§1 · Agent 1 · Literature and Table 1 audit · 2026-10-17T02:46:59Z

Proposition 1.2 · \(\SL_2/T\) versus \(\GL_2/T\): one variety, two Hecke algebras

proved audit: error

Error. Main claims 1–4 correct (re-derived: tree recursions, Hecke relation, coset bijection, determinant of stabilizers; dimension count rechecked for q = 2, 3, 4, 5, 7 in tree_G1.py). Two false statements. (i) Step 3: 'p(T₁²)φ₀ has support exactly {0,…,2 deg p}' is false — e.g. (T₁² − (q+3))φ₀ = 2·1_{d=1} + 1_{d=2}; only 'the support contains 2 deg p and lies in {0,…,2 deg p}' holds, which is all the argument needs. (ii) Step 4: the remark presented as proved rests on 'IC₀ is a direct summand of V_{(1,0)}∗IC₀'. This is false: Gr^{(1,0)} ×̃ L⁺X → closure(O₁) is finite, two-to-one over O₀ and an isomorphism over O₁, so V_{(1,0)}∗IC₀ ≅ IC(closure(O₁)) is simple. The remark's conclusion (no parity decomposition of the GL₂ category) is true for another reason: the nonzero map IC₁ → IC₀^{⊕2}[1] of the localisation triangle joins the would-be odd and even parts. The conclusion still holds by another argument.
Used later by: Summary 1.9 · Classification after the status audit, and the cells where a divergence can still hide.
Corrected by: Objection 7.15 · Objection: the parity count of the \(\SL_2/T\) analysis holds, but '\(\IC_0\) is a summand of \(V_{(1,0)}\ast\IC_0\)' is false, Summary 9.2 · Registry after the first audit round: the first-round results survive with their hypotheses, three cosmetic corrections, and \(E_6/F_4\) the only live cell.

The variety \(\GL_2/T=\SL_2/T\) satisfies (SG) for \(\GL_2\) but not for \(\SL_2\), whatever the residue field: the \(\SL_2\) Hecke algebra produces only even polynomials in the \(\GL_2\) adjacency operator. A closing sheaf-level remark argues that this parity cannot lift to the \(\GL_2\) category because \(\IC_0\) is a summand of \(V_{(1,0)}\ast\IC_0\); that argument is false (see the audit).

Proposition 1.2.

Let \(T\subset\GL_2\) be the diagonal torus, \(X=\GL_2/T=\SL_2/T_{\SL}\) (also \(=\PGL_2/\overline T\)), \(F\) a non-archimedean local field, \(K_{\GL}=\GL_2(\cO)\), \(K_{\SL}=\SL_2(\cO)\), \(\varphi_0=\mathbf 1_{X(\cO)}\), \(T_1=\mathbf 1_{K_{\GL}\operatorname{diag}(\varpi,1)K_{\GL}}\), and let \(d(x)\) be the distance from the apartment of \(T\) of the tree vertex attached to \(x\); write \(\mathbf 1_d=\mathbf 1_{\{d\}}\).

  1. \(X(F)=\GL_2(F)/T(F)=\SL_2(F)/T_{\SL}(F)\).
  2. \(K_{\GL}\backslash X(F)=\{d\ge0\}\), and \(C_{K_{\GL}}=\C[T_1]\varphi_0\) is a free \(\C[T_1]\)-module of rank one, for every \(q\).
  3. For \(q\) odd, \(K_{\SL}\backslash X(F)=\{0\}\sqcup\{(d,\epsilon):d\ge1,\ \epsilon\in\cO^\times/\cO^{\times2}\}\).
  4. \(\mathcal H_{K_{\SL}}\cdot\varphi_0=\C[T_1^2]\varphi_0\subsetneq\C[T_1]\varphi_0=C_{K_{\GL}}\subseteq C_{K_{\SL}}\). In particular \(\mathbf 1_1\notin\mathcal H_{K_{\SL}}\varphi_0\): (SG) fails for \(\SL_2/T\) for every \(q\) and every \(F\).
  5. Over \(\C(\!(t)\!)\) the \(L^+\GL_2\)- and \(L^+\SL_2\)-orbits on \(LX\) coincide (\(\mathcal O_d\), \(d\ge0\)), with \(\pi_0\Stab_{L^+\SL_2}(x_d)=\mu_2\) and \(\pi_0\Stab_{L^+\GL_2}(x_d)=1\) for \(d\ge1\). The Satake objects are \(V_{(k,0)}\ast\IC_0\) (top orbit \(\mathcal O_k\)) for \(\GL_2\), and \(V_{k\alpha^\vee}\ast\IC_0=\pi^{*}(V_{(2k,0)}\ast\IC_0)\) (top orbit \(\mathcal O_{2k}\)) for \(\SL_2\). Remark, stated as proved: the function-level grading \(y\mapsto-y\) does not lift to the \(\GL_2\) category, because \(\IC_0\) is a direct summand of the "odd" object \(V_{(1,0)}\ast\IC_0\).
  1. Points and \(\GL_2\)-orbits. Hilbert 90 for \(T\) and for \(T_{\SL}\) gives \(X(F)=\GL_2(F)/T(F)\); since \(\det:T(F)\to F^\times\) is onto, \(\SL_2(F)\) acts transitively with stabilizer \(T_{\SL}(F)\). The two rows of the table differ only in the acting group, hence in \(K\), \(\mathcal H_K\) and \(C_K\). The \(K_{\GL}\)-orbits on \(X(F)\) are the \(T(F)\)-orbits on vertices of the tree of \(\PGL_2(F)\): \(T(\cO)\) permutes transitively the vertices at distance \(d\) above a fixed apartment vertex (by \(x\mapsto u_1u_2^{-1}x\) on \((\cO/\varpi^d)^\times\)), and \(T(F)\) translates the apartment. The operator \(T_1\) is the adjacency operator, and counting neighbours gives
$$T_1\mathbf 1_0=2\cdot\mathbf 1_0+\mathbf 1_1,\qquad T_1\mathbf 1_1=(q-1)\mathbf 1_0+\mathbf 1_2,\qquad T_1\mathbf 1_d=q\,\mathbf 1_{d-1}+\mathbf 1_{d+1}\quad(d\ge2).$$

Hence \(T_1^d\varphi_0=\mathbf 1_d+(\text{terms with smaller }d)\), so \(\C[T_1]\varphi_0=C_{K_{\GL}}\), freely. No hypothesis on \(F\) or \(q\) enters.

  1. Square classes for \(\SL_2\), \(q\) odd. For a point \(y_d\) at distance \(d\ge1\), \(\Stab_{K_{\GL}}(y_d)\) consists of the conjugates \(u\operatorname{diag}(a,b)u^{-1}\) with \(a,b\in\cO^\times\), \(b\equiv a\bmod\varpi^d\); its determinants form \(\cO^{\times2}(1+\varpi^d\cO)=\cO^{\times2}\). So each \(K_{\GL}\)-orbit \(\{d\}\), \(d\ge1\), splits into two \(K_{\SL}\)-orbits. The argument of Proposition J(2) is correct: \(s=\operatorname{diag}(u,1)\) with \(u\) a non-square unit normalizes \(K_{\SL}\), \(T\) and every double coset and fixes \(x_0\), so the automorphism \(\sigma\) it induces commutes with \(\mathcal H_{K_{\SL}}\) and fixes \(\varphi_0\); the explicit matrix computation shows that \(\sigma\) exchanges the two classes at \(d=1\).
  1. Structural relation. Since \(K_{\GL}=K_{\SL}\cdot\{\operatorname{diag}(u,1)\}\), \(C_{K_{\GL}}\) is the subspace of \(C_{K_{\SL}}\) fixed by all \(\sigma_u\), \(u\in\cO^\times\); the argument of step 2 works for every unit and every \(q\), so \(\mathcal H_{K_{\SL}}\varphi_0\subseteq C_{K_{\GL}}\). For \(t=\operatorname{diag}(\varpi,\varpi^{-1})\) the coset spaces \(K_{\SL}tK_{\SL}/K_{\SL}\) and \(K_{\GL}tK_{\GL}/K_{\GL}\) agree, so on \(C_{K_{\GL}}\)
$$T^{\SL}_{(1,-1)}=T^{\GL}_{(1,-1)}=T_{(2,0)}=T_1^2-(q+1),$$

using that the centre acts trivially on \(X\) and the Hecke relation \(T_1^2=T_{(2,0)}+(q+1)T_{(1,1)}\). As \(\mathcal H_{K_{\SL}}=\C[T^{\SL}_{(1,-1)}]\), this gives \(\mathcal H_{K_{\SL}}\varphi_0=\C[T_1^2]\varphi_0=\C[y^2]\) inside \(C_{K_{\GL}}\cong\C[y]\), \(y=T_1\). A degree argument — \(p(T_1^2)\varphi_0\) has support exactly \(\{0,\dots,2\deg p\}\) — shows \(\mathbf 1_1\notin\mathcal H_{K_{\SL}}\varphi_0\). Two independent obstructions are visible: the parity \(y\mapsto-y\) of the Hecke action, already on \(\GL_2(\cO)\)-invariant functions (the shadow of \(\varphi\) doubling), for every \(q\); and for \(q\) odd the square classes (the shadow of \(Z_{G^\theta}(S)=\{\pm I\}\)).

  1. Consistency and the sheaf level. The rows are consistent: (SG) is a statement about the pair \((G,X)\), and \(\mathcal H_{K_{\GL}}\) contains the operator \(T_1\), of odd determinant valuation, which \(\mathcal H_{K_{\SL}}\) lacks. Over \(\C(\!(t)\!)\) all units are squares, so the \(L^+\GL_2\)- and \(L^+\SL_2\)-orbits coincide; the label of \(\mathcal O_d\) is \(d\alpha^\vee\). For \(d\ge1\), \(\pi_0\Stab_{L^+\SL_2}(x_d)=\mu_2\), while \(\Stab_{L^+\GL_2}(x_d)=\Stab_{L^+\SL_2}(x_d)\cdot Z(\cO)\) is connected; so the \(\GL_2\) category has only constant local systems, and the \(\SL_2\) category also has the sign local systems \(\mathcal L^-_d\). Because \(L\GL_2^0=L\SL_2\cdot L^+\GL_2\) and \(L^+\GL_2\cap L\SL_2=L^+\SL_2\), the convolution diagrams agree, and with \(\pi:L^+\SL_2\backslash LX\to L^+\GL_2\backslash LX\) one gets \(V_{k\alpha^\vee}\ast\IC_0=\pi^{*}(V_{(k,-k)}\ast\IC_0)=\pi^{*}(V_{(2k,0)}\ast\IC_0)\), the last equality because the centre acts trivially. Remark: a grading of the \(\GL_2\) category matching \(y\mapsto-y\) would make \(V_{(1,0)}\ast\IC_0\) odd; since \(\IC_0\) is a summand of it, any grading compatible with retracts would make \(\IC_0\) both even and odd. So the parity obstruction is not by itself a sheaf-level obstruction, and the sheaf-level verdict for \(\SL_2/\SO_2\) is drawn from the sign local systems instead (Proposition 1.7).

Verified by computation: checks.py, tree_check.py.

Builds on

  • Setup 1.1 · Table 1 of the Summary, re-derived row by row

Built on by

  • Proposition 1.7 · \(\SL_2/\SO_2\): sign local systems never occur in Satake objects (ℓ-adic proof)
  • Summary 1.9 · Classification after the status audit, and the cells where a divergence can still hide
  • Objection 7.15 · Objection: the parity count of the \(\SL_2/T\) analysis holds, but '\(\IC_0\) is a summand of \(V_{(1,0)}\ast\IC_0\)' is false
  • Objection 7.24 · Objection: the support-bound check passes; the \(E_6/F_4\) expectation and two table cells are stale
  • Lemma X2.1 · \(\sigma_n\) is translation by \(h\in L^+\mathrm{SL}_2(\overline{\mathbb F}_q)\) with \(h^{-1}\mathrm{Fr}^n(h)=-I\)
  • Proposition X2.6 · (SG) fails for \(\mathrm{SL}_2/\mathrm{SO}(x^2+y^2)\) over every \(F\), \(q\) odd
  • Audit X2.7 · No \(\mathrm{SL}_2/\mathrm{SO}_2\) conclusion of the run rests on the false §8.2 remark
  • Summary X2.8 · Re-derivation of Summary 4.13, Summary 7.19, Summary 7.25, Summary 7.32 from the repaired steps
  • Proposition X3.5 · Only \(\mathbb C^\times\varphi_0\) generates; J-model fails at residue characteristic 2
  • Correction X3.6 · Correction of A7-24: its two objections stand; its claim that A5-8 holds needs a new proof
Original record, as the agent wrote it at 02:46 UTC

SL2/T versus GL2/T: same variety, different Hecke algebras; SL2/T fails for every q

Setting (classical.html Prop J): \(T\subset\mathrm{GL}_2\) diagonal, \(X=\mathrm{GL}_2/T=\mathrm{SL}_2/T_{\mathrm{SL}_2}\) (equal as varieties: \(\mathrm{GL}_2=\mathrm{SL}_2\cdot Z\), \(Z\subset T\)); also \(=\mathrm{PGL}_2/T\). Over any non-archimedean \(F\), \(X(F)=\mathrm{GL}_2(F)/T(F)=\mathrm{SL}_2(F)/T_{\mathrm{SL}}(F)\) (Hilbert 90 twice; \(\det:T(F)\to F^\times\) is onto). The rows differ only in the acting group, hence in \(K\), \(\mathcal H_K\), \(C_K\).

  1. Verification of J(1) (GL): \(K_{\mathrm{GL}}\backslash X(F)=\{d\ge0\}\) (distance to the apartment, \(T(\mathcal O)\) transitive on the \((\mathcal O/\varpi^d)^\times\) vertices above a fixed apartment vertex, \(\overline T(F)\) translating the apartment); \(T_1=T_{(1,0)}\) is the adjacency operator; the three recursions \(T_1\mathbf 1_0=2\mathbf 1_0+\mathbf 1_1\), \(T_1\mathbf 1_1=(q-1)\mathbf 1_0+\mathbf 1_2\), \(T_1\mathbf 1_d=q\mathbf 1_{d-1}+\mathbf 1_{d+1}\) are the neighbour counts of a tree; \(T_1^d\varphi_0=\mathbf 1_d+\) lower, so \(\mathbb C[T_1]\varphi_0=C_{K_{\mathrm{GL}}}\) and \(C_{K_{\mathrm{GL}}}\cong\mathbb C[y]\) as a free rank-one \(\mathbb C[y]\)-module, \(y=T_1\) (numerically confirmed for \(q=3,5\)). Char-free: no hypothesis on \(F\) or \(q\).
  1. Verification of J(2) (SL, \(q\) odd): \(s=\operatorname{diag}(u,1)\), \(u\) a non-square unit, normalizes \(K_{\mathrm{SL}}\), \(T\) and every \(K_{\mathrm{SL}}t^\lambda K_{\mathrm{SL}}\) and fixes \(x_0\), so \(\sigma(f\ast\varphi)=f\ast\sigma\varphi\) (change of variables \(g\mapsto s^{-1}gs\), using \(f(s^{-1}gs)=f(g)\)); the matrix computation \(k=\begin{pmatrix}a^{-1}&xa^{-1}(ua^2-1)\\0&a\end{pmatrix}\) is correct and \(k\in K\) forces \(u\equiv a^{-2}\bmod\varpi\). Correct. Moreover \(K_{\mathrm{SL}}\backslash X(F)=\{0\}\sqcup\{(d,\epsilon):d\ge1,\epsilon\in\mathcal O^\times/\mathcal O^{\times2}\}\): \(\det\operatorname{Stab}_{K_{\mathrm{GL}}}(y_d)=\mathcal O^{\times2}(1+\varpi^d\mathcal O)=\mathcal O^{\times 2}\) for \(d\ge1\), \(q\) odd.
  1. Structural relation (new). \(K_{\mathrm{GL}}=K_{\mathrm{SL}}\cdot\{\operatorname{diag}(u,1)\}\), so \(C_{K_{\mathrm{GL}}}=\) the \(\sigma_u\)-invariants of \(C_{K_{\mathrm{SL}}}\) for all \(u\), and step 2's argument (valid for every \(q\)) gives \(\mathcal H_{K_{\mathrm{SL}}}\cdot\varphi_0\subseteq C_{K_{\mathrm{GL}}}\). On \(C_{K_{\mathrm{GL}}}\), \(T^{\mathrm{SL}}_{(1,-1)}\) acts as \(T^{\mathrm{GL}}_{(1,-1)}\) (the coset spaces \(K_{\mathrm{SL}}tK_{\mathrm{SL}}/K_{\mathrm{SL}}\to K_{\mathrm{GL}}tK_{\mathrm{GL}}/K_{\mathrm{GL}}\) agree for \(\det t\in\mathcal O^\times\)), \(=T_{(2,0)}\) (centre acts trivially on \(X\)) \(=T_1^2-(q+1)\) (Hecke relation). Hence
$$\mathcal H_{K_{\mathrm{SL}}}\cdot\varphi_0=\mathbb C[T_1^2]\,\varphi_0=\mathbb C[y^2]\subsetneq\mathbb C[y]=C_{K_{\mathrm{GL}}}\subseteq C_{K_{\mathrm{SL}}}.$$

A degree argument (\(p(T_1^2)\varphi_0\) has support exactly \(\{0,\dots,2\deg p\}\)) shows \(\mathbf 1_{\{d=1\}}\notin\mathcal H_{K_{\mathrm{SL}}}\cdot\varphi_0\) (confirmed numerically). So (SG) for \(\mathrm{SL}_2/T\) fails for EVERY \(q\) and every \(F\), by two independent obstructions: the \(y\mapsto-y\) "parity" of the Hecke algebra (visible already on \(\mathrm{GL}_2(\mathcal O)\)-invariant functions; the function-level shadow of \(\varphi\) doubling) and, for \(q\) odd, the square classes (shadow of \(Z_{G^\theta}(S)=\{\pm I\}\)).

  1. Consistency and sheaf level. The two rows are consistent: (SG) is a statement about \((G,X)\), and \(\mathcal H_{K_{\mathrm{GL}}}\) contains the odd-determinant operator \(T_1\) that \(\mathcal H_{K_{\mathrm{SL}}}\) lacks. At sheaf level over \(\mathbb C(\!(t)\!)\) (units are squares) the \(L^+\mathrm{GL}_2\)- and \(L^+\mathrm{SL}_2\)-orbits on \(LX\) coincide (\(\mathcal O_d\), \(d\ge0\); \(\mathcal O_d\leftrightarrow d\alpha^\vee\) by the relative position of \(L\) and \(\epsilon L\), \(\epsilon=\operatorname{diag}(i,-i)\)), but the stabilizers differ: \(\pi_0\operatorname{Stab}_{L^+\mathrm{SL}_2}(x_d)=\mu_2\) (\(d\ge1\)), \(\pi_0\operatorname{Stab}_{L^+\mathrm{GL}_2}(x_d)=1\) (\(\operatorname{Stab}_{\mathrm{GL}}=\operatorname{Stab}_{\mathrm{SL}}\cdot Z(\mathcal O)\)). So \(\mathcal C_{\mathrm{GL}}=\operatorname{Shv}(L^+\mathrm{GL}_2\backslash LX)\) has only constant local systems, \(\mathcal C_{\mathrm{SL}}=\operatorname{Shv}(L^+\mathrm{SL}_2\backslash LX)\) also has the sign local systems \(\mathcal L^-_d\). The Satake objects: \(\mathrm{GL}\): \(V_{(k,0)}\ast\mathrm{IC}_0\), top orbit \(\mathcal O_k\) (every \(k\)); \(\mathrm{SL}\): \(V_{k\alpha^\vee}\ast\mathrm{IC}_0=\pi^*(V_{(k,-k)}\ast\mathrm{IC}_0)=\pi^*(V_{(2k,0)}\ast\mathrm{IC}_0)\), top orbit \(\mathcal O_{2k}\) (\(\pi:L^+\mathrm{SL}_2\backslash LX\to L^+\mathrm{GL}_2\backslash LX\); \(LGL_2^{0}\times^{L^+GL_2}LX=LSL_2\times^{L^+SL_2}LX\) since \(LGL_2^0=LSL_2\cdot L^+GL_2\) and \(L^+GL_2\cap LSL_2=L^+SL_2\)). Remark (proved): the function-level \(\mathbb Z/2\)-grading does not lift to \(\mathcal C_{\mathrm{GL}}\): \(V_{(1,0)}\ast\mathrm{IC}_0\) would be odd, but any grading compatible with retracts makes its summand \(\mathrm{IC}_0\) both even and odd. So the parity obstruction is not by itself a sheaf-level obstruction; the sheaf-level verdict for \(\mathrm{SL}_2/\mathrm{SO}_2\) comes from the sign local systems (A1-7).

Sources