Proof PDF · original proposition numbering →

Proof correspondence: (a) = V, (b) = VI, (c) = the multiplicative-group corollary.

Proof

Step 1 states the strategy. Steps 2–3 set up the criterion for the group, steps 4–7 supply the trace and weight inputs, step 8 records why the trace alone does not suffice, and step 9 proves the sheaf half of (a); steps 10–11 finish (a). Steps 12–17 prove (b), step 18 proves (c).

1. Purpose Reduce the rows to summand sets

   By Proposition I a simple object that is no perverse constituent of a generator is not Iwahori-generated, and one that is a generator up to shift is:

   $$S\notin\mathcal M_I^+\ \Longrightarrow\ S\notin\mathcal C_I^{\mathrm{Iwa}},\qquad S\in\mathcal M_I\ \Longrightarrow\ S\in\mathcal C_I^{\mathrm{Iwa}}.$$
   Plan

   • Group. Show \(\Omega\cap\mathcal M_I^+=\varnothing\) (steps 2–9), then decide (IG) and \(\mathcal M_I\) (steps 10–11).
   • \(\mathbb A^n\), \(\mathbb G_m\). Show that every simple object is a generator up to shift (steps 12–18).
   • Not used. The affine Whittaker functor of the pages’ Proposition II and the I3 dévissage of their IV.1.
2. Result Group: levels, simple objects, generators

   For \(X=H\) under \(H\times H\), with \(\Omega\)-truncated levels, the inputs of Proposition I hold, \(\mathcal S_I=\{\mathrm{IC}_z:z\in\widetilde W\}\), and

   $$\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{M_{x,y}:x,y\in\widetilde W\},\qquad M_{x,y}\simeq\bigoplus_nP_n[-n],\quad P_n\simeq\bigoplus_z\mathrm{IC}_z\otimes E_{z,n}.$$
   Proof

   • Levels. \(Y_N=\bigcup\{\overline{IwI}:\ell(w)\le N,\ w\in W_{\mathrm{aff}}\Omega_N\}\), \(\Omega_N\) finite and exhausting \(\Omega\), \(Y_{N,M}=Y_N/I_M\). Each level has finitely many pieces, the double cosets \(I_HvI_H\subseteq Y_N\), so I1 holds; the transitions are torsors under vector groups, trivial on affine opens, so (Lev) holds.
   • Repair. The pages’ levels \(\bigcup_{\ell(w)\le N}\overline{IwI}\) contain \(\bigsqcup_{\omega\in\Omega}I\omega I\), infinitely many pieces when \(\Omega\) is infinite (\(H=\mathrm{GL}_n\)).
   • Stabilizers. Images of \(I_H\cap vI_Hv^{-1}=T(\mathcal O)\cdot(\text{pro-unipotent})\), connected, so I2 holds. So the heart has finite length, \(\mathcal S_I=\{\mathrm{IC}_z\}\), \(\operatorname{End}\mathrm{IC}_z=k\), and \(\mathrm{IC}_\omega=\Delta_\omega\) is a skyscraper.
   • Generators. \(\mathcal H_{c,I}=\operatorname{Thick}\{\mathrm{IC}_x\boxtimes\mathrm{IC}_y\}\), \((\mathcal A\boxtimes\mathcal B)\ast_I\mathcal F=\mathcal A\ast\mathcal F\ast\iota^*\mathcal B\), \(\iota^*\mathrm{IC}_y=\mathrm{IC}_{y^{-1}}\), \(\mathrm{IC}_0=\underline k_{K_H/I_H}=\mathrm{IC}_{w_0}[-\ell(w_0)]\); then (T2).
   • Splitting. \(M_{x,y}=m_!(\mathrm{IC}_x\mathbin{\widetilde\boxtimes}\mathrm{IC}_{w_0}\mathbin{\widetilde\boxtimes}\mathrm{IC}_y)\), \(m\) proper from a projective twisted product; the input is pure of weight \(\ell(x)+\ell(w_0)+\ell(y)\), so BBD 5.4.5 splits it geometrically, and Sun, Theorem 1.2, on the finite-type quotient stack, equivariantly: (DT).
   • Run A4-1, A2-8, A1-2; audit A8-1 (repairs R1–R3).
3. Method Constituent criterion

   The constituent subcategories are thick, and a simple object outside \(\mathcal M_I^+\) is not Iwahori-generated:

   $$\mathcal C_I^{\mathrm{Iwa}}\subseteq\mathcal D_{\mathcal M_I^+}=\operatorname{Thick}\{\mathrm{IC}_z:\mathrm{IC}_z\in\mathcal M_I^+\},\qquad\mathrm{IC}_z\notin\mathcal M_I^+\ \Longrightarrow\ \mathrm{IC}_z\notin\mathcal C_I^{\mathrm{Iwa}}.$$
   Proof

   • Premises. Finite-length heart, simple objects \(\mathrm{IC}_z\) (step 2).
   • Thick. In an exact triangle the constituents of \({}^pH^i\) of the middle term lie among those of the outer terms; a retract has \({}^pH^i\) a summand (Stacks 05RD, with the Serre quotient).
   • Output. The generators lie in \(\mathcal D_{\mathcal M_I^+}\) by definition; \({}^pH^0(\mathrm{IC}_z)=\mathrm{IC}_z\) is its own unique constituent. No decomposition is used. With (DT), \(\mathcal M_I=\mathcal M_I^+\) and \(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathrm{IC}_z:\mathrm{IC}_z\in\mathcal M_I\}\).
   • Run A4-2; audit A8-2. This is Proposition I for the group.
4. Method Lemma: the sign character

   Over every non-archimedean \(F\), the sign character is a ring homomorphism that kills \(\mathbf 1_{K_H}\):

   $$\varepsilon:\mathcal H_{I_H}\to\mathbb Z,\quad\varepsilon(T_w)=(-1)^{\ell(w)},\qquad\varepsilon(T_\omega)=1,\qquad\varepsilon(\mathbf 1_{K_H})=\sum_{w\in W}(-1)^{\ell(w)}=0 .$$
   Proof

   • Relations. \(T_xT_s=T_{xs}\) if \(x\lt xs\); \(T_xT_s=qT_{xs}+(q-1)T_x\) if \(xs\lt x\); \(T_xT_\sigma=T_{x\sigma}\), \(\sigma\in\Omega\).
   • Check. \(q(-1)^{\ell(x)-1}+(q-1)(-1)^{\ell(x)}=(-1)^{\ell(x)+1}\); induction on a reduced word gives multiplicativity. This is Borel’s special character.
   • \(\mathbf 1_{K_H}\). \(K_H=\bigsqcup_{w\in W}I_HwI_H\), so \(\mathbf 1_{K_H}=\sum_WT_w\); for a finite simple reflection \(s\), \(w\mapsto ws\) is a fixed-point-free involution of \(W\) changing the parity of \(\ell\), and \(W\ne1\).
   • Run A4-3 (a), (b), A2-9 step 4; audit A8-3.
5. Method Lemma: traces of convolutions

   Trace of Frobenius turns convolution of Weil objects into convolution in \(\mathcal H_{I,m}\), with the values used below:

   $$\operatorname{tr}_m(\mathcal A\ast\mathcal B)=\operatorname{tr}_m\mathcal A\ast\operatorname{tr}_m\mathcal B,\qquad\operatorname{tr}_m\mathrm{IC}_\omega=T_\omega,\quad\operatorname{tr}_m\mathrm{IC}_{w_0}=(-1)^{\ell(w_0)}\mathbf 1_{K_H},\quad\operatorname{tr}_m\mathrm{IC}_s=-(T_e+T_s).$$
   Proof

   • Inference. Proper base change and the Grothendieck–Lefschetz formula on the fibres \(\{gI:g^{-1}x\in\mathrm{Fl}_{\le u'}\}\) of the convolution map.
   • Points. The \(\mathbb F_{q^m}\)-points of \(LH\times^I\mathrm{Fl}\) are \(H(F_m)\times^{I(\mathcal O_m)}\mathrm{Fl}(\mathbb F_{q^m})\): Lang for each connected \(I/I_n\), and a limit over \(n\).
   • Values. A rational point; the smooth \(K_H/I_H\); \(\overline{\mathrm{Fl}_s}\cong\mathbb P^1\); \(\operatorname{tr}_m(\mathcal F[1])=-\operatorname{tr}_m\mathcal F\).
   • Run A4-3 (c), (d), A2-9 step 3; audit A8-3.
6. Result The sign character kills the IC traces off Ω

   In the IC-trace basis only the length-zero coefficients survive \(\varepsilon\), each with value \(1\), and \(\operatorname{tr}\mathrm{IC}_{w_0}\) is killed:

   $$\varepsilon(\operatorname{tr}_m\mathrm{IC}_z)=\begin{cases}0,&z\notin\Omega,\\1,&z\in\Omega,\end{cases}\qquad\varepsilon(\operatorname{tr}_m\mathrm{IC}_{w_0})=(-1)^{\ell(w_0)}\varepsilon(\mathbf 1_{K_H})=0\qquad(m\ge1).$$
   Proof

   • Descent. For \(z=\omega z'\notin\Omega\) choose a simple affine \(s\) with \(zs\lt z\); then \(\overline{IzI}\) is right \(P_s\)-stable.
   • \(P_s\)-invariance. \(\mathrm{IC}_z\) is pulled back, with its Weil structure, along the \(\mathbb P^1\)-bundle \(\mathrm{Fl}\to LH/P_s\); so \(f=\operatorname{tr}_m\mathrm{IC}_z\) is right \(P_s(\mathcal O_m)\)-invariant and \(f\ast T_s=q^mf\).
   • Output. Applying \(\varepsilon\): \(-\varepsilon(f)=q^m\varepsilon(f)\), so \(\varepsilon(f)=0\). For \(z=\omega\), \(\varepsilon(T_\omega)=1\); for \(\mathrm{IC}_{w_0}\), step 4.
   • Run A4-3 (e), A2-9 step 4, A3-9 step 2; audit A8-3 (also checked on the Kazhdan–Lusztig bases of \(\widetilde A_1\), \(\widetilde A_2\)).
7. Result Purity fixes the weight of each multiplicity space

   The trace of a generator expands in IC traces with Frobenius traces of the multiplicity spaces as coefficients, and each multiplicity space has a single weight, fixed by its shift:

   $$\operatorname{tr}_mM_{x,y}=\sum_{z,n}(-1)^n\operatorname{tr}\big(A_{z,n}^m\big)\operatorname{tr}_m\mathrm{IC}_z,\qquad|\iota\alpha|=q^{(w+n-\ell(z))/2},\quad w=\ell(x)+\ell(w_0)+\ell(y).$$
   Proof

   • Expansion. The evaluation map \(\bigoplus_z\mathrm{IC}_z\otimes E_{z,n}\to P_n\) is an isomorphism of Weil perverse sheaves (BBD 5.3.8); traces are additive on the Frobenius-stable perverse truncation triangles, and \(\operatorname{tr}(P[-n])=(-1)^n\operatorname{tr}P\).
   • Weight. \(P_n\) is pure of weight \(w+n\) (BBD 5.4.1–5.4.4); its isotypic part \(\mathrm{IC}_z\otimes(E_{z,n},A_{z,n})\) is a Weil summand, hence pure; on the open cell it is \(E_{z,n}\otimes\overline{\mathbb Q}_\ell[\ell(z)]\), so \(A_{z,n}\) is pure of weight \(w+n-\ell(z)\).
   • Equivalently. The stalk at \(\dot zI\) bounds the weights from above, Verdier duality \(\mathbb D\mathrm{IC}_z\simeq\mathrm{IC}_z(\ell(z))\) from below. Only purity is used, not Tate type or parity.
   • Run A4-4, A2-9 step 1; audit A8-4.
8. Ruled out The trace alone

   \(\varepsilon\circ\operatorname{tr}_m\) is additive on exact triangles, not on retracts, so vanishing on generator traces cannot exclude a summand:

   $$\varepsilon\big(\operatorname{tr}_m(\Delta_e\oplus\Delta_e[1])\big)=1-1=0\quad(m\ge1),\qquad\Delta_e\ \text{a retract of}\ \Delta_e\oplus\Delta_e[1].$$
   Proof

   • Consequence. The function failure (step 10(a)) does not give \(\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\); the weights of step 7 fix the sign of every possible contribution of \(\Delta_\omega\), which is what step 9 uses.
   • Comparison. The pages’ Whittaker functor \(\Phi\) is exact, so it sees retracts; here exactness is replaced by weights.
   • Run A4-5, A4-6 (d); audit A8-5 step 4, A8-6 step 5.
9. Result No cancellation; no \(\Delta_\omega\)

   Over \(\overline{\mathbb F}_q\), no \(\Delta_\omega\) is a perverse constituent of a generator, so the unit is not Iwahori-generated:

   $$\operatorname{Hom}\big(\Delta_\omega,{}^pH^n(M_{x,y})\big)=0\quad(x,y\in\widetilde W,\ \omega\in\Omega,\ n\in\mathbb Z),\qquad\Delta_e\notin\mathcal C_I^{\mathrm{Iwa}}\subseteq\operatorname{Thick}\{\mathrm{IC}_z:z\notin\Omega\}\subsetneq\mathcal C_I .$$
   Proof

   • Vanishing. Steps 4–6: \(\varepsilon(\operatorname{tr}_mM_{x,y})=\varepsilon(\operatorname{tr}_m\mathrm{IC}_x)\,\varepsilon(\operatorname{tr}_m\mathrm{IC}_{w_0})\,\varepsilon(\operatorname{tr}_m\mathrm{IC}_y)=0\); steps 6–7 rewrite it as \(\sum_{\omega,n}(-1)^n\operatorname{tr}(A_{\omega,n}^m)\). So this sum vanishes for every \(m\ge1\).
   • Separation. Put \(c_\alpha=\sum_{\omega,n}(-1)^n\mu_{\omega,n}(\alpha)\), \(\mu\) the multiplicity of the eigenvalue \(\alpha\). Then \(\sum_\alpha c_\alpha\alpha^m=0\) for \(m=1,\dots,|\Lambda|\), and the Vandermonde matrix is invertible: every \(c_\alpha=0\).
   • Weights. By step 7 with \(\ell(\omega)=0\), every eigenvalue of \(A_{\omega,n}\) has weight \(w+n\): \(\alpha\) determines \(n\), and \(c_\alpha=(-1)^{n(\alpha)}\sum_\omega\mu_{\omega,n(\alpha)}(\alpha)\) is a sum of non-negative integers with one sign. So every \(\mu=0\) and \(E_{\omega,n}=0\).
   • Output. \(\Omega\cap\mathcal M_I^+=\varnothing\); step 3 excludes every \(\mathrm{IC}_\omega=\Delta_\omega\). Over \(\mathbb C\), (C) transports \(E_{\omega,n}=0\) from \(\overline{\mathbb F}_p\), \(p\gg0\).
   • Four versions in the run: A4-5, A2-9, A3-9, A6-9; audit A8-5, A8-13.
10. Result (IG) fails, by the character and by Proposition II

    The functions fail over every \(F\), and over \(\mathbb F_q(\!(t)\!)\) the sheaf statement forces the same failure:

    $$\mathcal H_I\ast\varphi_0=\mathcal H_{I_H}\mathbf 1_{K_H}\mathcal H_{I_H}\subseteq\ker\varepsilon\not\ni T_e .$$
    Proof

    • (a) Every \(F\). \(C_I=\mathcal H_{I_H}\), with \(((f_1\otimes f_2)\ast\varphi)=f_1\ast\varphi\ast f_2^\vee\), \(\varepsilon(f^\vee)=\varepsilon(f)\) and \(\varphi_0=\mathbf 1_{K_H}\); step 4 gives the display (the pages’ Proposition E).
    • (b) \(\mathbb F_q(\!(t)\!)\). (Dict) holds: rational cells and Lang, step 5, unitriangular IC traces, step 7. If (IG) held, Proposition II would put \(\Delta_e\) in \(\mathcal C_I^{\mathrm{Iwa}}\), against step 9.
    • (c) The converse inference fails (step 8).
    • Run A4-6, A2-9 step 8; audit A8-6.
11. Result The summand set of the group

    Identifying \(\mathrm{IC}_z\) with \(z\), the summand set contains the top cells, misses \(\Omega\), is \(\widetilde W\setminus\Omega\) in type \(A_1\), and is the lowest two-sided cell times \(\Omega\) in general (cited):

    $$\{xw_0y:\ \ell(xw_0y)=\ell(x)+\ell(w_0)+\ell(y)\}\subseteq\mathcal M_I\subseteq\widetilde W\setminus\Omega,\qquad\mathcal M_I=c_0\Omega .$$
    Proof

    • Top cell. \(\operatorname{supp}M_{x,y}=\overline{\mathrm{Fl}_{xw_0y}}\) and \(m\) is an isomorphism over \(\mathrm{Fl}_{xw_0y}\): \(\mathrm{IC}_{xw_0y}\) occurs once, in \({}^pH^0\).
    • Type \(A_1\). \(w_0=s_1\); every \(z\ne e,s_0\) of \(W_{\mathrm{aff}}\) has a reduced word through \(s_1\); \(\mathrm{IC}_{s_0}\) is a summand of \(M_{s_0s_1,s_1s_0}\) because the Bott–Samelson map for \(s_0s_1s_0\) is semismall with relevant strata \(\mathrm{Fl}_{s_0s_1s_0}\), \(\mathrm{Fl}_{s_0}\). With (DT), \(\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathrm{IC}_z:\ell(z)\gt0\}\).
    • General (cited). Positivity gives \(\mathcal M_I=\{z\le_{LR}w_0\}\), the lowest two-sided cell of Lusztig and Shi; so \(\mathrm{IC}_s\notin\mathcal C_I^{\mathrm{Iwa}}\) for \(H\) simple of rank \(\ge2\). A computation in \(\widetilde A_2\) agrees up to length 6.
    • Run A4-7, A1-2; audit A8-7.
12. Result 𝔸ⁿ: the lattice chain is one Ω-orbit

    The \(I\)-stable lattices are the translates of \(\mathcal O^n\) by one length-zero element, and every level satisfies (Lev), I1, I2:

    $$\tau\Lambda_{m,j}=\Lambda_{m,j-1},\qquad\Lambda_{m,j}=\tau^{nm+n-j}\mathcal O^n,\qquad\tau I\tau^{-1}=I,\quad\ell(\tau)=0,\quad\Omega=\langle\tau\rangle .$$
    Proof

    • Lattices. \(\tau\) sends the basis \(t^me_1,\dots,t^me_j,t^{m+1}e_{j+1},\dots,t^{m+1}e_n\) of \(\Lambda_{m,j}\) to one of \(\Lambda_{m,j-1}\); with \(\Lambda_{m,0}=\Lambda_{m+1,n}\), \((m,j)\mapsto nm+n-j\) is a bijection onto \(\mathbb Z\).
    • Length zero. \(I\) is the stabilizer of the chain \(\{\Lambda_{m,j}\}\), which \(\tau\) permutes; \(\operatorname{val}\det\tau=1\).
    • Orbits. \(\{0\}\) and \(o_k=\tau^k\mathcal O^n\setminus\tau^{k+1}\mathcal O^n\); at level \((N,M)\), \(n(N+M)+1\) orbits with linear closures \(\tau^k\mathcal O^n/t^M\mathcal O^n\) of dimension \(nM-k\).
    • Stabilizers. Nonempty open subsets of affine-linear spaces, hence connected.
    • Run A5-1, A2-6; audit A8-8 (orbit counts recomputed).
13. Result \(\mathbb A^n\): the simple objects \(S_k\)

    With the perverse t-structure shifted by \(nM\) at level \((N,M)\), the simple objects of \(\mathcal C_I\) are the shifted lattice sheaves:

    $$\mathcal S_I=\{S_k=\underline k_{\tau^k\mathcal O^n}[-k]:k\in\mathbb Z\},\qquad S_0=\mathrm{IC}_0,\qquad\mathcal C_I=\operatorname{Tri}\{S_k\}.$$
    Proof

    • Transitions. \(q^*[n]\) is t-exact for the smooth \(\mathbb A^n\)-fibration \(V_{N,M+1}\to V_{N,M}\); with the shift \(nM\), \(q^*\) and the closed \(i_*\) are t-exact and fully faithful with images closed under subquotients.
    • Simple objects. \(\mathrm{IC}(\tau^k\mathcal O^n/t^M\mathcal O^n)=\underline k[nM-k]\), which is \(S_k\) after the shift.
    • Output. Every object is an iterated extension of shifted \(S_k\); no object is supported on \(\{0\}\), whose finite-level skyscraper is \(S_{nM}\).
    • Run A5-2, A2-6; audit A8-8.
14. Method 𝔸ⁿ: length-zero translates

    Convolution with a length-zero skyscraper is a translation, so each simple object is a generator up to shift:

    $$\Delta_{\tau^k}\ast_I\mathcal F\simeq(\tau^k)_!\mathcal F,\qquad\Delta_{\tau^k}\ast_I\mathrm{IC}_0=\underline k_{\tau^k\mathcal O^n}=S_k[k].$$
    Proof

    • Inference. \(I\tau^kI/I\) is a point and \(\tau^kI\times^ILX\to LX\), \([\tau^ki,y]\mapsto\tau^kiy\), is an isomorphism with inverse \(x\mapsto[\tau^k,\tau^{-k}x]\).
    • Output. \((\tau^k)_!\) preserves \(I\)-equivariance; \((\tau^k)_!S_l=S_{k+l}[k]\). No decomposition theorem, no weights.
    • Run A5-3 (a), A2-7 (c); audit A8-9.
15. Result 𝔸ⁿ: sheaves generate

    Every simple object is a summand of a generator, and extensions and shifts suffice:

    $$\mathcal M_I=\mathcal S_I,\qquad\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I=\operatorname{Tri}\{\Delta_{\tau^k}\ast_I\mathrm{IC}_0:k\in\mathbb Z\}.$$
    Proof

    • Inference. \(\mathcal C_I=\operatorname{Tri}\{S_k\}\subseteq\operatorname{Tri}\{\Delta_{\tau^k}\ast_I\mathrm{IC}_0\}\subseteq\mathcal C_I^{\mathrm{Iwa}}\) (steps 13–14).
    • Fields. No weights: the same proof over \(\mathbb C\) and \(\overline{\mathbb F}_q\), for every \(n\ge1\).
    • Run A5-5, A2-7 (d); audit A8-9.
16. Result 𝔸ⁿ: functions generate

    The length-zero Hecke operators carry \(\varphi_0\) to a basis of \(C_I\), over every non-archimedean \(F\):

    $$T_{\tau^k}\ast\varphi_0=b_k,\qquad C_I=\bigoplus_{k\in\mathbb Z}\mathbb C\,b_k,\qquad\mathbb C[\Omega]\xrightarrow{\ \sim\ }C_I,\ \ h\mapsto h\ast\varphi_0 .$$
    Proof

    • Translates. \((T_{\tau^k}\ast f)(x)=\int_{\tau^kI}f(g^{-1}x)\,dg=f(\tau^{-k}x)\), with \(\operatorname{vol}(I)=1\).
    • Basis. Independent since the lattices strictly decrease; spanning since \(\varphi\in C_I\) is supported in some \(\tau^{-nN}\mathcal O^n\), constant on some \(\tau^{nM}\mathcal O^n\) and on each orbit: \(\varphi=\sum_k\varphi(o_k)(b_k-b_{k+1})+\varphi(0)b_{nM}\).
    • Also. \(T_{s_0}\ast\varphi_0=b_{-1}+q\,b_1-b_0\) (recomputed pointwise), and \(\operatorname{tr}S_k=(-1)^kb_k\).
    • Run A5-6, A2-7 (e); audit A8-10.
17. Result 𝔸ⁿ: sheaves again, from functions

    Over \(\mathbb F_q(\!(t)\!)\) the inputs of Proposition II hold for \(\mathbb A^n\), so (IG) gives the sheaf statement a second time:

    $$\text{(IG)}\ \Longrightarrow\ \mathcal M_I=\mathcal S_I,\quad\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\quad\text{over }\overline{\mathbb F}_q .$$
    Proof

    • (Dict). Levels and orbits defined over \(\mathbb F_q\), with rational representatives \(t^me_j\); Lang for the connected \(I/I_{N,M}\).
    • (DT). \(\mathrm{IC}_w\ast_I\mathrm{IC}_0=\pi_{N,M}^*m_{w!}\mathrm{pr}_1^*\mathrm{IC}_w\) for the proper lattice bundle \(m_w:E_w\to V_{N,M}\); pure, hence split (Sun, Theorem 1.2).
    • Scope. This route needs purity and is \(\ell\)-adic only; step 15 needs neither.
    • Run A5-8, A3-8; audit A8-10.
18. Result 𝔾ₘ: both levels generate

    For \(\mathbb G_m\) every simple object is a length-zero translate of \(\mathrm{IC}_0\), and the translates of \(\varphi_0\) span \(C_I\):

    $$\mathcal S_I=\{\underline k_{t^m\mathcal O^\times}\}=\Omega\cdot\mathrm{IC}_0,\qquad\Delta_{\omega^m}\ast_I\mathrm{IC}_0=\underline k_{t^m\mathcal O^\times},\qquad T_{\omega^m}\ast\varphi_0=\mathbf 1_{t^m\mathcal O^\times}.$$
    Proof

    • Orbits. \(\det(I)=\mathcal O^\times\), so the \(I\)-orbits on \(F^\times\) are the \(t^m\mathcal O^\times\), open and closed; \(I\) acts on \(t^m(\mathcal O/t^M)^\times\) through the connected \((\mathcal O/t^M)^\times\), simply transitively.
    • Translates. \(\omega\) normalizes \(I\) and \(\det\omega=-t\), so \((\omega^m)_!\underline k_{\mathcal O^\times}=\underline k_{t^m\mathcal O^\times}\). Over \(\mathbb C\), \(\overline{\mathbb F}_q\) and every \(F\).
    • Run A2-10 (c), A1-4; audit A8-11. This re-proves the pages’ IV.3 by the route of step 15.
19. Conclusion Rows 1–2 and 𝔾ₘ: the levels agree

    Steps 9–11 prove (a), steps 12–17 prove (b), step 18 proves (c):

    $$\text{group: }\ \Delta_e\notin\mathcal C_I^{\mathrm{Iwa}},\ \ \mathcal H_I\ast\varphi_0\ne C_I;\qquad\mathbb A^n,\ \mathbb G_m:\ \ \mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I,\ \ \mathcal H_I\ast\varphi_0=C_I .$$
    Scope

    • In each row the two levels agree, as in the verdict’s table.
    • Conditional parts: the group over \(\mathbb C\), on (C); the summand set beyond type \(A_1\), on cited cell theory.