mathcal C_k\), and (SG) fails too. One side remark transfers a statement from \(\mathbb C\) to \(\overline{\mathbb F}_q\) without justification, but it does not affect the conclusion.

Audit 4.10.

Let \(q\) be odd, \(T_E=\mathbb F_q[s_0]^\times\subset\mathrm{GL}_2\) with \(s_0^2=\varepsilon\) a non-square, and \(T_E^1=\ker(\det|_{T_E})\).

  1. \(X=\mathrm{GL}_2/T_E\) (symmetric, \(\theta=\mathrm{Ad}(s_0)\)): \(K\backslash X(F)=\{c\ge0\}\), the distance in the tree from the vertex \([\mathcal O_E]\), and \(T_{(1,0)}\ast f_0=f_1\), \(T_{(1,0)}\ast f_1=f_2+(q+1)f_0\), \(T_{(1,0)}\ast f_j=f_{j+1}+qf_{j-1}\) (\(j\ge2\)); (SG) holds.
  2. \(X=\mathrm{GL}_2/T_E^1\) (spherical, not symmetric): \(\mathcal C_k=\mathcal C^+\times\mathcal C^-\) by the action of \(-I\), \(\mathcal C_k^{\mathrm{Sat}}\subseteq\mathcal C^+\) and \(\mathcal C^-\ne0\), so \(\mathcal C_k^{\mathrm{Sat}}\ne\mathcal C_k\); \(X(F)\) has two \(G(F)\)-orbits and (SG) fails.
  3. Up to conjugacy these are all non-split \(\mathbb F_q\)-subtori of \(\mathrm{GL}_2\). Neither case is a counterexample.

Outcome: holds with a minor defect. The side claim "\(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) for \(\mathrm{GL}_2/T_E\) (the pages' family \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\))" cites pages written over \(\mathbb C\) ("\(H\) is a connected complex reductive group"). Over \(k=\overline{\mathbb F}_q\) it needs Proposition III rerun with Chen–Yi's D0–D2, or D5, which covers only \(p\) outside a finite set. It is irrelevant to the dead end, since (SG) holds there anyway.

  1. \(\mathrm{GL}_2/T_E\): points. \(X\) is the conjugacy class \(\{y:y^2=\varepsilon\}\); \(H^1(F,T_E)=H^1(E,\mathbb G_m)=1\), so \(X(F)=G(F)x_0\). \(K\backslash X(F)\cong E^\times\backslash\mathrm{GL}_2(F)/K\), the \(\mathcal O_E^\times\)-orbits on the vertices of the \((q+1)\)-regular tree (\(t^{\mathbb Z}\subseteq F^\times\) acts by homothety). \(\mathcal O_E^\times\) fixes \(v_0=[\mathcal O_E]\), and the stabilizer of the vertex \([\mathcal O+t^c\mathcal O_E]\) at distance \(c\ge1\) is \((\mathcal O+t^c\mathcal O_E)^\times\), of index \((q^2-1)q^{2(c-1)}/\big((q-1)q^{c-1}\big)=(q+1)q^{c-1}\), the size of the sphere: so the action on each sphere is transitive.
  2. Hecke recursion. \(T_{(1,1)}\) is trivial (the centre lies in \(T_E\)); \((T_{(1,0)}\ast f_j)(y)\) counts the neighbours of the standard vertex at distance \(j\) from \(v_y\): all \(q+1\) neighbours at distance \(1\) if \(d(y)=0\); one at \(d-1\) and \(q\) at \(d+1\) if \(d(y)=d\ge1\). This gives the three formulas, and induction puts every \(f_j\) in \(\mathbb C[T_{(1,0)}]\varphi_0\).
  3. \(\mathrm{GL}_2/T_E^1\): splitting. Over \(k\), \(T_E^1\cong\{\operatorname{diag}(a,a^{-1})\}=\mathrm{SO}(J)\). \(-I\in\mathrm{SO}(J)\) is central in \(\mathrm{GL}_2\), so it acts trivially on \(LX\) and on \(\operatorname{Gr}\) and defines an involution \(\varepsilon\) of the identity functor of \(\mathcal C_k\). The idempotents \((1\pm\varepsilon)/2\) split every object (the category is Karoubian, the coefficients have characteristic \(0\)), and \(\operatorname{Hom}(\mathcal C^+,\mathcal C^-)=0\) by naturality.
  4. Satake objects in \(\mathcal C^+\). \(-I\) lies in the connected group \(L^+\mathbb G_m\) of scalars, which acts trivially on \(\operatorname{Gr}\); so \(\varepsilon=1\) on every \(L^+G\)-equivariant perverse sheaf on \(\operatorname{Gr}\) (Zhu, Lemma A.1.4(2)). On \(\mathrm{IC}_0\), \(-I\in L^+\mathrm{SO}(J)\), which is connected, acts trivially on the stalk at \(x_0\). Since \(\varepsilon\) commutes with \(m_!\), \(\mathcal C_k^{\mathrm{Sat}}\subseteq\mathcal C^+\).
  5. \(\mathcal C^-\ne0\). For \(S=\operatorname{diag}(t^{2m},1)=hJh^{\mathsf T}\) (\(-t^{2m}\) is a square in \(k(\!(t)\!)\)), the stabilizer is \(\mathrm{SO}(S)(k[\![t]\!])\). Through the isotropic forms \(b\pm it^ma\), \(\mathrm{SO}(S)\cong\mathbb G_m\), and \(z\) has matrix entries \((z+z^{-1})/2\), \(it^m(z-z^{-1})/2\), \((z-z^{-1})/(2it^m)\). These are integral iff \(z\in k[\![t]\!]^\times\) and \(z\equiv\pm1\bmod t^m\) (\(z-1\) and \(z+1\) differ by the unit \(2\)). So \(\pi_0=\{\pm1\}\) and \(-I\leftrightarrow z=-1\) lies in the non-identity component (\(m\ge1\)). The sign character gives an equivariant local system \(\mathcal L^-\) with \(\varepsilon=-1\), and \(j_!\mathcal L^-\in\mathcal C^-\) is nonzero (supported on one orbit closure).
  6. Functions. \(H^1(F,T_E^1)=F^\times/NE^\times\cong\mathbb Z/2\) and \(H^1(F,\mathrm{GL}_2)=1\), so \(X(F)\supsetneq G(F)x_0\), and (SG) fails by Audit 4.4(a) (\(L^+T_E^1\) is connected, so \(X(\mathcal O)=Kx_0\)).
  7. Completeness of the case list. \(X_*(T_E)=\mathbb Z^2\) with \(\sigma\) the swap; the \(\sigma\)-stable saturated rank-one sublattices are \(\mathbb Z(1,1)\) (the split centre) and \(\mathbb Z(1,-1)\) (\(T_E^1\)). So \(T_E\) and \(T_E^1\) are, up to conjugacy, all non-split subtori. The extension to a general split \(G\) is labeled heuristic in Dead end 2.11(3), correctly.

Builds on

  • Dead end 2.11 · Dead end: \(G=\mathrm{GL}_2\) split, \(H=T_E\) or \(T_E^1\) non-split

Built on by

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

Sources