§6 · Agent 6 · GL₂/T: the Iwahori module · 2026-10-18T18:45:11Z

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

proved

Identifies LX for X = GL2/T with oriented geodesics of the Bruhat–Tits tree, lists all I-orbits (two closed ones A+, A− and two infinite families), their dimensions, connected stabilizers and closures; every level Y_N has finitely many orbits.

Setup 6.1.

Let \(k=\overline{\mathbb F}_q\) (\(p\) odd) or \(k=\mathbb C\), \(F=k((t))\), \(O=k[[t]]\), \(G=\mathrm{GL}_2\), \(T\) the diagonal torus, \(I\) the upper-triangular-mod-\(t\) Iwahori, \(\omega=\begin{pmatrix}0&1\\ t&0\end{pmatrix}\). Let \(\mathcal T\) be the tree of \(\mathrm{PGL}_2(F)\) (vertices = homothety classes of \(O\)-lattices in \(F^2\)), \(v_0=[O^2]\), \(v_1=[O\oplus tO]\), \(C_0=(v_0,v_1)\) the base chamber (an oriented edge; \(G(F)/I=\) oriented edges).

(a) \(X\cong(\mathbb P^1\times\mathbb P^1)\smallsetminus\Delta\cong\{x^2+yz=1\}\subset\mathbb A^3\) via \(gT\mapsto g\,\mathrm{diag}(1,-1)g^{-1}=\begin{pmatrix}x&y\\ z&-x\end{pmatrix}\); \(LX=X(F)=\{\)oriented geodesics \(\gamma=(a,b)\) of \(\mathcal T\}\); the involution \(M_\gamma\) preserves a lattice \(L\) iff \([L]\in\gamma\), and \(-\mathrm{ord}_t(M_\gamma)=d(v_0,\gamma)\). Hence $$Y_n:=LX\cap t^{-n}\mathfrak{gl}_2(O)=\{\gamma:\ d(v_0,\gamma)\le n\},\qquad Y_0=L^+X=\{\gamma\ni v_0\},\qquad LX=\mathrm{colim}_n\,Y_n .$$ (b) The \(I\)-orbits on \(LX\) are $$A_\pm=\{\gamma\supset\{v_0,v_1\},\ \gamma\text{ runs }v_0\to v_1\ (+)\text{ or }v_1\to v_0\ (-)\},\qquad O_{i,m}=\{d(v_i,\gamma)=m,\ d(v_{1-i},\gamma)=m+1\}\quad(i\in\{0,1\},\ m\ge0).$$ (c) \(\mathrm{Stab}_I(\gamma)=Z(O)\cdot\{\mathrm{diag}_\gamma(a,1):a\in1+t^{m'}O\}\) (\(m'=\max_i d(v_i,\gamma)\); \(1+t^0O:=O^\times\)), connected; a dimension theory is \(\delta(A_\pm)=0\), \(\delta(O_{i,m})=m+1\). (d) \(\overline{O_{0,n}}=Y_n\), \(\overline{O_{1,n}}=\omega Y_n=\{d(v_1,\gamma)\le n\}\), \(A_\pm\) closed; \(Y_n\) is the union of the \(2n+3\) orbits \(A_\pm,O_{0,m}\ (m\le n),O_{1,m}\ (m\le n-1)\). \(\omega\) normalizes \(I\), swaps \(v_0\leftrightarrow v_1\) and acts by \(A_+\leftrightarrow A_-\), \(O_{0,m}\leftrightarrow O_{1,m}\). (e) The same list is the set of \(I\)-orbits on \(X(F)\) for \(F=\mathbb F_q((t))\).

  1. Model. \(\mathrm{PGL}_2\) acts simply transitively modulo \(T\) on ordered pairs of distinct points of \(\mathbb P^1\), stabilizer of \((\infty,0)\) is \(T\), so \(X=(\mathbb P^1)^2\smallsetminus\Delta\), affine (complement of an ample divisor), and \(g\mapsto g\,\mathrm{diag}(1,-1)g^{-1}\) identifies it with the quadric of trace-zero involutions (\(p\ne2\)). \(H^1(F,T)=0\) (Hilbert 90), so \(X(F)=G(F)/T(F)\) (Chen–Yi, Proposition 17(ii)). Points of \(\mathbb P^1(F)\) are the ends of \(\mathcal T\), so \(X(F)\) is the set of oriented geodesics.
  2. Lattices on \(\gamma\). \(M=M_\gamma\) has eigenlines \(\ell_\pm\) (the ends). Since \(2\in O^\times\), \(M(L)=L\) iff the projectors \((1\pm M)/2\) preserve \(L\) iff \(L=(L\cap\ell_+)\oplus(L\cap\ell_-)\) iff \([L]\) lies on the apartment \(\gamma\). For \(\gamma=(\infty,u)\), \(u=t^{-m}\): \(M_\gamma=\begin{pmatrix}1&-2u\\0&-1\end{pmatrix}\) has \(\mathrm{ord}=-m\) and \(d(v_0,\gamma)=m\); \(K=G(O)\) preserves both quantities and is transitive on \(\{d(v_0,\gamma)=m\}\) (step 3 with \(v_0\) in place of the edge), so \(-\mathrm{ord}\,M_\gamma=d(v_0,\gamma)\) and \(Y_n\) is the \(n\)-th term of the ind-structure of \(LX\subset\mathbb A^3(F)\).
  3. Orbits. \(IgT\mapsto Tg^{-1}I\) identifies \(I\backslash X(F)\) with \(T(F)\)-orbits of oriented edges, \(T=T_{\gamma_0}\), \(\gamma_0=(0,\infty)\). \(T(F)\) translates \(\gamma_0\) by \(\mathrm{val}(a/b)\) (transitive on its edges of each orientation: \(A_\pm\)); \(T(O)\) fixes \(\gamma_0\) pointwise and \(\{a/b\equiv1\bmod t^m\}\) acts simply transitively modulo \(\{a/b\equiv1\bmod t^{m+1}\}\) on the \(q\) (resp. \(q-1\) at \(m=0\)) neighbours one step farther from \(\gamma_0\) of a vertex at distance \(m\). Hence \(T(F)\) is transitive on edges with given \((d(u_0,\gamma_0),d(u_1,\gamma_0))=(m,m+1)\) or \((m+1,m)\): the orbits \(O_{0,m},O_{1,m}\). The stabilizer of an edge at distance \(m'\) (farther endpoint) is \(Z\cdot(1+t^{m'}O)\): (c). The same argument works over \(\mathbb F_q\): (e).
  4. Dimensions. \(\dim(I\gamma)-\dim(I\gamma')=\mathrm{codim}\,\mathrm{Stab}_I(\gamma)-\mathrm{codim}\,\mathrm{Stab}_I(\gamma')\) inside the rank-one torus \(O^\times\), giving \(\delta\); Chen–Yi, Proposition 63, guarantees a dimension theory with \(\delta(v)-\delta(u)=\mathrm{codim}\). Check: \(A_+\subset Y_0\) is the preimage under \(L^+X\to X(k)\) of the closed \(B\)-orbit \(\{\infty\}\times\mathbb A^1\) of codimension \(1\).
  5. Closures. \(Y_n\) is closed (step 2) and \(K\)-stable; \(K\)-orbits in it are \(O_m=\{d(v_0,\gamma)=m\}=O_{0,m}\sqcup O_{1,m-1}\) (\(O_0=A_+\sqcup A_-\sqcup O_{0,0}\)). The curve \(s\mapsto M=\begin{pmatrix}1&-2(st^{-n}+t^{-m})\\0&-1\end{pmatrix}\) lies in \(O_n\) for \(s\ne0\) and in \(O_m\) at \(s=0\), so \(O_m\subset\overline{O_n}\) (\(m<n\)). \(O_n\) is irreducible (\(K\) connected) and \(O_{0,n}\) is its top-dimensional \(I\)-orbit, hence dense: \(\overline{O_{0,n}}=Y_n\). Apply \(\omega\) for \(O_{1,n}\). \(A_\pm\) are preimages of closed \(B\)-orbits of \(X(k)\) inside the closed \(L^+X\).
  6. Check. A brute-force count on the ball of radius \(6\) (\(q=2\)) and \(5\) (\(q=3\)) of the \((q+1)\)-regular tree confirms that relative position \((C_0,\gamma)\) takes exactly the values in (b) and that Hecke fibre counts are constant on each class (Computation 6.2).

Compared with \(\mathbb A^n\) (checklist 1): here every \(Y_N\) is a finite union of orbits, so pieces of large level are orbits and every simple object of \(\mathcal C_I\) is an \(\mathrm{IC}\) of an orbit with constant coefficients.

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Original record, as the agent wrote it at 18:45 UTC

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

  1. Model. \(\mathrm{PGL}_2\) acts simply transitively modulo \(T\) on ordered pairs of distinct points of \(\mathbb P^1\), stabilizer of \((\infty,0)\) is \(T\), so \(X=(\mathbb P^1)^2\smallsetminus\Delta\), affine (complement of an ample divisor), and \(g\mapsto g\,\mathrm{diag}(1,-1)g^{-1}\) identifies it with the quadric of trace-zero involutions (\(p\ne2\)). \(H^1(F,T)=0\) (Hilbert 90), so \(X(F)=G(F)/T(F)\) (Chen–Yi, Proposition 17(ii)). Points of \(\mathbb P^1(F)\) are the ends of \(\mathcal T\), so \(X(F)\) is the set of oriented geodesics.
  2. Lattices on \(\gamma\). \(M=M_\gamma\) has eigenlines \(\ell_\pm\) (the ends). Since \(2\in O^\times\), \(M(L)=L\) iff the projectors \((1\pm M)/2\) preserve \(L\) iff \(L=(L\cap\ell_+)\oplus(L\cap\ell_-)\) iff \([L]\) lies on the apartment \(\gamma\). For \(\gamma=(\infty,u)\), \(u=t^{-m}\): \(M_\gamma=\begin{pmatrix}1&-2u\\0&-1\end{pmatrix}\) has \(\mathrm{ord}=-m\) and \(d(v_0,\gamma)=m\); \(K=G(O)\) preserves both quantities and is transitive on \(\{d(v_0,\gamma)=m\}\) (step 3 with \(v_0\) in place of the edge), so \(-\mathrm{ord}\,M_\gamma=d(v_0,\gamma)\) and \(Y_n\) is the \(n\)-th term of the ind-structure of \(LX\subset\mathbb A^3(F)\).
  3. Orbits. \(IgT\mapsto Tg^{-1}I\) identifies \(I\backslash X(F)\) with \(T(F)\)-orbits of oriented edges, \(T=T_{\gamma_0}\), \(\gamma_0=(0,\infty)\). \(T(F)\) translates \(\gamma_0\) by \(\mathrm{val}(a/b)\) (transitive on its edges of each orientation: \(A_\pm\)); \(T(O)\) fixes \(\gamma_0\) pointwise and \(\{a/b\equiv1\bmod t^m\}\) acts simply transitively modulo \(\{a/b\equiv1\bmod t^{m+1}\}\) on the \(q\) (resp. \(q-1\) at \(m=0\)) neighbours one step farther from \(\gamma_0\) of a vertex at distance \(m\). Hence \(T(F)\) is transitive on edges with given \((d(u_0,\gamma_0),d(u_1,\gamma_0))=(m,m+1)\) or \((m+1,m)\): the orbits \(O_{0,m},O_{1,m}\). The stabilizer of an edge at distance \(m'\) (farther endpoint) is \(Z\cdot(1+t^{m'}O)\): (c). The same argument works over \(\mathbb F_q\): (e).
  4. Dimensions. \(\dim(I\gamma)-\dim(I\gamma')=\mathrm{codim}\,\mathrm{Stab}_I(\gamma)-\mathrm{codim}\,\mathrm{Stab}_I(\gamma')\) inside the rank-one torus \(O^\times\), giving \(\delta\); Chen–Yi, Proposition 63, guarantees a dimension theory with \(\delta(v)-\delta(u)=\mathrm{codim}\). Check: \(A_+\subset Y_0\) is the preimage under \(L^+X\to X(k)\) of the closed \(B\)-orbit \(\{\infty\}\times\mathbb A^1\) of codimension \(1\).
  5. Closures. \(Y_n\) is closed (step 2) and \(K\)-stable; \(K\)-orbits in it are \(O_m=\{d(v_0,\gamma)=m\}=O_{0,m}\sqcup O_{1,m-1}\) (\(O_0=A_+\sqcup A_-\sqcup O_{0,0}\)). The curve \(s\mapsto M=\begin{pmatrix}1&-2(st^{-n}+t^{-m})\\0&-1\end{pmatrix}\) lies in \(O_n\) for \(s\ne0\) and in \(O_m\) at \(s=0\), so \(O_m\subset\overline{O_n}\) (\(m<n\)). \(O_n\) is irreducible (\(K\) connected) and \(O_{0,n}\) is its top-dimensional \(I\)-orbit, hence dense: \(\overline{O_{0,n}}=Y_n\). Apply \(\omega\) for \(O_{1,n}\). \(A_\pm\) are preimages of closed \(B\)-orbits of \(X(k)\) inside the closed \(L^+X\).
  6. Check. A brute-force count on the ball of radius \(6\) (\(q=2\)) and \(5\) (\(q=3\)) of the \((q+1)\)-regular tree confirms that relative position \((C_0,\gamma)\) takes exactly the values in (b) and that Hecke fibre counts are constant on each class (Computation 6.2).

Compared with \(\mathbb A^n\) (checklist 1): here every \(Y_N\) is a finite union of orbits, so pieces of large level are orbits and every simple object of \(\mathcal C_I\) is an \(\mathrm{IC}\) of an orbit with constant coefficients.

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