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Definitions

• Groups, the tree and the base edge

  \(G=\mathrm{GL}_2\supset T\) the diagonal torus, \(X=G/T\), \(K=G(\mathcal O)\), \(I\subset K\) the matrices upper triangular modulo \(t\), \(\omega=\begin{pmatrix}0&1\\t&0\end{pmatrix}\), \(K'=\omega K\omega^{-1}\). Sheaves: \(k=\overline{\mathbb F}_q\) (\(p\) odd) or \(\mathbb C\), \(F=k(\!(t)\!)\), \(\mathcal O=k[\![t]\!]\). Functions: \(F\) a non-archimedean local field with residue field \(\mathbb F_q\).

  \(\mathcal T\) is the tree of \(\mathrm{PGL}_2(F)\): vertices are lattice classes, ends are the points of \(\mathbb P^1(F)\). A point of \(LX=X(F)=G(F)/T(F)\) is an oriented geodesic between two ends, the eigenlines of \(M_\gamma=g\,\mathrm{diag}(1,-1)g^{-1}\):

  $$v_0=[\mathcal O^2],\quad v_1=[\mathcal O\oplus t\mathcal O],\quad C_0=(v_0,v_1),\qquad\gamma=(a,b),\qquad Y_n=\{\gamma:d(v_0,\gamma)\le n\}=LX\cap t^{-n}\mathfrak{gl}_2(\mathcal O).$$

  \(Y_0=L^+X\), \(LX=\varinjlim_nY_n\). \(K\) fixes \(v_0\), \(K'\) fixes \(v_1\), \(I\) fixes \(C_0\); \(P_{s_1}=K\), \(P_{s_0}=K'\).

• The \(I\)-orbits and their closures

  Relative positions of a geodesic and the base edge (Figure 1, step 1), for \(i\in\{0,1\}\), \(m\ge0\):

  $$A_\pm=\{\gamma\supset C_0\ \text{running}\ v_0\to v_1\ (+)\ \text{or}\ v_1\to v_0\ (-)\},\qquad O_{i,m}=\{\gamma:\ d(v_i,\gamma)=m,\ d(v_{1-i},\gamma)=m+1\},$$ $$\overline{O_{0,n}}=Y_n,\qquad\overline{O_{1,n}}=\omega Y_n,\qquad A_\pm\ \text{closed},\qquad\delta(A_\pm)=0,\qquad\delta(O_{i,m})=m+1 .$$

  \(\omega\) exchanges \(A_+\leftrightarrow A_-\) and \(O_{0,m}\leftrightarrow O_{1,m}\). The \(K\)-orbits are \(O_m=\{d(v_0,\cdot)=m\}=O_{0,m}\sqcup O_{1,m-1}\).

• Levels, pieces and the categories

  A level presentation is \(LX=\varinjlim_NY_N\), \(Y_N=\varprojlim_MY_{N,M}\), with \(Y_N\) closed and \(I\)-stable, \(Y_{N,M}\) of finite type, \(\pi_{N,M}:Y_N\to Y_{N,M}\) smooth, affine and surjective, and \(I\) acting on \(Y_{N,M}\) through a finite-dimensional quotient \(I/I_{N,M}\):

  $$\mathcal P_{N,M}=\{\pi_{N,M}^{-1}(o):\ o\ \text{an}\ I/I_{N,M}\text{-orbit}\},\qquad\mathcal C_I^{(N,M)}=\pi_{N,M}^*\,D^b_c\big(I/I_{N,M}\backslash Y_{N,M}\big).$$

  \(\mathcal C_I=\operatorname{Shv}_c(I\backslash LX)\); \(\mathcal C_K\), \(\mathcal C_{K'}\) map to it by forgetting equivariance. With \(\widetilde W=\mathbb Z^2\rtimes S_2\) and \(\operatorname{Thick}\) the closure under isomorphisms, shifts, cofibres and retracts ((T1), (T2)):

  $$\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathrm{IC}_w\ast_I\mathrm{IC}_0:w\in\widetilde W\},\qquad\mathrm{IC}_{0,n}=\mathrm{IC}(Y_n),\quad\mathrm{IC}_{1,n}=\mathrm{IC}(\omega Y_n),\quad\mathrm{IC}_{A_\pm}=\underline k_{A_\pm}.$$

  \(\mathcal M_I\) is the set of simple objects that are shifted direct summands of some \(\mathrm{IC}_w\ast_I\mathrm{IC}_0\).

• Functions and the Hecke algebra

  \(C_I=C_c^\infty(X(F))^I\), with \(K\)- and \(K'\)-invariant subspaces \(C_K\), \(C_{K'}\); \(\mathcal H_I\) with \(\operatorname{vol}I=1\), \(T_w=\mathbf 1_{IwI}\), \((T_s-q)(T_s+1)=0\), acting by \((f\ast\varphi)(\gamma)=\int f(g)\varphi(g^{-1}\gamma)\,dg\); \(s_1\) is the panel of \(C_0\) at \(v_0\), \(s_0\) the panel at \(v_1\); \(\varphi_0=\mathbf 1_{Y_0}\).

  $$f_n(\gamma)=\max\big(0,\,n+1-d(v_0,\gamma)\big),\qquad f^{(1)}_n=T_\omega f_n=\max\big(0,\,n+1-d(v_1,\gamma)\big),\qquad f_n-f_{n-1}=\mathbf 1_{Y_n}.$$
• The functionals \(\lambda\), \(\psi\) and the characters \(\varepsilon_\pm\)

  With \(f(O)\) the value of \(f\) on the orbit \(O\):

  $$\lambda(f)=f(A_+)-f(A_-),\qquad\psi(f)=\tfrac12\big(f(A_+)+f(A_-)\big)+\sum_{i\in\{0,1\},\,m\ge0}(-1)^{m+1}f(O_{i,m}),$$ $$\varepsilon_\pm(T_{s_0})=\varepsilon_\pm(T_{s_1})=-1,\qquad\varepsilon_\pm(T_\omega)=\pm1 .$$

  \(\lambda\) is the Steinberg functional of the classical page's Proposition F (\(\mathrm{St}^I=\varepsilon_-\)); \(\psi\), with \(T_\omega\mapsto+1\), is the twisted one (heuristically \(\mathrm{St}\otimes(-1)^{\operatorname{val}\det}\); not used).

• Classes of Weil objects

  Over \(\overline{\mathbb F}_q\), with every orbit defined over \(\mathbb F_q\): \(\mathcal M_A\) is the Grothendieck group of Weil objects of \(\mathcal C_I\) (Chen–Yi §8.2), a module over \(\mathbb Z[A]\), \(A=\overline{\mathbb Q}_\ell^\times/\mu_\infty\), and over the affine Hecke algebra:

  $$\mathbb Z[v^{\pm1}]\subset\mathbb Z[A],\ \ v\mapsto-[q^{-1/2}],\qquad\mathbf 1_O=[j_{O!}\underline k_O],\qquad\underline H_s=v(T_s+1)=[\mathrm{IC}_s],\qquad[\mathcal G\ast_I\mathcal F]=[\mathcal G]\cdot[\mathcal F].$$

  The trace specializes \(v\mapsto-q^{-1/2}\); \(\lambda\) and \(\psi\) extend \(\mathbb Z[A]\)-linearly from their values on the \(\mathbf 1_O\).

Hypothesis I1

The sheaf half assumes I1; the function half assumes nothing. \(p\) odd enters only through the involution model of the levels.

• I1 · Finite levels.

  \(LX\) has a level presentation \(LX=\varinjlim_NY_N\), \(Y_N=\varprojlim_MY_{N,M}\), such that every object of \(\mathcal C_I\) is pulled back from some \(Y_{N,M}\) and each level has finitely many pieces: $$\mathcal C_I=\bigcup_{N,M}\mathcal C_I^{(N,M)},\qquad|\mathcal P_{N,M}|<\infty .$$

  What is proved and what is assumed

  • Proved in step 1. \(Y_N\) is the union of \(2N+3\) orbits and each closure is cut out by finitely many coefficient conditions, so at large \(M\) the pieces of \(Y_N\) are its orbits: the finiteness half. At \(N=0\), \(L^+X=\varprojlim_MX(\mathcal O/t^M)\) with affine-space transitions: the strict form.
  • Literature. Over \(\overline{\mathbb F}_p\), \(p\ne2\), Chen–Yi give an \(I_0\)-placid presentation \(LX=\varinjlim_v\overline{O_v}\) (Theorem 36, Proposition 39(i)). Placidness is not the strict form, in which \(I\) acts on finite-type \(Y_{N,M}\) through \(I/I_{N,M}\).
  • Open. The strict form for \(N\ge1\), and anything over \(\mathbb C\) beyond assuming it (audit A9-5). The sheaf half is conditional on it exactly as the pages' Proposition IV.1 for \(\mathbb A^n\).
  • Where it is used. Finite-type models of \(\nu_n\), \(\nu'\) and the convolution maps, the perverse t-structure level by level, and Weil structures: steps 3–7, 9, 11, 12. Step 8 does not use it.

Proof

Steps 1–2 describe the orbits and the Hecke action. Steps 3–7 prove (b): words, the two IC computations, constituent subcategories. Step 8 proves (a) with no sheaves. Steps 9–11 prove (c): positive certificates, why \(\lambda\) is not one, and \(2\psi\). Step 12 compares the two levels.

1. Method Tree model and I-orbits

   Points of \(LX\) are oriented geodesics; the \(I\)-orbits are their relative positions with the base edge:

   $$I\backslash LX=\{A_+,A_-\}\sqcup\{O_{i,m}:i\in\{0,1\},\ m\ge0\},\qquad\operatorname{Stab}_I(\gamma)\ \text{connected},\qquad\overline{O_{0,n}}=Y_n .$$
   Proof

   • Model. \(\mathrm{PGL}_2\) acts simply transitively modulo \(T\) on ordered pairs of distinct points of \(\mathbb P^1\), so \(X\cong(\mathbb P^1\times\mathbb P^1)\setminus\Delta\). As \(2\in\mathcal O^\times\), \(M_\gamma L=L\iff L=(L\cap a)\oplus(L\cap b)\iff[L]\in\gamma\) (Casselman 7.2); for \(\gamma=(\infty,t^{-m})\), \(\operatorname{ord}_tM_\gamma=-m=-d(v_0,\gamma)\), and \(K\) is transitive on each \(\{d(v_0,\gamma)=m\}\).
   • Orbits. \(G(F)/I\to\{\text{oriented edges}\}\) is a torsor under \(Z(F)/Z(\mathcal O)\cong\mathbb Z\); since \(Z\subset T\), \(IgT\mapsto Tg^{-1}I\) identifies \(I\backslash X(F)\) with the \(T(F)\)-orbits of oriented edges (repair of A9-1). \(T(F)\) translates the torus apartment \(\gamma_0\) (orbits \(A_\pm\)); \(T(\mathcal O)/Z(\mathcal O)\cong\mathcal O^\times\) is simply transitive on the \((q-1)q^{m-1}\) vertices at distance \(m\ge1\) above a vertex of \(\gamma_0\) (Casselman 7.4). Endpoint distances \((m,m+1)\) or \((m+1,m)\) give \(O_{0,m}\), \(O_{1,m}\).
   • Stabilizers. \(\operatorname{Stab}_I(\gamma)=Z(\mathcal O)\cdot\{\operatorname{diag}_\gamma(a,1):a\in1+t^{m'}\mathcal O\}\), \(m'=\max_id(v_i,\gamma)\): connected, and connected at every finite level when \(\pi_{N,M}\) has connected fibres. So every equivariant local system on an orbit is constant.
   • Dimensions. \(T_s\) is self-adjoint for an \(I\)-invariant measure \(\mu\), which forces \(\mu(O_{i,m})=(q-1)q^m\mu(A_\pm)\), the point count of \(\mathbb G_m\times\mathbb A^m\): \(\delta(A_\pm)=0\), \(\delta(O_{i,m})=m+1\) (Chen–Yi, Proposition 41, normalizes by the closed orbits).
   • Closures. The curve \(s\mapsto\begin{pmatrix}1&-2(st^{-n}+t^{-m})\\0&-1\end{pmatrix}\) lies in \(O_n\) for \(s\ne0\) and in \(O_m\) at \(s=0\), so \(O_m\subset\overline{O_n}\) (\(m\lt n\)); \(O_{0,n}\) is open in the irreducible \(O_n\), so \(\overline{O_{0,n}}=Y_n\); apply \(\omega\). \(A_\pm\) are preimages of closed \(B\)-orbits of \(X(k)\) in \(L^+X\): closed (Chen–Yi, Lemma 31).
   • Levels. \(Y_N\) is the union of the \(2N+3\) orbits \(A_\pm\), \(O_{0,m}\) (\(m\le N\)), \(O_{1,m}\) (\(m\lt N\)): the finiteness half of I1. Only the tree is used, so the list holds over every non-archimedean \(F\). Run nodes A6-1, A9-1.
2. Method Hecke action on orbit indicators

   Each \(T_s\) counts the chambers of one panel by relative position; on the orbit basis, for \(m\ge1\),

   $$\begin{aligned}T_{s_1}\mathbf 1_{A_\pm}&=\mathbf 1_{A_\mp}+\mathbf 1_{O_{0,0}},&T_{s_1}\mathbf 1_{O_{0,0}}&=(q-1)(\mathbf 1_{A_+}+\mathbf 1_{A_-})+(q-2)\mathbf 1_{O_{0,0}},\\T_{s_1}\mathbf 1_{O_{1,m-1}}&=\mathbf 1_{O_{0,m}},&T_{s_1}\mathbf 1_{O_{0,m}}&=q\,\mathbf 1_{O_{1,m-1}}+(q-1)\mathbf 1_{O_{0,m}};\end{aligned}$$

   \(T_{s_0}\): exchange the indices \(0\leftrightarrow1\) of the orbits; \(T_\omega\mathbf 1_O=\mathbf 1_{\omega O}\); and \(T_{s_1}\varphi_0=q\varphi_0\) for \(\varphi_0=\mathbf 1_{A_+}+\mathbf 1_{A_-}+\mathbf 1_{O_{0,0}}\).

   Proof

   • Count. \((T_s\mathbf 1_O)(\gamma)\) is the number of chambers \(C'\ne C_0\) of the \(s\)-panel of \(C_0\) in relative position \(O\) to \(\gamma\); the \(s_1\)-panel is the \(q+1\) edges \((v_0,u)\), \(u\sim v_0\).
   • Panel at \(v_0\). If \(\gamma\ni v_0\): the neighbour towards \(b\) gives \(A_+\), towards \(a\) gives \(A_-\), the other \(q-1\) give \(O_{0,0}\). If \(d(v_0,\gamma)=m\ge1\): the neighbour towards \(\gamma\) gives \(O_{1,m-1}\), the other \(q\) give \(O_{0,m}\).
   • Checks. \(T_s^2=(q-1)T_s+q\) on every block, \(T_\omega T_{s_0}T_\omega^{-1}=T_{s_1}\), \(T_\omega^2=1\) on \(C_I\) (\(\omega^2=t\cdot\mathrm{Id}\) is central). The audit recomputed every coefficient by explicit coset representatives.
   • Sheaf reading. Over \(\overline{\mathbb F}_q\) the panel is cut into points, affine lines and punctured affine lines, with Tate classes \(1,q,q-1\): the same matrices with \(q=v^{-2}\) act on \(\mathcal M_A\). This is Chen–Yi (29), with types IIIa (\(A_\pm\)), IIIb (\(O_{0,0}\)), IIa, IIb for \(s_1\). Run nodes A6-2, A9-2.
3. Method Generators by words

   The generated part is spanned by words in \(\mathrm{IC}_{s_0}\), \(\mathrm{IC}_{s_1}\), \(\mathrm{IC}_\omega\) applied to \(\mathrm{IC}_0\): for a reduced \(w=s_{i_1}\cdots s_{i_r}\omega^j\),

   $$\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathrm{IC}_w\ast_I\mathrm{IC}_0\},\qquad\mathrm{IC}_w\ast_I\mathrm{IC}_0\ \text{is a retract of}\ B_w\simeq\bigoplus_y\mathrm{IC}_y\ast_I\mathrm{IC}_0[s_y]\in\mathcal C_I^{\mathrm{Iwa}}.$$
   Proof

   • Cells. \(IwI/I\cong\mathbb A^{\ell(w)}\) with connected stabilizers, so \(\mathcal H_{c,I}=\operatorname{Thick}\{\Delta_w\}\); the cone of \(\Delta_w\to\mathrm{IC}_w[-\ell(w)]\) lives on lower cells, so \(\mathcal H_{c,I}=\operatorname{Thick}\{\mathrm{IC}_w\}\) by induction on \(\ell(w)\).
   • Exactness. \(-\ast_I\mathrm{IC}_0\) is exact, so (T2) gives \(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathrm{IC}_w\ast_I\mathrm{IC}_0\}\). The action is one-sided.
   • Bott–Samelson. \(\mathrm{IC}_{s_{i_1}}\ast\cdots\ast\mathrm{IC}_{s_{i_r}}\ast\mathrm{IC}_{\omega^j}\) is a direct sum of shifted \(\mathrm{IC}_y\) containing \(\mathrm{IC}_w\) once: the decomposition theorem for the proper Bott–Samelson map of finite-type quotient stacks in its equivariant form (Sun, Theorem 1.2, over \(\overline{\mathbb F}_q\); Bernstein–Lunts over \(\mathbb C\)). A non-equivariant splitting would not suffice (audit A7-3). Run nodes A6-5 (steps 1–2), A9-6.
4. Result IC(Yₙ) is a finite pushforward

   The closures \(Y_n\) are images of a finite map from a rationally smooth space, so their IC sheaves are pushforwards of constant sheaves:

   $$\mathrm{IC}_{0,n}=\mathrm{IC}(Y_n)\cong\nu_{n*}k[n+1],\qquad\mathcal H^{-n-1}(\mathrm{IC}_{0,n})_\gamma=k^{f_n(\gamma)},\qquad[\mathrm{IC}_{0,n}]=v^{n+1}f_n,\qquad\mathrm{IC}_{1,n}=\omega_*\mathrm{IC}_{0,n}.$$
   Proof

   • The map. \(\nu_n:\widetilde Y_n=LG_{\le n}\times^KL^+X\to LX\), \([g,x]\mapsto gx\), with \(\mathrm{Gr}_{\le n}=\overline{Kt^{(n,0)}K}/K\) the vertices \(u\) with \(d(v_0,u)\le n\), \(d(v_0,u)\equiv n\bmod2\). Since \(\nu_n[g,x]=\gamma\iff\gamma\ni gv_0\), the fibre is the set of vertices of \(\gamma\) in \(\mathrm{Gr}_{\le n}\): \(\max(0,n+1-d(v_0,\gamma))\) points.
   • Finite. Proper and quasi-finite, hence finite, with image \(Y_n\), and a universal homeomorphism over the open \(K\)-orbit \(O_n\).
   • Source. \(\mathrm{Gr}_{\le n}\) is rationally smooth (the Kazhdan–Lusztig polynomials of the infinite dihedral group are \(1\)), so \(\mathrm{IC}_{\widetilde Y_n}=k[n+1]\).
   • Small map. A finite map is t-exact and commutes with duality, so \(\nu_{n*}\) carries the strict support and cosupport conditions of \(k[n+1]\) to those of an intersection complex (BBD 1.4.24; de Cataldo–Migliorini 4.2.7: only the open stratum is relevant).
   • Class. Stalks are \(k^{f_n}\) in one degree, pointwise pure Tate, so the trace is \(v^{n+1}f_n\), Chen–Yi's Kazhdan–Lusztig element \(C_{0,n}\).
   • Spherical level. \(\nu_{n*}(k_{\mathrm{Gr}_{\le n}}[n]\mathbin{\widetilde\boxtimes}k_{L^+X}[1])\) is the Satake action, so \(\mathrm{IC}(Y_n)\cong\mathrm{IC}^{\mathrm{Gr}}_{(n,0)}\ast_K\mathrm{IC}_0[1]\). I1 enters only through finite-type models of \(\nu_n\); \(\mathrm{IC}_{0,0}=\mathrm{IC}_0[1]\) changes no thick span. Run nodes A6-3, A9-3.
5. Result Convolution rules

   Convolution with a simple reflection or with \(\omega\) keeps the non-closed intersection complexes among themselves (\(\mathrm{IC}_{i,-1}=0\); the same with \(0\leftrightarrow1\); up to Tate twists):

   $$\mathrm{IC}_{s_0}\ast_I\mathrm{IC}_{0,n}\cong\mathrm{IC}_{1,n+1}\oplus\mathrm{IC}_{1,n-1},\qquad\mathrm{IC}_{s_1}\ast_I\mathrm{IC}_{0,n}\cong\mathrm{IC}_{0,n}[1]\oplus\mathrm{IC}_{0,n}[-1],\qquad\mathrm{IC}_\omega\ast_I\mathrm{IC}_{i,n}\cong\mathrm{IC}_{1-i,n}.$$
   Proof

   • The map. By step 4, \(\mathrm{IC}_{s_0}\ast\mathrm{IC}_{0,n}=\nu'_*k[n+2]\) for \(\nu':P_{s_0}\times^I\widetilde Y_n\to LX\), a \(\widetilde Y_n\)-bundle over \(P_{s_0}/I=\mathbb P^1\).
   • Fibres. For \(u\in\gamma\) with \(r=d(v_1,u)\), the neighbours \(u'\) of \(v_1\) are at distance \(r\pm1\) from \(u\); the condition \(d(u',u)\le n\), \(d(u',u)\equiv n\) forces \(r\equiv n+1\) and holds for all \(u'\) if \(r\le n-1\), for one if \(r=n+1\), for none otherwise: $$\nu'^{-1}(\gamma)=\bigsqcup_{r(u)\le n-1}\mathbb P^1\ \sqcup\bigsqcup_{r(u)=n+1}\mathrm{pt},\qquad H^*(\nu'^{-1}\gamma)=k^{f^{(1)}_{n+1}(\gamma)}\oplus k^{f^{(1)}_{n-1}(\gamma)}[-2].$$
   • Decomposition. \(k\) is pure and \(\nu'\) proper, so \(\nu'_*k[n+2]\) is pure (BBD 5.1.14) and an equivariant direct sum of shifted orbit ICs with constant coefficients (Sun, Theorem 1.2; over \(\mathbb C\), Bernstein–Lunts).
   • Stalk ranks. On \(O_{1,n+1}\) the stalk is \(k[n+2]\): \(\mathrm{IC}_{1,n+1}\) occurs once, unshifted. The remainder has stalks \(k^{f^{(1)}_{n-1}}[n]\): \(\mathrm{IC}_{1,n-1}\) occurs once, and what is left has zero stalks, so it is zero.
   • The other rules. \(\mathrm{IC}_{0,n}\) is \(P_{s_1}\)-equivariant (\(P_{s_1}=K\)), so \(P_{s_1}\times^ILX\cong\mathbb P^1\times LX\) and \(\mathrm{IC}_{s_1}\ast\mathrm{IC}_{0,n}\cong R\Gamma(\mathbb P^1,k)[1]\otimes\mathrm{IC}_{0,n}\); \(I\omega I=\omega I\) gives \(\mathrm{IC}_\omega\ast_I-=\omega_*\). Run nodes A6-4, A9-4.
6. Method Constituent subcategories are thick

   Objects whose perverse constituents lie in a fixed set \(\Sigma\) of simple objects form a thick subcategory:

   $$\mathcal S_\Sigma=\{\mathcal F\in\mathcal C_I:\ \text{every Jordan–Hölder factor of every}\ {}^pH^i(\mathcal F)\ \text{lies in}\ \Sigma\}\ \ \text{is thick}.$$
   Proof

   • Heart. Each finite level has a finite-length perverse heart (BBD 4.3.1); smooth pullback shifted by the relative dimension and closed pushforward \(Y_N\hookrightarrow Y_{N+1}\) are t-exact and preserve intersection complexes (I1). By step 1 the simple objects are the \(\mathrm{IC}_{i,n}\) and \(\underline k_{A_\pm}\).
   • Shifts and retracts. \({}^pH^i\) of a shift is a shifted \({}^pH\); \({}^pH^i\) of a retract is a retract.
   • Cones. For an exact triangle \(A\to B\to C\xrightarrow{+1}\), \({}^pH^i(C)\) is an extension of a subobject of \({}^pH^{i+1}(A)\) by a quotient of \({}^pH^i(B)\); Jordan–Hölder factors of subquotients and extensions stay in \(\Sigma\).
   • Reference. Equivalently, \(\mathcal S_\Sigma\) is the kernel of a cohomological functor to a Serre quotient (Stacks, Lemma 13.6.3). Run nodes A6-5 (step 5), A9-6 (step 4).
7. Result Sheaves fail at the closed orbits

   The summands of the generators are exactly the non-closed intersection complexes, so the two closed orbits are missed:

   $$\mathcal M_I=\{\mathrm{IC}_{i,n}\},\qquad\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathrm{IC}_{i,n}\}=\operatorname{Thick}(\mathcal C_K\cup\mathcal C_{K'})\subseteq\mathcal S_{\{\mathrm{IC}_{i,n}\}}\not\ni\underline k_{A_\pm}.$$
   Proof

   • Words close up. By step 5 and induction on the length \(r\) of the word, starting from \(\mathrm{IC}_0=\mathrm{IC}_{0,0}[-1]\), every \(B_w\) lies in \(\operatorname{add}\{\mathrm{IC}_{i,n}[m]\}\).
   • Upper bound. A finite level is Krull–Schmidt (finite-dimensional Hom spaces, split idempotents) and \(\operatorname{End}\mathrm{IC}_{i,n}=k\); so every summand of the retract \(\mathrm{IC}_w\ast\mathrm{IC}_0\) of \(B_w\) (step 3) is some \(\mathrm{IC}_{i,n}[m]\), and \(\mathcal M_I\subseteq\{\mathrm{IC}_{i,n}\}\).
   • Occurrence. \(B_{s_0}=\mathrm{IC}_{1,1}[-1]\), \(B_{s_1s_0}=(\mathrm{IC}_{0,2}\oplus\mathrm{IC}_{0,0})[-1]\), and the alternating word of length \(n\) gives \(\mathrm{IC}_{i,n}\) plus lower terms; as \(B_w\simeq\bigoplus_y\mathrm{IC}_y\ast\mathrm{IC}_0[s_y]\), Krull–Schmidt puts \(\mathrm{IC}_{i,n}\) into one \(\mathrm{IC}_y\ast\mathrm{IC}_0\); \(\omega\) gives the other index.
   • Spans. \(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}(\mathcal M_I)\); \(\mathcal C_K=\operatorname{Thick}\{\mathrm{IC}(Y_n)\}\) by dévissage over the finitely many \(K\)-orbits of a level with connected stabilizers, and \(\omega_*\mathcal C_K=\mathcal C_{K'}\).
   • Exclusion. By step 6, \(\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal S_{\{\mathrm{IC}_{i,n}\}}\); \(\underline k_{A_\pm}\) is simple with support \(A_\pm\ne Y_n,\omega Y_n\), so it is not there. Only perverse constituents are used here, so the non-equivariant splitting of BBD 5.4.5 suffices.
   • The rest. \(\mathcal C_I=\operatorname{Thick}(\mathcal C_I^{\mathrm{Iwa}}\cup\{\underline k_{A_+},\underline k_{A_-}\})\) by dévissage over the \(2N+3\) orbits of each level. This proves (b). Run nodes A6-5, A9-6.
8. Result Functions: codimension two

   The Hecke translates of \(\varphi_0\) are the functions invariant under \(K\) or \(K'\), and the quotient is two sign characters:

   $$\mathcal H_I\ast\varphi_0=\operatorname{span}\{f_n,f^{(1)}_n\}=C_K+C_{K'},\qquad C_I/\mathcal H_I\ast\varphi_0\ \xrightarrow[(\lambda,\psi)]{\ \sim\ }\ \varepsilon_-\oplus\varepsilon_+ .$$
   Proof

   • Shadow identities. \((T_{s_0}+1)f_n(\gamma)=\sum_{u\sim v_1}\max(0,n+1-d(u,\gamma))\). If \(r_1=d(v_1,\gamma)\ge1\), one neighbour is at distance \(r_1-1\) and \(q\) at \(r_1+1\); if \(r_1=0\), two are at \(0\) and \(q-1\) at \(1\). In both cases $$(T_{s_0}+1)f_n=f^{(1)}_{n+1}+q\,f^{(1)}_{n-1},\qquad(T_{s_1}+1)f_n=(q+1)f_n,\qquad T_\omega f_n=f^{(1)}_n .$$
   • Generation. \(\mathcal H_I\) is generated by \(T_{s_0}\), \(T_{s_1}\), \(T_\omega^{\pm1}\) (Iwahori–Matsumoto, Theorem 3.5), so the span is \(\mathcal H_I\)-stable and contains \(f_0=\varphi_0\). Conversely, by induction on \(n\), \(f^{(1)}_0=T_\omega\varphi_0\), \(f^{(1)}_{n+1}=(T_{s_0}+1)f_n-qf^{(1)}_{n-1}\), \(f_{n+1}=T_\omega f^{(1)}_{n+1}\). As \(f_n-f_{n-1}=\mathbf 1_{Y_n}\), \(\operatorname{span}\{f_n\}=C_K\); apply \(\omega\).
   • Codimension two. Orbits are compact and open, so the \(\mathbf 1_O\) form a basis of \(C_I\), and \(C_I=C_K+C_{K'}+\mathbb C\mathbf 1_{A_+}+\mathbb C\mathbf 1_{A_-}\). A functional kills \(C_K+C_{K'}\) iff \(\mu(O_{0,0})=\mu(O_{1,0})=-\mu(A_+)-\mu(A_-)\) and \(\mu(O_{i,m})=-\mu(O_{1-i,m-1})\): a plane spanned by \(\lambda,\psi\), whose pairing with \(\mathbf 1_{A_\pm}\) has determinant \(1\cdot\tfrac12-(-1)\cdot\tfrac12=1\).
   • The quotient. On the blocks of step 2, \(\lambda\circ T_s=-\lambda\), \(\psi\circ T_s=-\psi\), \(\lambda\circ T_\omega=-\lambda\), \(\psi\circ T_\omega=\psi\); e.g. \(\psi(T_{s_1}\mathbf 1_{O_{0,m}})=q(-1)^m+(q-1)(-1)^{m+1}=-\psi(\mathbf 1_{O_{0,m}})\). \(\lambda\) is, up to \(\operatorname{vol}(A_+)\), the restriction of the Steinberg map \(h\mapsto h([1:0])-h([0:1])\), and \(\mathrm{St}^I=\varepsilon_-\). No sheaves and no I1 are used. Run nodes A6-6, A9-7.
9. Method Positive certificates

   A functional on classes of Weil objects that kills the generators and is nonnegative on every simple class excludes every simple object on which it is nonzero (Proposition V(b)):

   $$\Lambda\big([\mathrm{IC}_w\ast_I\mathrm{IC}_0]\big)=0\ \ \forall w,\qquad\Lambda([S'])\in\mathbb Z_{\ge0}[v^{\pm1}]\ \ \forall S'\qquad\Longrightarrow\qquad\Lambda([S])\ne0\ \Rightarrow\ S\notin\mathcal M_I,\ S\notin\mathcal C_I^{\mathrm{Iwa}}.$$
   Proof

   • Setting. Over \(\overline{\mathbb F}_q\), given I1: finitely many orbits per level, every simple object Frobenius-stable with its weight-\(0\) structure \(j_{!*}k[\delta](\delta/2)\), \(\mathrm{IC}_w\) and \(\mathrm{IC}_0\) pure of weight \(0\); \(\Lambda:\mathcal M_A\to\mathbb Z[A]\) is \(\mathbb Z[A]\)-linear.
   • Killing the generators. If \(\Lambda\circ\underline H_s=0\) for every simple affine \(s\), \(\Lambda\circ T_\omega=\pm\Lambda\), and \(W\ne1\): for a finite \(s\), \(\underline H_s[\mathrm{IC}_0]=(v+v^{-1})[\mathrm{IC}_0]\), and \(\mathbb Z[A]\) is a domain, so \(\Lambda([\mathrm{IC}_0])=0\); \([\mathrm{IC}_w]\) is a \(\mathbb Z[v^{\pm1}]\)-combination of monomials \(\underline H_{s_{i_1}}\cdots\underline H_{s_{i_r}}T_\omega^j\). (\(W\ne1\) is the repair of A9-8.)
   • Nonnegative multiplicities. \(\mathcal F=\mathrm{IC}_w\ast\mathrm{IC}_0\) is pure (BBD 5.1.14), so \(\mathcal F\simeq\bigoplus_n{}^pH^n(\mathcal F)[-n]\) with \({}^pH^n(\mathcal F)\cong\bigoplus_SS\otimes V^n_S\), \(V^n_S\) pure of weight \(n\) with eigenvalues \(q^{n/2}\) modulo \(\mu_\infty\) (BBD 5.4.5–5.4.6; the class lies in Chen–Yi's module): \([\mathcal F]=\sum_Sm_S[S]\), \(m_S=\sum_nd^n_Sv^{-n}\), \(d^n_S\ge0\).
   • No cancellation. \(0=\sum_Sm_S\Lambda([S])\) is a sum of Laurent polynomials with nonnegative coefficients, so every term vanishes, and \(\Lambda([S])\ne0\) forces all \(d^n_S=0\): \(S\) is no perverse constituent of a generator, hence \(S\notin\mathcal C_I^{\mathrm{Iwa}}\) by step 6. Weights are used, so this is over \(\overline{\mathbb F}_q\) only. Run nodes A6-7, A9-8.
10. Ruled out The Steinberg functional as a certificate

    The Steinberg functional kills the generators but has mixed sign on the closed orbits, so it gives only a relation between multiplicities:

    $$\lambda([\mathrm{IC}_{i,n}])=0,\qquad\lambda([\underline k_{A_\pm}])=\pm1,\qquad0=\sum_Sm_S\lambda([S])\ \Longrightarrow\ m_{A_+}=m_{A_-}\ \text{only}.$$
    Proof

    • Generators. \(\lambda\circ T_s=-\lambda\) and \(\lambda\circ T_\omega=-\lambda\) (step 8), so \(\lambda\circ\underline H_s=0\) and step 9 kills every generator class; \(\lambda([\mathrm{IC}_{i,n}])=0\) by the argument of step 11.
    • Mixed sign. \(\lambda([\underline k_{A_\pm}])=\pm1\), so the positivity hypothesis of step 9 fails: a summand \(\underline k_{A_+}\oplus\underline k_{A_-}\) of a generator would have \(\lambda\)-value \(0\) and is invisible to \(\lambda\).
    • Function level. The classical page's Corollary D needs no positivity, so there \(\lambda\) suffices: that is Proposition F. At sheaf level, weights make positivity decisive. Run nodes A6-8 (c), A9-9.
11. Result Second proof: 2ψ is a certificate

    The twisted Steinberg functional is a positive certificate, nonzero exactly on the two closed orbits:

    $$2\psi\circ\underline H_s=0,\quad2\psi\circ T_\omega=2\psi,\qquad2\psi([\mathrm{IC}_{i,n}])=0,\qquad2\psi([\underline k_{A_\pm}])=1\quad\Longrightarrow\quad\underline k_{A_\pm}\notin\mathcal C_I^{\mathrm{Iwa}}.$$
    Proof

    • Hecke equivariance. \(\psi\circ T_s=-\psi\) and \(\psi\circ T_\omega=\psi\) (step 8) are polynomial in \(q\), so they hold in \(\mathcal M_A\) with \(q=v^{-2}\); then \(2\psi(\underline H_sm)=v\big(2\psi(T_sm)+2\psi(m)\big)=0\), and step 9 kills every generator class.
    • Non-closed orbits. Every orbit other than \(A_\pm\) has closure \(Y_n\) or \(\omega Y_n\), stable under \(K\) or \(K'\) (step 1), so \(\mathrm{IC}_{i,n}\) is \(P_{s_1}\)- or \(P_{s_0}\)-equivariant, \(\underline H_s[\mathrm{IC}_{i,n}]=(v+v^{-1})[\mathrm{IC}_{i,n}]\) for that \(s\), and \(2\psi([\mathrm{IC}_{i,n}])=0\). Directly: \(\psi(\mathbf 1_{Y_j})=1-1=0\).
    • Closed orbits. \(\underline k_{A_\pm}=j_{A_\pm!}k\) has class \(\mathbf 1_{A_\pm}\), so \(2\psi=1\). These are all the simple objects, so step 9 gives \(\underline k_{A_\pm}\notin\mathcal M_I\) and \(\underline k_{A_\pm}\notin\mathcal C_I^{\mathrm{Iwa}}\), without steps 4 and 5.
    • Uniqueness. Among \(\Lambda\) with \(\Lambda\circ\underline H_s=0\) and \(\Lambda\circ T_\omega=\pm\Lambda\), the positive ones are the nonnegative multiples of \(\psi\); \(\lambda+2\psi\) (values \(2,0\)) and \(2\psi-\lambda\) (values \(0,2\)) certify one closed orbit each. Run nodes A6-8, A9-9.
12. Conclusion Both levels fail at A±

    The trace of the generated category is the generated function space, and the two missing simple objects trace its complement:

    $$\operatorname{tr}\big(\mathcal C_I^{\mathrm{Iwa}}\big)=\mathcal H_I\ast\varphi_0=C_K+C_{K'},\qquad\operatorname{tr}\underline k_{A_\pm}=\mathbf 1_{A_\pm},\qquad C_I=\mathcal H_I\ast\varphi_0\oplus\mathbb C\mathbf 1_{A_+}\oplus\mathbb C\mathbf 1_{A_-}.$$

    Both levels fail, at the same two orbits: the row agrees and is not a divergence, and functions ⇒ sheaves (Proposition II) is vacuous here because (IG) fails. One mechanism: every generator is a sum of objects equivariant for \(K\) or \(K'\), and \(A_\pm\) are the only orbits whose closure is stable under neither \(P_{s_0}\) nor \(P_{s_1}\) (type IIIa for both reflections).

    Proof

    • Traces. Over \(\mathbb F_q(\!(t)\!)\), given I1: a Weil object whose underlying object lies in \(\mathcal C_I^{\mathrm{Iwa}}\) has class in the span of the \([\mathrm{IC}_{i,n}\otimes V]\), \(\dim V=1\) (step 7; the repair of A6-6(e) in A9-7), whose traces are multiples of \(f_n\), \(f^{(1)}_n\) (step 4); traces are additive (Laumon 1.1.1).
    • Complement. \(\operatorname{tr}\underline k_{A_\pm}=\mathbf 1_{A_\pm}\), and by step 8 these span a complement of \(\mathcal H_I\ast\varphi_0\) (pairing determinant \(1\)).
    • Audit. The run's checklist (A6-11) and the audit verdict: no step fails; I1 is assumed for \(N\ge1\), not checked, which repairs the overclaim of A6-12(4). Run nodes A6-12, A9-11, A9-12.