§2 · Agent 2 · SL₂/SO₂: central inertia · 2026-10-17T02:33:26Z

Computation 2.5 · Fibres of the convolution map are finite sets of tree vertices; the map is finite and an isomorphism over the top orbit

proved audit: correct

Second audit: correct. Fibre count and labelling re-derived (the label of u(x)T with val x = −d is dα∨, since τ(u(x)T) = u(2x) has elementary divisors t^{∓d}).

For \(\SL_2/\SO_2\) the arc space of \(X\) is a line in the Bruhat–Tits tree, so the fibres of the convolution map \(\Gr^{\le n\alpha^\vee}\widetilde\times L^+X\to LX\) are finite sets of vertices on a line. The map is finite and is an isomorphism over the open orbit \(\mathcal O_{2n}\).

Computation 2.5.

Over \(k=\C\), \(X\simeq\SL_2/T\) with \(T\) diagonal. Let \(d(y)\) be the distance from the apartment \(A_T\) of the vertex attached to \(y\in X(F)\), \(F=\C(\!(t)\!)\), and let $$m_n:Z_n:=\Gr^{\le n\alpha^\vee}\mathbin{\widetilde\times}L^+X\longrightarrow LX,\qquad[g,x]\longmapsto gx .$$

  1. The \(L^+G\)-orbits are \(\{d=m\}=\mathcal O_m\), \(m\ge0\), with \(\{d=0\}=L^+X\).
  2. The image of \(m_n\) is \(\overline{\mathcal O_{2n}}\). For \(y=y_0T\) with \(d(y)=m'\), the fibre \(m_n^{-1}(y)\) is the set of type-0 vertices \(w\) of the line \(y_0A_T\) with \(\operatorname{dist}(w,v_0)\le2n\), and
$$s_n(m'):=\#m_n^{-1}(y)=\#\{k:m'\le2k\le2n\}+\#\{k:m'<2k\le2n\};$$

in particular \(s_n(2n)=1\), \(s_n(2n-1)=2\), \(s_n(0)=2n+1\), and \(s_1=(3,2,1)\).

  1. \(m_n\) is finite on the finite-type slice models, and an isomorphism over \(\mathcal O_{2n}\).
  1. Tree model. Over \(\C\) all involutions of \(\SL_2\) with a torus as fixed group are conjugate (\(g\mapsto g^{-\mathsf T}=\operatorname{Ad}(J)\) with \(J\sim\operatorname{diag}(i,-i)\)), so \(X\simeq\SL_2/T\). As in Proposition J(1), \(K\backslash X(F)=T(F)\backslash\SL_2(F)/K\) is the set of \(T(F)\)-orbits on type-0 vertices of the tree of \(\PGL_2(F)\) (of valence \(\lvert\mathbb P^1(\C)\rvert\)). \(T(F)\) translates \(A_T\) by even steps and \(T(\cO)\) acts on the vertices above a fixed apartment vertex by \(x\mapsto a^2x\), transitively because units are squares. So the orbits are the sets \(\{d=m\}\). They carry the labels of Computation 2.1: \(\mathcal O_0\) has codimension one in \(\overline{\mathcal O_1}\) (Chen–Yi, Remark 89), and the point \(u(x)T\) with \(\val x=-1\) has \(d=1\), since \(u(x)\) fixes exactly a half-apartment.
  2. Fibres. The image of \(m_n\) is closed, \(L^+G\)-stable, contains \(t^{n\alpha^\vee}x_0=x_{2n}\) and lies in \(\overline{\mathcal O_{2n}}\) by the support bound (Proposition III, step 2); so it is \(\overline{\mathcal O_{2n}}\). For \(y=y_0T\) with vertex \(v_y=y_0^{-1}v_0\),
$$m_n^{-1}(y)=\{gK\in\Gr^{\le n\alpha^\vee}:g^{-1}y\in X(\cO)\}=\{w=gv_0:\operatorname{dist}(w,v_0)\le2n,\ w\in y_0A_T\},$$

because \(g^{-1}y_0\in KT(F)\) exactly when \(gv_0\in y_0A_T\). If \(d(y)=m'\), the line \(y_0A_T\) is at distance \(m'\) from \(v_0\), and it contains one vertex at distance \(r=m'\) from \(v_0\), two at each distance \(r>m'\), and none at distance \(r<m'\); type-0 vertices are those at even distance. This gives \(s_n(m')\).

  1. Finiteness and the top orbit. \(m_n\) is proper (Proposition III, step 3) with finite fibres, hence finite on the slice models \(Z_S:=m_n^{-1}(Y_S)\to Y_S:=S^\mu\cap\overline{\mathcal O_{2n}}\), which are affine of finite type (Chen–Yi, Lemma 45(iii)). In characteristic \(0\), the generic fibre of this finite map of reduced schemes is reduced (a localisation), and \(Y_S\cap\mathcal O_{2n}\) is dense in every component of \(Y_S\) (the map \(L^+G\times Y_S\to\overline{\mathcal O_{2n}}\) is formally smooth, Chen–Yi, Prop. 47(iii), hence open). So each component of \(Z_S\) maps birationally onto a component of \(Y_S\), distinct components to distinct components. By equivariance all fibres over \(\mathcal O_{2n}\) are isomorphic to \(m_n^{-1}(x_{2n})\), which is therefore a single reduced point; hence \(m_n^{-1}(\mathcal O_{2n})\simeq L^+G\times^{\Stab(x_{2n})}\mathrm{pt}=\mathcal O_{2n}\), i.e. \(m_n\) is an isomorphism over the open dense orbit.

Verified by computation: tree.py.

Builds on

  • Computation 2.1 · Stabilizers for \(\SL_2/\SO_2\): \(\mu_2\) times a pro-unipotent group

Built on by

  • Lemma 2.6 · Each Satake object is a single IC sheaf: \(V_{n\alpha^\vee}\ast\IC_0\simeq\IC(\overline{\mathcal O_{2n}})\)
  • Audit 7.2 · Audit: the \(\SL_2/\SO_2\) verdict holds at both levels; recursions and fibre counts re-derived
Original record, as the agent wrote it at 02:33 UTC

Fibres of the convolution map are finite sets of tree vertices; the map is finite and birational

Tree model over \(\mathbb C\). Over \(k=\mathbb C\) all involutions of \(\mathrm{SL}_2\) with a torus as fixed group are conjugate (\(g\mapsto g^{-\mathsf T}=\mathrm{Ad}(J)\), \(J=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\sim\operatorname{diag}(i,-i)\)), so \(X\simeq\mathrm{SL}_2/T\), \(T\) diagonal. As in classical.html J(1): \(K\backslash X(F)=T(F)\backslash\mathrm{SL}_2(F)/K\) = \(T(F)\)-orbits on type-0 vertices of the Bruhat–Tits tree of \(\mathrm{PGL}_2(F)\), \(F=\mathbb C((t))\) (valence \(|\mathbb P^1(\mathbb C)|\)); invariant \(d(v)=\) distance to the apartment \(A_T\). \(T(F)\) translates \(A_T\) by even steps and \(T(\mathcal O)\) acts on the vertices above a fixed apartment vertex by \(x\mapsto a^2x\), transitively (units are squares). So the \(L^+G\)-orbits are \(\{d=m\}\), \(m\ge0\), with \(\{d=0\}=L^+X\); \(\{d=m\}=\mathcal O_m\) (same labelling as A2-1: \(\mathcal O_0\) has codimension 1 in \(\overline{\mathcal O_1}\), Chen–Yi Remark 89; the pages' \(u(x)T\), \(\operatorname{val}x=-1\), has \(d=1\): \(u(x)\) fixes exactly the half-apartment \(a-b\le-1\)).

Fibres (proved). Let \(m_n:Z_n:=\mathrm{Gr}^{\le n\alpha^\vee}\mathbin{\widetilde\times}L^+X\to LX\), \([g,x]\mapsto gx\). Its image is \(\overline{\mathcal O_{2n}}\) (support bound of criterion.html step 2, and \(t^{n\alpha^\vee}x_0=x_{2n}\)). For \(y=y_0T\in X(F)\) with vertex \(v_y=y_0^{-1}v_0\): $$F_y:=m_n^{-1}(y)=\{gK\in\mathrm{Gr}^{\le n\alpha^\vee}:g^{-1}y\in X(\mathcal O)\}=\{w=gv_0:\ \operatorname{dist}(w,v_0)\le2n,\ w\in y_0A_T\},$$ since \(g^{-1}y_0\in KT(F)\iff gv_0\in y_0A_T\). Thus \(F_y(\mathbb C)\) is the set of vertices of the line \(y_0A_T\) at even distance \(\le2n\) from \(v_0\). If \(d(y)=m'\) (so \(\operatorname{dist}(v_0,y_0A_T)=m'\)), the line has one vertex at distance \(2k\) if \(2k=m'\), two if \(2k>m'\), none if \(2k<m'\): $$s_n(m'):=|F_y|=\#\{k:m'\le 2k\le 2n\}+\#\{k:m'<2k\le2n\},\qquad s_n(2n)=1,\ s_n(2n-1)=2,\ s_n(0)=2n+1;\quad n=1:\ (3,2,1).$$

Consequences (proved). (i) \(m_n\) is proper (criterion.html step 3: closed in \(\mathrm{Gr}^{\le n\alpha^\vee}\times LX\), \(\mathrm{Gr}^{\le n\alpha^\vee}\) projective) with finite fibres (each \(F_y\) is a closed subscheme of a projective variety with finitely many points), hence finite on the finite-type slice models \(Z_S:=m_n^{-1}(Y_S)\to Y_S:=S^{\mu}\cap\overline{\mathcal O_{2n}}\) (Chen–Yi Lemma 45(iii): finite type affine). (ii) Over \(\mathcal O_{2n}\) the fibre is one point. Char 0: the generic fibre of the finite map of reduced schemes \(Z_S\to Y_S\) is reduced (a localisation), and \(Y_S\cap\mathcal O_{2n}\) is dense in every component of \(Y_S\) (\(L^+G\times Y_S\to\overline{\mathcal O_{2n}}\) is formally smooth, Chen–Yi Prop. 47(iii), hence open); so each component of \(Z_S\) maps birationally onto a component of \(Y_S\), and distinct components to distinct components. (iii) By \(L^+G\)-equivariance all fibres over \(\mathcal O_{2n}\) are isomorphic to \(F:=m_n^{-1}(x_{2n})\); (ii) forces \(F\) to be a reduced point, so \(m_n^{-1}(\mathcal O_{2n})\simeq L^+G\times^{\operatorname{Stab}(x_{2n})}\mathrm{pt}=\mathcal O_{2n}\): \(m_n\) is an isomorphism over the open dense orbit.

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