Proposition III as a paper · statement and complete proof (PDF) →

Definitions

• The space, its orbits and labels

  Work over \(k=\mathbb C\) with complex coefficients. \(G=\mathrm{SL}_2\), \(\theta(g)=(g^{\mathsf T})^{-1}\), \(G^\theta=\mathrm{SO}_2\), which is connected. The Cartan embedding \(\tau(g\,\mathrm{SO}_2)=g\theta(g)^{-1}=gg^{\mathsf T}\) identifies

  $$X=G/G^\theta\cong\{A\in\mathrm{Mat}_2:A=A^{\mathsf T},\ \det A=1\},\qquad g\cdot A=gAg^{\mathsf T},\qquad x_0=I.$$

  Labels. The diagonal torus is both \(S\) and \(T\); \(\theta\) inverts it, so the leading-term map doubles:

  $$\Lambda_T^+=\Lambda_S^+=\mathbb Z_{\ge0}\alpha^\vee,\qquad\varphi(n\alpha^\vee)=n\alpha^\vee-\theta(n\alpha^\vee)=2n\alpha^\vee.$$

  Orbits. With \(F=k(\!(t)\!)\), \(LX=X(F)\) and \(L^+G=\mathrm{SL}_2(k[\![t]\!])\):

  $$x_m=\operatorname{diag}(t^m,t^{-m})=t^{m\alpha^\vee},\qquad\mathcal O_m=L^+G\cdot x_m\ (m\ge0),\qquad t^{n\alpha^\vee}\cdot x_0=x_{2n}.$$

  Sign local systems. For \(m\ge1\), \(\mathcal L^-_m\) is the rank-one equivariant local system on \(\mathcal O_m\) on which \(-I\in\operatorname{Stab}_{L^+G}(x_m)\) acts by \(-1\) (Step 1 shows \(\pi_0\operatorname{Stab}(x_m)=\{\pm I\}\)); its standard object is \(\Delta_m^-=a_{m!}\mathcal L^-_m\).

  Example. \(\operatorname{diag}(t^{-1},t)=w\cdot x_1\in\mathcal O_1\), and \(\begin{pmatrix}1&t^{-1}\\t^{-1}&1+t^{-2}\end{pmatrix}\in\mathcal O_2\), since its entries have minimal valuation \(-2\).

• Categories, Hecke action, Satake span

  \(\mathcal C_K=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash LX)\) is the pages' small, idempotent-complete stable \(\infty\)-category of bounded \(L^+G\)-equivariant constructible complexes on \(LX\), each supported on finitely many orbit closures; \(a_m:\mathcal O_m\hookrightarrow LX\), \(\mathrm{IC}_0=a_{0!}\underline k_{\mathcal O_0}\). The Hecke category acts by convolution:

  $$\mathcal H=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash\operatorname{Gr}),\qquad\mathcal A\ast\mathcal F=m_!(\mathcal A\mathbin{\widetilde\boxtimes}\mathcal F),\qquad m:LG\times^{L^+G}LX\to LX,$$ $$\mathcal C_K^{\mathrm{Sat}}=\operatorname{Thick}_{\mathcal C_K}\{V_\lambda\ast\mathrm{IC}_0:\lambda\in\Lambda_T^+\},\qquad V_\lambda=\mathrm{IC}(\operatorname{Gr}^{\le\lambda}).$$

  \(\mathrm{IC}_{2n}=\mathrm{IC}(\overline{\mathcal O_{2n}},\underline k)\). A thick span is closed under isomorphisms, shifts, cofibers and retracts.

• Central inertia and the two parts

  The central element \(-I\in L^+G\) acts trivially on \(LX\) and on \(\operatorname{Gr}\), so it is a 2-automorphism of the identity of the quotient stack and gives a natural involution of every object:

  $$\varepsilon\in\operatorname{Aut}(\mathrm{id}_{\mathcal C_K}),\qquad\varepsilon^2=\mathrm{id},\qquad\mathcal C_K^\pm=\{F\in\mathcal C_K:\varepsilon_F=\pm\mathrm{id}\}.$$

  Properties used: (E1) naturality; (E2) compatibility with shifts and truncations; (E3) \(f^*,f_!,f_*\) commute with \(\varepsilon\) for equivariant \(f\); (E4) \(\varepsilon=\mathrm{id}\) on sheaves pulled back from a point; (E5) \(\varepsilon=\rho(-I)\) on a local system with monodromy \(\rho\).

  Example. \(\varepsilon_{\mathrm{IC}_0}=\mathrm{id}\) and \(\varepsilon_{\Delta_m^-}=-\mathrm{id}\).

• Function level

  \(F\) is a non-archimedean local field with ring of integers \(\mathcal O\), uniformizer \(\varpi\), residue field \(\mathbb F_q\). At function level \(X=\mathrm{SL}_2/T\), \(T\) diagonal (over \(\mathbb C\) this is \(\mathrm{SL}_2/\mathrm{SO}_2\); over \(\mathbb F_q\) the two are different forms).

  $$K=\mathrm{SL}_2(\mathcal O),\quad K_{\mathrm{GL}}=\mathrm{GL}_2(\mathcal O),\quad\mathcal H_K=C_c(K\backslash\mathrm{SL}_2(F)/K)\ (\operatorname{vol}K=1),\quad\varphi_0=\mathbf 1_{X(\mathcal O)},\quad C_K=C_c^\infty(X(F))^K.$$

  For \(x=gT\), \(d(x)\) is the distance of the vertex \(g^{-1}[\mathcal O^2]\) of the tree of \(\mathrm{PGL}_2(F)\) from the apartment of \(T\); \(\mathbf 1_d=\mathbf 1_{\{d(x)=d\}}\) and \(T_1=\mathbf 1_{K_{\mathrm{GL}}\operatorname{diag}(\varpi,1)K_{\mathrm{GL}}}\).

Inputs (P) and D0–D3

The main claims (c) and (f) use neither. Proposition I (I · summand criterion) is not used.

• (P) · Perverse t-structure.

  On each closed finite union of orbits \(Z\), \(\mathcal C_Z\) has a bounded perverse t-structure whose heart has finite length, with simple objects $$\mathrm{IC}(\overline{\mathcal O_m},\underline k),\qquad\mathrm{IC}(\overline{\mathcal O_m},\mathcal L^-_m)\qquad(\mathcal O_m\subseteq Z).$$ This is the standing input of the pages' Proposition III, step 6 (BBD, Theorem 4.3.1 on finite-type models). Used only in part (e).
• D0–D3 · ℓ-adic inputs.

  Over \(\overline{\mathbb F}_q\), \(p\ne2\): Chen–Yi's orbit geometry (D0); proper base change and Grothendieck–Lefschetz (D1); the decomposition of \(V_\lambda\ast\mathrm{IC}_0\) into shifted \(\mathrm{IC}_{\chi,\nu}\), here with disconnected stabilizers (D2); Lang's theorem for \(L^+G\) and the stabilizers (D3). See the verdict's obligations. Used only in the trace remark (Step 12).

Proof

The question and why the pages' criterion is silent: Purpose; the model: Assumption. Steps 1–6 prove \(\mathcal C_K^{\mathrm{Sat}}\ne\mathcal C_K\) by central inertia, steps 7–9 describe the span exactly, steps 10–12 treat functions, and step 13 concludes.

1. Result Orbits and stabilizers

   The orbits are the minimal-valuation shells, and every stabilizer off the base orbit has component group generated by the central element:

   $$\operatorname{Stab}_{L^+G}(x_m)\cong\mu_2\times t^mk[\![t]\!],\qquad\pi_0=\{\pm I\}\ (m\ge1),\qquad\mathcal C_m^\heartsuit\simeq\operatorname{Rep}(\mathbb Z/2).$$
   Proof

   • Premises. The matrix model; units of \(k[\![t]\!]\) are squares; equivariant local systems on an orbit are representations of \(\pi_0\) of the stabilizer.
   • Inference. A form \(t^{-m}A'\) with a unit entry in \(A'\) is congruent to \(\operatorname{diag}(t^{-m},t^m)\) under \(\mathrm{SL}_2(k[\![t]\!])\). The stabilizer of \(x_m\) is \(\{g_{a,b}:a^2+b^2=1,\ b\in t^mk[\![t]\!]\}\), and \(a=\pm\sqrt{1-b^2}\) for \(m\ge1\).
   • Output. Two components, \(-I\) in the non-identity one; for \(m=0\) the stabilizer \(\mathrm{SO}_2(k[\![t]\!])\) is connected. The simples of \(\mathcal C_m^\heartsuit\) are \(\underline k\) and \(\mathcal L^-_m\).
2. Method Central inertia

   The central element \(-I\), acting trivially on \(LX\), defines a natural involution of every object:

   $$\varepsilon\in\operatorname{Aut}(\mathrm{id}_{\mathcal C_K}),\qquad\varepsilon^2=\mathrm{id},\qquad\varepsilon|_{\text{local system}}=\rho(-I).$$
   Proof

   • Premises. \(-I\) is central in \(L^+G\) and acts trivially on \(LX\) and \(\operatorname{Gr}\); the sheaf theory of \(\mathcal C_K\) is functorial in 2-morphisms of stacks.
   • Inference. Multiplication by \(-I\) is an automorphism of every \(L^+G\)-torsor commuting with its map to \(LX\), a 2-automorphism of the identity of \(L^+G\backslash LX\), defined on each finite-type model \(G(k[t]/t^N)\backslash Z^{(N)}\).
   • Output. \(\varepsilon\) with (E1)–(E5); on the stalk at \(y\) the inertia acts through \(-I\in\operatorname{Stab}(y)\).
3. Result Two orthogonal thick parts

   Every object splits into an even and an odd part, with no maps from even to odd:

   $$\mathcal C_K=\mathcal C_K^+\oplus\mathcal C_K^-,\qquad\operatorname{Hom}(\mathcal C_K^+,\mathcal C_K^-[n])=0,\qquad\mathcal C_K^\pm\ \text{thick}.$$
   Proof

   • Premises. (E1), (E2), \(\varepsilon^2=\mathrm{id}\), \(2\ne0\) in \(k\), \(\mathcal C_K\) idempotent complete.
   • Inference. For \(f:F\to G[n]\), \(F\in\mathcal C_K^+\), \(G\in\mathcal C_K^-\): \(f=f\varepsilon_F=\varepsilon_{G[n]}f=-f\). The idempotent \((1+\varepsilon_F)/2\) splits in \(h\mathcal C_K\). An object with no maps to \(\mathcal C_K^-[n]\) has zero odd part.
   • Output. \(\mathcal C_K^+\) is the left orthogonal of \(\mathcal C_K^-\), hence thick, and symmetrically; \(\mathcal C_K^+\cap\mathcal C_K^-=0\).
4. Result The even part is the PGL₂ category

   The inertia-trivial objects are exactly those pulled back from \(L^+\mathrm{PGL}_2\backslash LX\), whose stabilizers are connected:

   $$p^*:\operatorname{Shv}_{\mathrm{cons}}(L^+\mathrm{PGL}_2\backslash LX)\xrightarrow{\ \sim\ }\mathcal C_K^+.$$
   Proof

   • Premises. \(\mathrm{PGL}_2\) acts on \(X\) by \(gAg^{\mathsf T}/\det g\); \(L^+\mathrm{PGL}_2=L^+\mathrm{SL}_2/\mu_2\) since units are squares.
   • Inference. \(p\) is a \(\mu_2\)-gerbe and \(p_*p^*=(-)\otimes H^*(B\mu_2,k)=\mathrm{id}\) in characteristic \(0\).
   • Output. \(p^*\) is fully faithful with essential image the objects with trivial \(\mu_2\)-inertia, namely \(\mathcal C_K^+\). Steps 5–6 do not use this.
5. Result Satake objects are even

   The base object and all Hecke kernels have trivial inertia, and convolution multiplies inertia:

   $$\varepsilon_{\mathrm{IC}_0}=\mathrm{id},\qquad\varepsilon_{V\ast F}=\varepsilon_V\ast\varepsilon_F,\qquad\mathcal C_K^{\mathrm{Sat}}\subseteq\mathcal C_K^+.$$
   Proof

   • Premises. (E3)–(E5); stabilizers of \(L^+G\) on \(\operatorname{Gr}\) are connected; \(\mathcal C_K^+\) is thick (Step 3).
   • Inference. \(\mathrm{IC}_0\) is the extension of a sheaf pulled back from a point. The odd part of \(V\in\mathcal H\) has cohomology sheaves that are local systems with \(\varepsilon=-1\) on orbits with connected stabilizers, hence vanishes. On \(LG\times^{L^+G}LX\), \((-I)[g,x]=[g(-I),x]=[g,(-I)x]\), so the inertia on \(V_g\otimes F_x\) is \(\varepsilon_V\otimes\varepsilon_F\), and \(m_!\) commutes with \(\varepsilon\).
   • Output. \(\mathcal H\ast\mathcal C_K^+\subseteq\mathcal C_K^+\), so every \(V_\lambda\ast\mathrm{IC}_0\) and their thick span lie in \(\mathcal C_K^+\).
6. Result The sign standard object is not generated

   The standard object of a sign local system is nonzero and odd, so it is not in the Satake span:

   $$0\ne\Delta_m^-=a_{m!}\mathcal L^-_m\in\mathcal C_K^-\quad(m\ge1),\qquad\mathcal C_K^{\mathrm{Sat}}\ne\mathcal C_K.$$
   Proof

   • Premises. Step 1 (existence of \(\mathcal L^-_m\)), (E3), (E5), Steps 3 and 5.
   • Inference. \(\Delta_m^-\) is bounded, constructible, equivariant and supported on \(\overline{\mathcal O_m}\), with \(a_m^*\Delta_m^-=\mathcal L^-_m\ne0\); \(\varepsilon_{\Delta_m^-}=a_{m!}(\rho(-I))=-\mathrm{id}\).
   • Output. \(\Delta_m^-\in\mathcal C_K^-\setminus\{0\}\), while \(\mathcal C_K^{\mathrm{Sat}}\subseteq\mathcal C_K^+\) and \(\mathcal C_K^+\cap\mathcal C_K^-=0\). Only \(2\ne0\) in \(k\) is used: no weights, decomposition theorem, H1–H5 or D5.
7. Method Convolution on a line of the tree

   The convolution map of a Satake object has fibres of size \(s_n(m')\) and a single point over the top orbit:

   $$m_n:\operatorname{Gr}^{\le n\alpha^\vee}\mathbin{\widetilde\times}L^+X\to\overline{\mathcal O_{2n}},\qquad s_n(m')=\#\{j:m'\le2j\le2n\}+\#\{j:m'\lt2j\le2n\},\qquad s_n(2n)=1.$$
   Proof

   • Premises. \(X\cong\mathrm{SL}_2/T\) over \(\mathbb C\); orbits are distance classes from an apartment; the support bound; Chen–Yi's finite-type slices.
   • Inference. The fibre over \(y_0T\) is the set of even-distance vertices of the line \(y_0A_T\) within distance \(2n\) of \(v_0\). A proper map with finite fibres is finite on the slice models; in characteristic \(0\) the generic fibre is reduced.
   • Output. \(m_n\) is finite and an isomorphism over the dense orbit \(\mathcal O_{2n}\).
8. Result Each Satake object is one IC sheaf

   A finite birational pushforward of an IC sheaf is the IC sheaf of the image:

   $$V_{n\alpha^\vee}\ast\mathrm{IC}_0=m_{n*}\mathrm{IC}(Z_n)\simeq\mathrm{IC}(\overline{\mathcal O_{2n}})=\mathrm{IC}_{2n}.$$
   Proof

   • Premises. Step 7; \(V_{n\alpha^\vee}\mathbin{\widetilde\boxtimes}\mathrm{IC}_0=\mathrm{IC}(Z_n)\); finite pushforward is t-exact; BBD, Corollary 1.4.24.
   • Inference. \(i^*m_{n*}\mathrm{IC}(Z_n)\) lies in perverse degrees \(\le-1\) and \(i^!m_{n*}\mathrm{IC}(Z_n)\) in degrees \(\ge1\), by base change along the boundary.
   • Output. \(m_{n*}\mathrm{IC}(Z_n)=j_{!*}(\cdot)=\mathrm{IC}_{2n}\), componentwise on the slice models. No decomposition theorem.
9. Result The exact Satake span, under (P)

   Thick spans of simples are detected by composition factors, so the span misses the odd labels and the sign coefficients:

   $$\mathcal C_K^{\mathrm{Sat}}=\operatorname{Thick}\{\mathrm{IC}_{2n}\}\subsetneq\mathcal C_K^+=\operatorname{Thick}\{\mathrm{IC}(\overline{\mathcal O_m},\underline k)\}\subsetneq\mathcal C_K.$$
   Proof

   • Premises. (P); Step 8; (E2); Step 3.
   • Inference. \(\mathcal D_\Sigma=\operatorname{Thick}(\Sigma)\) for a set \(\Sigma\) of simples; \(\varepsilon\) is \(\mathrm{id}\) on a simple exactly when its coefficients are constant, and is inherited by subquotients.
   • Output. Witnesses \(\mathrm{IC}(\overline{\mathcal O_1},\underline k)\in\mathcal C_K^+\setminus\mathcal C_K^{\mathrm{Sat}}\) (coverage failure \(\varphi(\Lambda_T^+)=2\Lambda_S^+\)) and \(\mathrm{IC}(\overline{\mathcal O_1},\mathcal L^-_1)\in\mathcal C_K\setminus\mathcal C_K^+\) (failure of H3).
10. Result Function level: the translates are even polynomials

    The \(\mathrm{SL}_2\) Hecke algebra acts on \(\mathrm{GL}_2(\mathcal O)\)-invariant functions through \(T_1^2\), so the translates of \(\varphi_0\) miss \(\mathbf 1_1\):

    $$\mathcal H_K\varphi_0=\mathbb C[T_1^2]\varphi_0,\qquad\mathbf 1_1=(T_1-2)\varphi_0\notin\mathcal H_K\varphi_0,\qquad\dim\big(\mathcal H_K\varphi_0\cap C_c(\{d\le2N\})\big)=N+1.$$
    Proof

    • Premises. \(K_{\mathrm{GL}}\backslash X(F)=\{d\ge0\}\); \(T_1\) is the adjacency operator; \(\mathcal H_K=\mathbb C[S]\), \(S=\mathbf 1_{K\operatorname{diag}(\varpi,\varpi^{-1})K}\).
    • Inference. \(T_1^d\varphi_0=\mathbf 1_d+(\text{lower})\), so \(C_{K_{\mathrm{GL}}}\cong\mathbb C[y]\). The coset bijection \(KtK/K\to K_{\mathrm{GL}}tK_{\mathrm{GL}}/K_{\mathrm{GL}}\) gives \(S=T_1^2-(q+1)\) on \(C_{K_{\mathrm{GL}}}\).
    • Output. \(\mathcal H_K\varphi_0\leftrightarrow\mathbb C[y^2]\not\ni y-2\leftrightarrow\mathbf 1_1\), for every \(F\) and every \(q\).
11. Result Square classes, q odd

    For \(q\) odd each distance class \(d\ge1\) is two \(K\)-orbits, exchanged by \(\sigma\), which fixes every translate:

    $$\mathcal H_K\varphi_0\subseteq C_K^\sigma\subsetneq C_K.$$
    Proof

    • Premises. Determinants of \(K_{\mathrm{GL}}\)-stabilizers; \(s=\operatorname{diag}(u,1)\) normalizes \(K\), \(T\), every \(Kt^\lambda K\), and fixes \(x_0\).
    • Inference. The stabilizer at distance \(d\ge1\) has determinants \(\mathcal O^{\times2}\); \(K_{\mathrm{GL}}=K\cdot\{\operatorname{diag}(u,1)\}\). Proposition J(2) checks \(d=1\) by a matrix computation.
    • Output. A second obstruction for \(q\) odd, independent of the parity count.
12. Result σ is the Frobenius trace of ε, under D0–D3

    Over \(\mathbb F_{q^n}(\!(t)\!)\), \(\sigma_n\) multiplies the trace of each simple IC object by the value of its coefficient character at \(-I\):

    $$\operatorname{tr}_n(\mathrm{IC}_{\chi,\nu})\circ\sigma_n=\chi(-1)\operatorname{tr}_n(\mathrm{IC}_{\chi,\nu}).$$
    Proof

    • Premises. D0–D3 with disconnected stabilizers; \(\operatorname{Aut}\mathrm{IC}_{\chi,\nu}\) is scalar.
    • Inference. \(\sigma_n\) is the action of \(h=\operatorname{diag}(\sqrt u,\sqrt u^{-1})\in L^+\mathrm{SL}_2(\overline{\mathbb F}_q)\); \(h^*M\simeq M\) gives a second Weil structure differing by a scalar, read off at \(x_\nu\), where the Lang cocycle of \(hx_\nu\) is \(-I\).
    • Output. \(\mathcal C_K^\pm\) trace to the \(\sigma\)-even and \(\sigma\)-odd functions. This repairs the circular step of the run's first \(\ell\)-adic proof: the audit's objection.
13. Conclusion Both levels fail

    Steps 6 and 10 give the two failures; the levels agree, and this row gives no divergence:

    $$\mathcal C_K^{\mathrm{Sat}}\ne\mathcal C_K\ \text{ over }\mathbb C,\qquad\mathcal H_K\cdot\varphi_0\ne C_K\ \text{ over every }F.$$

    One mechanism at both levels: \(\mu_2=\ker(\mathrm{SL}_2\to\mathrm{PGL}_2)\) is inertia over \(\mathbb C\) and the square classes \(H^1(\mathbb F_q,\mu_2)\) over \(\mathbb F_q\); the Hecke action, being \(\mathrm{PGL}_2\)-equivariant, sees neither. The only divergence among the pages' rows is VIII · E₆/F₄; the reverse direction is excluded in II · functions ⇒ sheaves.