§2 · Agent 2 · SL₂/SO₂: central inertia · 2026-10-17T02:39:50Z

Computation 2.8 · Function level for $\SL_2/T$: Lang count of $K$-orbits, the sign-class dictionary, and a Hecke dimension count for every $q$

proved audit: minor defect ★ key step

Minor defect (second audit). Counts (a) and (c) correct: recursions re-derived symbolically and confirmed on an explicit tree for q = 2, 3, 5; the dimension count was rechecked for q = 2, 3, 4, 5, 7 (tree_G1.py); the failure holds for every q because H_Kφ₀ consists of functions of d (σ_u-invariance for all units u, as in Proposition 1.2). The closing identification Tr(IC_{2n}) = Tr(V_{nα∨}∗IC₀) applies the Betti lemma Lemma 2.6 to ℓ-adic sheaves over 𝔽̄q without comment (its generic-reducedness step uses characteristic 0). The ℓ-adic statement does hold: in the model of symmetric matrices, with λ = nα∨ diagonal, LSO₂ ∩ t^{−λ}L⁺G t^{λ} ⊆ L⁺SO₂ (an element [[a,b],[−b,a]] conjugated into L⁺G has a ∈ 𝒪, b ∈ t^{2n}𝒪), so Stab([t^{λ}, x₀]) = Stab(x{2n}) and the convolution map is an isomorphism over O_{2n} in every odd characteristic. Likewise the correspondences 'C^± ↔ σ-even/odd functions' in (b) are statements about the ℓ-adic analogues of C^±.

The Hecke algebra of $\SL_2$ is generated by the distance-two operator of the tree, whose powers applied to $\varphi_0$ reach only one new orbit per two steps. So the span of the translates misses half of the distance classes, and (SG) fails for $\SL_2/T$ over every non-archimedean field — without square classes.

Computation 2.8.

Let $F$ be non-archimedean with residue field $\F_q$, $X=\SL_2/T$ the split form (the form of Proposition J(2); the $\SO_2$-form of $x^2+y^2$ is a different $\F_q$-form, non-split for $q\equiv3\bmod4$), $K=\SL_2(\cO)$, $\varphi_0=f_0$, and $f_d=\mathbf 1_{\{d\}}$, $d$ the distance from the apartment in the tree.

(a) For $q$ odd, $K\backslash X(F)=\{0\}\sqcup(\{d\ge1\}\times\cO^\times/\cO^{\times2})$; this is Lang's theorem for the orbits of Computation 2.1, since $H^1(\F_q,\mu_2)=\Z/2$.
(b) For $q$ odd, the trace of $\mathcal L_m^-$ at $y\in\mathcal O_m(F)$ is the sign of the class of $Ky$, so $\tr(\mathrm{Frob},\mathcal L_m^-)=\pm(\mathbf 1_{Ky_m}-\mathbf 1_{K\sigma y_m})$ is $\sigma$-odd, while constant coefficients give $\sigma$-even functions; hence $\mathcal C_K^+\leftrightarrow C_K^{\sigma\text{-even}}=C_{\PGL_2(\cO)}$ and $\mathcal C_K^-\leftrightarrow C_K^{\sigma\text{-odd}}$.
(c) For every $q$ and every $F$: $\mathcal H_K=\C[T]$ with $T=\mathbf 1_{Kt^{\alpha^\vee}K}=A^2-(q+1)$, $A$ the adjacency operator, and

$$Tf_0=2f_0+2f_1+f_2,\qquad Tf_1=2(q-1)f_0+(q-2)f_1+f_3,\qquad Tf_2=q(q-1)f_0+(q-1)f_2+f_4,$$ $$Tf_d=q^2f_{d-2}+(q-1)f_d+f_{d+2}\quad(d\ge3).$$

Hence $\dim\big(\mathcal H_K\varphi_0\cap C_c(\{d\le2N\})\big)=N+1<2N+1$; in particular $f_1=\mathbf 1_{\mathcal O_1(F)}\notin\mathcal H_K\varphi_0$, so (SG) fails for $\SL_2/T$ for every $q$.

(a) Orbits. In the tree of valence $q+1$, $T(\cO)$ acts on the vertices above a fixed apartment vertex by $x\mapsto a^2x$, with two orbits (the square classes) at each distance $d\ge1$. In terms of Computation 2.1: the $K$-orbits in $\mathcal O_m(F)$ correspond to $H^1(\F_q,\pi_0\Stab(x_m))=H^1(\F_q,\mu_2)=\Z/2$ for $m\ge1$, and to a point for $m=0$ (connected stabilizer). The point $y=u(x)T$ with $\val x=-1$ of Proposition J(2) lies in $\mathcal O_1$, and $\sigma(y)$ is the other class.
(b) Dictionary. For the Lang torsor, the trace of $\mathcal L_m^\chi$ at $y\in\mathcal O_m(F)$ is $\chi$ of the class of $Ky$. The automorphism $\sigma$ is the action of a generator of $\PGL_2(\cO)/\operatorname{im}\SL_2(\cO)\cong\cO^\times/\cO^{\times2}\cong H^1(\F_q,\mu_2)$: the $\F_q$-shadow of $1\to\mu_2\to\SL_2\to\PGL_2\to1$, whose kernel gives the inertia $\varepsilon$ over $\C$ (where the cokernel is trivial). Thus the statement of Proposition J(2) that $\sigma$ commutes with $\mathcal H_K$ and fixes $\varphi_0$ corresponds to Lemma 2.3, and the statement that $\sigma$ moves $Ky$ corresponds to Theorem 2.4: the sheaf argument is the geometric form of J(2).
(c) Hecke operator. By the Satake isomorphism $\mathcal H_K\cong\C[y^{\pm1}]^W=\C[y+y^{-1}]$, $\mathcal H_K=\C[T]$. As a function of type-0 vertices, $(Tf)(v)=\sum_{\operatorname{dist}(w,v)=2}f(w)$, and since a path of length two either returns ($q+1$ ways) or reaches a vertex at distance two (one way), $T=A^2-(q+1)$. With the adjacency recursions of Proposition J(1) ($Af_0=2f_0+f_1$, $Af_1=(q-1)f_0+f_2$, $Af_d=qf_{d-1}+f_{d+1}$), one obtains the four displayed formulas.
(c) Dimension count. By induction $T^nf_0\in\operatorname{span}(f_0,\dots,f_{2n})$ with coefficient $1$ on $f_{2n}=f_{\varphi(n\alpha^\vee)}$. So a polynomial $p(T)f_0$ is supported in $\{d\le2N\}$ exactly when $\deg p\le N$, and $\mathcal H_K\varphi_0\cap C_c(\{d\le2N\})=\operatorname{span}\{T^nf_0:n\le N\}$, of dimension $N+1$, against $\dim\operatorname{span}(f_0,\dots,f_{2N})=2N+1$ (and $4N+1$ for all $K$-invariant functions on $\{d\le2N\}$ when $q$ is odd). In particular $f_1\notin\mathcal H_K\varphi_0$. This strengthens Proposition J(2), which needs $q$ odd.
Trace of the sheaf picture. The nesting $\mathcal H_K\varphi_0\subsetneq C_K^{\sigma\text{-even}}\subsetneq C_K$ is the trace of $\Csat\subsetneq\mathcal C_K^+\subsetneq\mathcal C_K$ (Proposition 2.7): by Lemma 2.6, $\tr(\IC_{2n})=\tr(V_{n\alpha^\vee}\ast\IC_0)\in\mathcal H_K\varphi_0$, and these span it (one per even label), while the traces $\tr(\IC(\overline{\mathcal O_m},\underline k))=f_m+(\text{lower})$ span the $\sigma$-even functions (one per label).

Verified by computation: tree.py (recursions for symbolic $q$).

Builds on

Computation 2.1 · Stabilizers for $\SL_2/\SO_2$: $\mu_2$ times a pro-unipotent group
Proposition 2.7 · The Satake span is exactly the even-label, constant-coefficient part, strictly inside the even part

Built on by

Summary 2.9 · Verdict for $\SL_2/\SO_2$: both levels fail, and the sheaf argument is the geometric form of Proposition J(2)
Audit 7.2 · Audit: the $\SL_2/\SO_2$ verdict holds at both levels; recursions and fibre counts re-derived
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
Computation X2.3 · Stalks for $\mu=\alpha^\vee,2\alpha^\vee$: $\mathcal L_{\mathrm{sgn}}$ is Kummer on each slice branch
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
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 02:39 UTC

Function level: Lang count of K-orbits, sign-class dictionary, and a Hecke dimension count for all q

Local field $F$ non-archimedean with residue field $\mathbb F_q$, $\mathcal O$, uniformiser $\varpi$; split form $X=\mathrm{SL}_2/T$ (the pages' model in J(2); note the $\mathrm{SO}_2$-form with $x^2+y^2$ is a different $\mathbb F_q$-form, non-split for $q\equiv3\bmod4$, with a different $K$-orbit structure; all statements here are for the split form). $K=\mathrm{SL}_2(\mathcal O)$, $\varphi_0=\mathbf 1_{X(\mathcal O)}=f_0$.

(a) $K$-orbits (proved, $q$ odd). Tree as in A2-5 with finite valence $q+1$: $T(\mathcal O)$ acts on the vertices above an apartment vertex by $x\mapsto a^2x$, with two orbits (square classes) at every distance $d\ge1$. So $K\backslash X(F)=\{0\}\sqcup(\{d\ge1\}\times\mathcal O^\times/\mathcal O^{\times2})$. This is Lang's theorem for the geometric orbits of A2-1: $K$-orbits in $\mathcal O_m(F)\leftrightarrow H^1(\mathbb F_q,\pi_0\operatorname{Stab}(x_m))=H^1(\mathbb F_q,\mu_2)=\mathbb Z/2$ for $m\ge1$, trivial for $m=0$ (connected stabiliser). The pages' $y=u(x)T$, $\operatorname{val}x=-1$, lies in $\mathcal O_1$ ($d=1$); $\sigma(y)$ is the other class.

(b) Dictionary (proved, $q$ odd). For the Lang torsor the trace function of $\mathcal L_m^\chi$ at $y\in\mathcal O_m(F)$ is $\chi(\text{class of }Ky)$: $\operatorname{Tr}(\mathrm{Frob},\mathcal L_m^-)=\pm(\mathbf 1_{Ky_m}-\mathbf 1_{K\sigma y_m})$, $\sigma$-odd, while constant coefficients give $\sigma$-even functions. $\sigma$ is the action of the generator of $\mathrm{PGL}_2(\mathcal O)/\operatorname{im}\mathrm{SL}_2(\mathcal O)\cong\mathcal O^\times/\mathcal O^{\times2}\cong H^1(\mathbb F_q,\mu_2)$: the $\mathbb F_q$-shadow of $1\to\mu_2\to\mathrm{SL}_2\to\mathrm{PGL}_2\to1$, whose kernel is the inertia $\varepsilon$ over $\mathbb C$ (where the cokernel is trivial). Hence: $\mathcal C^+\leftrightarrow C_K^{\sigma\text{-even}}=C_{\mathrm{PGL}_2(\mathcal O)}$, $\mathcal C^-\leftrightarrow C_K^{\sigma\text{-odd}}$; J(2)'s 'σ commutes with $\mathcal H_K$ and fixes $\varphi_0$' is A2-3, 'σ moves $Ky$' is A2-4. The sheaf argument is the geometric form of J(2).

(c) Dimension count (proved; new; every $q$, every $F$). $\mathcal H_K=\mathbb C[T]$, $T:=\mathbf 1_{Kt^{\alpha^\vee}K}$ (Satake: $\mathbb C[y^{\pm1}]^{W}=\mathbb C[y+y^{-1}]$). $(Tf)(v)=\sum_{\operatorname{dist}(w,v)=2}f(w)$, i.e. $T=A^2-(q+1)$ with $A$ the adjacency operator of J(1). On $f_d:=\mathbf 1_{\{d\}}\in C_K$ (exact computation, checked by script for symbolic $q$): $$Tf_0=2f_0+2f_1+f_2,\quad Tf_1=2(q-1)f_0+(q-2)f_1+f_3,\quad Tf_2=q(q-1)f_0+(q-1)f_2+f_4,\quad Tf_d=q^2f_{d-2}+(q-1)f_d+f_{d+2}\ (d\ge3).$$ So $T^nf_0\in\operatorname{span}(f_0,\dots,f_{2n})$ with coefficient $1$ on $f_{2n}=f_{\varphi(n\alpha^\vee)}$. Hence $$\mathcal H_K\varphi_0\cap C_c(\{d\le2N\})=\operatorname{span}\{T^nf_0:n\le N\},\qquad \dim=N+1\ <\ 2N+1=\dim\operatorname{span}(f_0,\dots,f_{2N})\ <\ 4N+1\ (q\text{ odd}).$$ In particular $f_1=\mathbf 1_{\mathcal O_1(F)}\notin\mathcal H_K\varphi_0$: (SG) fails for $X=\mathrm{SL}_2/T$ for every $q$, without square classes (strengthens J(2), which needs $q$ odd). The three-level nesting $\mathcal H_K\varphi_0\subsetneq C_K^{\sigma\text{-even}}\subsetneq C_K$ is the trace of $\mathcal C^{\mathrm{Sat}}\subsetneq\mathcal C^+\subsetneq\mathcal C$ (A2-7): by A2-6, $\operatorname{Tr}(\mathrm{IC}_{2n})=\operatorname{Tr}(V_{n\alpha^\vee}\ast\mathrm{IC}_0)\in\mathcal H_K\varphi_0$ and these span it (one per even label), while $\operatorname{Tr}(\mathrm{IC}(\overline{\mathcal O_m},\underline k))=f_m+\text{lower}$ span the $\sigma$-even functions (one per label).

Sources

classical.html Prop J(1),(2)
summary.html Table 1
sources/chen-yi-v4.pdf Prop 17 (orbits), Appendix A (Lang-type equivariant local systems)

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866

Computation 2.8 · Function level for \(\SL_2/T\): Lang count of \(K\)-orbits, the sign-class dictionary, and a Hecke dimension count for every \(q\)

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Function level: Lang count of K-orbits, sign-class dictionary, and a Hecke dimension count for all q

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