§9 · Agent 9 · Audit: GL₂/T · 2026-10-18T19:24:17Z

Audit 9.1 · Audit of A6-1: $I$-orbits on $L(\mathrm{GL}_2/T)$, closures, stabilizers

proved

Holds with a minor repair: the orbit list A_±, O_{i,m}, the closures Y_n, ωY_n, the connected stabilizers and δ are correct; G(F)/I is a Z-torsor over the oriented edges, not the set of oriented edges, which is harmless because the centre lies in T.

Audit 9.1.

Holds with a minor repair. Over \(k=\overline{\mathbb F}_q\) (\(p\) odd) or \(\mathbb C\), and over every non-archimedean \(F\) for the orbit list: (a)–(e) of Setup 6.1 are correct. Repair: \(G(F)/I\to\{\text{oriented edges}\}\) is a torsor under \(Z(F)/Z(\mathcal O)\cong\mathbb Z\); since \(Z\subset T\), \(I\backslash X(F)=T(F)\backslash G(F)/I=T(F)\backslash G(F)/Z(F)I\) is still the set of \(T(F)\)-orbits of oriented edges.

  1. Model and ends. \(X(F)=G(F)/T(F)\) by Hilbert 90 for the split torus \(T\); \(gT\mapsto(g\infty,g0)\) identifies it with ordered pairs of distinct ends. For \(\gamma=(\infty,u)\), \(u=t^{-m}\), the lattices on \(\gamma\) are \(t^a\mathcal O e_1\oplus t^b\mathcal O(u,1)\); a direct Smith-form computation gives distance \(b\) from \(v_0\) for \(b\ge m\) and \(2m-b\) for \(b<m\), so \(d(v_0,\gamma)=m=-\operatorname{ord}M_\gamma\), as claimed.
  2. Orbits. Recomputed: \(T(F)\) translates \(\gamma_0\) by \(\operatorname{val}(a/b)\in\mathbb Z\) (two orbits of edges on \(\gamma_0\): \(A_\pm\)); \(T(\mathcal O)/Z(\mathcal O)\cong\mathcal O^\times\) acts simply transitively on the \((q-1)q^{m-1}\) vertices at distance \(m\ge1\) projecting to a given point, with stabilizer \(1+t^m\mathcal O\). An edge off \(\gamma_0\) has endpoint distances \((m,m+1)\) or \((m+1,m)\). This is (b), (c).
  3. Stabilizers at finite level. The brief asks whether connectedness survives truncation. If \(\pi:Y_N\to Y_{N,M}\) has connected fibres and \(x=\pi(\gamma)\), then \(\mathrm{Stab}_I(x):=\{g:\pi(g\gamma)=x\}\) acts transitively on \(\pi^{-1}(x)\cap I\gamma=\pi^{-1}(x)\) (pieces are orbits at large level, step 5), so \(\pi^{-1}(x)=\mathrm{Stab}_I(x)/\mathrm{Stab}_I(\gamma)\); connected fibre and connected \(\mathrm{Stab}_I(\gamma)\cong\mathcal O^\times\times(1+t^{m'}\mathcal O)\) give connected \(\mathrm{Stab}_I(x)\). So every \(I\)-equivariant local system on an orbit is constant (checklist 4).
  4. Dimensions. Independently of Chen–Yi Proposition 63: \(T_s=\mathbf 1_{IsI}\) is self-adjoint for an \(I\)-invariant measure \(\mu\), so \(c_{O'O}\mu(O')=c_{OO'}\mu(O)\) for the coefficients of Computation 6.2. Solving gives \(\mu(O_{i,m})=(q-1)q^m\mu(A_\pm)\), the point count of \(\mathbb G_m\times\mathbb A^m\); hence \(\delta(O_{i,m})-\delta(A_\pm)=m+1\) (checked for \(q\in\{2,3,4,5,7,9\}\), all orbits with \(m\le28\); see Computation 9.2).
  5. Closures. The curve \(s\mapsto u=st^{-n}+t^{-m}\) puts \(O_m\) in \(\overline{O_n}\); \(O_n\) is irreducible (\(K\) connected) and \(O_{0,n}\) is its unique \(I\)-orbit of top dimension, hence dense: \(\overline{O_{0,n}}=Y_n\). Each closure \(Y_n\), \(\omega Y_n\), \(A_\pm\) is cut out by finitely many coefficient conditions, so it comes from a finite level; therefore at large \(M\) the pieces of \(Y_N\) are exactly its \(2N+3\) orbits (checklist 1 does not bite).
  6. Independent check. I recomputed the relative position of the base chamber and \(\gamma\) for about 10 000 random pairs of ends in \(\mathbb P^1(\mathbb F_p(t))\), \(p=2,3,5\), from \(t\)-adic valuations of differences in the disc model of the tree (vertex \(D(c,r)=c+t^r\mathcal O\), \(v_0=D(0,0)\), \(v_1=D(0,-1)\)). Every pair has \(|d(v_0,\gamma)-d(v_1,\gamma)|=1\) unless both distances vanish, so only the classes of (b) occur.

Builds on

  • Setup 6.1 · The $I$-orbits on $L(\mathrm{GL}_2/T)$ as relative positions of an edge and a geodesic

Built on by

Original record, as the agent wrote it at 19:24 UTC

Audit of A6-1: $I$-orbits on $L(\mathrm{GL}_2/T)$, closures, stabilizers

  1. Model and ends. \(X(F)=G(F)/T(F)\) by Hilbert 90 for the split torus \(T\); \(gT\mapsto(g\infty,g0)\) identifies it with ordered pairs of distinct ends. For \(\gamma=(\infty,u)\), \(u=t^{-m}\), the lattices on \(\gamma\) are \(t^a\mathcal O e_1\oplus t^b\mathcal O(u,1)\); a direct Smith-form computation gives distance \(b\) from \(v_0\) for \(b\ge m\) and \(2m-b\) for \(b<m\), so \(d(v_0,\gamma)=m=-\operatorname{ord}M_\gamma\), as claimed.
  2. Orbits. Recomputed: \(T(F)\) translates \(\gamma_0\) by \(\operatorname{val}(a/b)\in\mathbb Z\) (two orbits of edges on \(\gamma_0\): \(A_\pm\)); \(T(\mathcal O)/Z(\mathcal O)\cong\mathcal O^\times\) acts simply transitively on the \((q-1)q^{m-1}\) vertices at distance \(m\ge1\) projecting to a given point, with stabilizer \(1+t^m\mathcal O\). An edge off \(\gamma_0\) has endpoint distances \((m,m+1)\) or \((m+1,m)\). This is (b), (c).
  3. Stabilizers at finite level. The brief asks whether connectedness survives truncation. If \(\pi:Y_N\to Y_{N,M}\) has connected fibres and \(x=\pi(\gamma)\), then \(\mathrm{Stab}_I(x):=\{g:\pi(g\gamma)=x\}\) acts transitively on \(\pi^{-1}(x)\cap I\gamma=\pi^{-1}(x)\) (pieces are orbits at large level, step 5), so \(\pi^{-1}(x)=\mathrm{Stab}_I(x)/\mathrm{Stab}_I(\gamma)\); connected fibre and connected \(\mathrm{Stab}_I(\gamma)\cong\mathcal O^\times\times(1+t^{m'}\mathcal O)\) give connected \(\mathrm{Stab}_I(x)\). So every \(I\)-equivariant local system on an orbit is constant (checklist 4).
  4. Dimensions. Independently of Chen–Yi Proposition 63: \(T_s=\mathbf 1_{IsI}\) is self-adjoint for an \(I\)-invariant measure \(\mu\), so \(c_{O'O}\mu(O')=c_{OO'}\mu(O)\) for the coefficients of Computation 6.2. Solving gives \(\mu(O_{i,m})=(q-1)q^m\mu(A_\pm)\), the point count of \(\mathbb G_m\times\mathbb A^m\); hence \(\delta(O_{i,m})-\delta(A_\pm)=m+1\) (checked for \(q\in\{2,3,4,5,7,9\}\), all orbits with \(m\le28\); see Computation 9.2).
  5. Closures. The curve \(s\mapsto u=st^{-n}+t^{-m}\) puts \(O_m\) in \(\overline{O_n}\); \(O_n\) is irreducible (\(K\) connected) and \(O_{0,n}\) is its unique \(I\)-orbit of top dimension, hence dense: \(\overline{O_{0,n}}=Y_n\). Each closure \(Y_n\), \(\omega Y_n\), \(A_\pm\) is cut out by finitely many coefficient conditions, so it comes from a finite level; therefore at large \(M\) the pieces of \(Y_N\) are exactly its \(2N+3\) orbits (checklist 1 does not bite).
  6. Independent check. I recomputed the relative position of the base chamber and \(\gamma\) for about 10 000 random pairs of ends in \(\mathbb P^1(\mathbb F_p(t))\), \(p=2,3,5\), from \(t\)-adic valuations of differences in the disc model of the tree (vertex \(D(c,r)=c+t^r\mathcal O\), \(v_0=D(0,0)\), \(v_1=D(0,-1)\)). Every pair has \(|d(v_0,\gamma)-d(v_1,\gamma)|=1\) unless both distances vanish, so only the classes of (b) occur.

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