Proposition II, the group case

Hypotheses

Only S1 is used, through Proposition I(a); the rest is unconditional in characteristic zero.

• S2 · Nontrivial Weyl group.

  $$W\ne1,\quad\text{that is, }H\text{ is not a torus.}$$

  Discussion

  For a torus \(K=I\), \(c=\Delta_e\) and \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I\). The proof needs one finite simple reflection.

Proposition II

Proof

Step 1 is the function-level proof. Step 2 rewrites the ideal through left \(K\)-equivariant objects. Steps 3–4 compute \(\mathcal W\) and the action of \(\Delta_s\) on it. Step 5 is the shift trick, step 6 concludes.

1. Result Functions: the sign character.

   $$\varepsilon:\mathcal H_I\to\mathbb C,\ T_w\mapsto(-1)^{\ell(w)},\qquad\varepsilon(\mathbf 1_K)=\sum_{w\in W}(-1)^{\ell(w)}=0,\qquad\varepsilon(T_e)=1,\qquad\text{so}\quad\mathcal H_Ie_K\mathcal H_I\subseteq\ker\varepsilon\subsetneq\mathcal H_I .$$

   Proof

   • Proved on the Iwahori pages (Proposition E), with citations on the note's functions page, step 8: \(T_s\mapsto-1\), \(T_\omega\mapsto1\) respects the Iwahori–Matsumoto presentation, and \(w\mapsto ws\) pairs even with odd lengths in \(W\ne1\).
   • The three function-level spans agree with \(\mathcal H_Ie_K\mathcal H_I\) (note, step 7); the sheaf statement needs an argument that survives retracts, which \(K_0\)-classes do not: steps 2–6.
2. Method The ideal is the left span of the \(K\)-equivariant objects.

   $$\mathcal C_I^{\mathrm{Iwa}}=\operatorname{Thick}\{\mathcal A\ast\mathcal F:\ \mathcal A\in\mathcal H_{c,I},\ \mathcal F\in q^*\mathcal D_c(K\backslash LH/I)\},\qquad\Delta_s\ast\mathcal F\simeq\mathcal F[-2]\ \text{ for such }\mathcal F\text{ and finite simple }s .$$

   Proof

   • Left convolution with \(c=\underline k_{K/I}\) is left \(K\)-averaging: \(c\ast\mathcal B\simeq q^*q_*\mathcal B\) (pull–push along \(I\backslash LH/I\to K\backslash LH/I\), shifts suppressed). So \(\mathcal A\ast c\ast\mathcal B=\mathcal A\ast q^*(q_*\mathcal B)\) lies in the right-hand side.
   • Conversely, for \(M\in\mathcal D_c(K\backslash LH/I)\), the key lemma for \(q\) makes \(M\) a retract of \(q_*q^*M\); applying \(q^*\), \(q^*M\) is a retract of \(q^*q_*q^*M=c\ast q^*M\), hence \(\mathcal A\ast q^*M\) is a retract of \(\mathcal A\ast c\ast q^*M\in\mathcal C_I^{\mathrm{Iwa}}\).
   • \(q^*M\) is left \(K\)-equivariant, hence left \(P_s\)-equivariant for every finite simple \(s\), and the eigen-property of the definitions gives \(\Delta_s\ast q^*M\simeq q^*M[-2]\).
3. Result The affine Whittaker category is \(\mathrm{Vect}^{\Omega}\).

   $$\mathcal W=\mathcal D_c\big(I\backslash LH/(I_u^-,\psi)\big)\simeq\bigoplus_{\omega\in\Omega}\mathrm{Vect},\qquad E_\omega:=\Phi(\Delta_\omega)\ \text{ a generator of the }\omega\text{-summand},\qquad\Phi(\Delta_e)=E_e\ne0 .$$

   Proof

   • Orbits. \(LH=\bigsqcup_{w\in\widetilde W}IwI^-\) and \(IwI^-=IwI_u^-\) because \(I\supseteq T(\mathcal O)=wT(\mathcal O)w^{-1}\). So the \(I\times I_u^-\)-orbits on \(LH\) are indexed by \(\widetilde W\), and the stabilizer of the coset \(Iw\) in \(I_u^-\) is \(S_w:=I_u^-\cap w^{-1}Iw\), a connected pro-unipotent group.
   • Which orbits carry \(\psi\)-equivariant objects. On the orbit of \(w\), a \((I,\,(I_u^-,\psi))\)-equivariant object is a \((S_w,\psi|_{S_w})\)-equivariant object on a point, and these vanish unless \(\psi|_{S_w}=1\): a nonzero equivariant object would trivialize the character on the connected group \(S_w\). Now \(U_{-\alpha_i}\subseteq S_w\) iff \(w(-\alpha_i)\) is a positive affine root, iff \(w(\alpha_i)<0\). If \(w\notin\Omega\), some simple affine root is sent to a negative root (write \(w=\omega w'\), \(w'\in W_{\mathrm{aff}}\) of positive length, \(\omega\) permuting the simple affine roots), and \(\psi\) is nontrivial on that \(U_{-\alpha_i}\): the orbit contributes nothing. If \(w=\omega\in\Omega\), \(\omega\) permutes the positive affine roots, \(\psi|_{S_\omega}=1\), and the category on the orbit is \(\mathcal D_c(\mathbb BS_\omega)\simeq\mathrm{Vect}\). This is the stabilizer argument in the proof of Bezrukavnikov–Yun's Lemma 4.2.1 and in Arkhipov–Bezrukavnikov's Lemma 2, both stated for characters nondegenerate on a finite-type set of simple reflections only; for the fully affine character no published statement was found, so the lemma is proved here.
   • Gluing. The orbits \(I\omega I_u^-\), \(\omega\in\Omega\), lie in distinct connected components of \(LH\), and every other orbit carries the zero category; so \(\mathcal W\) is the direct sum of the \(\mathrm{Vect}\) summands. \(\Phi(\Delta_e)\) restricted to the orbit of \(e\) is \(H_c^*(S_e,\psi)=H_c^*(\mathrm{pt})=k\) since \(S_e=I_u^-\cap I=1\); likewise \(\Phi(\Delta_\omega)\) is a generator of the \(\omega\)-summand.
4. Result Every \(\Delta_s\) acts on \(\mathcal W\) by the shift \([-1]\): the categorical sign character.

   $$\Phi(c_s)=0,\qquad\Phi(\Delta_s)\simeq E_e[-1],\qquad\Delta_s\ast E_\omega\simeq E_\omega[-1]\quad(s\ \text{any simple affine reflection},\ \omega\in\Omega).$$

   Proof

   • \(\Phi(c_s)\) restricted to the orbit of \(e\) is \(H_c^*(I_u^-\cap P_s,\psi)\). The group \(I_u^-\cap P_s\) contains \(U_{-\alpha_s}\cong\mathbb A^1\) as a direct factor on which \(\psi\) is nontrivial, so this is \(H_c^*(\mathbb A^1,\exp)\otimes(\cdots)=0\); the same on the other orbits. Hence \(\Phi(c_s)=0\). Vanishing: the de Rham complex of the exponential D-module \((\mathcal O_{\mathbb A^1},d+dz)\) is \(k[z]\to k[z]\), \(f\mapsto f'+f\), an isomorphism (triangular for the degree filtration, identity on the graded pieces), so \(H^*_{\mathrm{dR}}(\mathbb A^1,\exp)=0\); the same computation with \(f'-f\) handles the Verdier dual \(\exp(-z)\), hence \(H_c^*=0\). The ℓ-adic counterpart, \(R\Gamma_c(\mathbb A^1,\mathcal L_\psi)=R\Gamma(\mathbb A^1,\mathcal L_\psi)=0\) for the Artin–Schreier sheaf, is Laumon (1.1.3.4), attributed to Deligne.
   • \(c_s\) restricted to the open cell \(IsI/I\cong\mathbb A^1\) and to the point \(e\) is constant, giving the cofibre sequence \(\Delta_s\to c_s\to\Delta_e\). Apply the exact functor \(\Phi\): \(\Phi(\Delta_s)\to0\to E_e\), so \(\Phi(\Delta_s)\simeq E_e[-1]\).
   • Module property: \(\Delta_s\ast E_e=\Delta_s\ast\Phi(\Delta_e)=\Phi(\Delta_s\ast\Delta_e)=\Phi(\Delta_s)=E_e[-1]\). For \(\omega\in\Omega\), \(s\omega=\omega s'\) with \(s'=\omega^{-1}s\omega\) again simple, and \(\Delta_s\ast\Delta_\omega=\Delta_{s\omega}=\Delta_\omega\ast\Delta_{s'}\) (lengths add), so \(\Delta_s\ast E_\omega=\Delta_\omega\ast\Delta_{s'}\ast E_e=E_\omega[-1]\).
5. Method The shift trick: \(\Phi\) kills every left \(K\)-equivariant object, hence the ideal.

   $$\mathcal F\ \text{left }K\text{-equivariant}\ \Longrightarrow\ \Phi(\mathcal F)[-2]\simeq\Phi(\Delta_s\ast\mathcal F)\simeq\Phi(\mathcal F)[-1]\ \Longrightarrow\ \Phi(\mathcal F)=0;\qquad\mathcal C_I^{\mathrm{Iwa}}\subseteq\ker\Phi .$$

   Proof

   • Take a finite simple reflection \(s\in W\) (S2). Step 2 gives \(\Delta_s\ast\mathcal F\simeq\mathcal F[-2]\); step 4 and \(\mathcal H_{c,I}\)-linearity give \(\Phi(\Delta_s\ast\mathcal F)=\Delta_s\ast\Phi(\mathcal F)\simeq\Phi(\mathcal F)[-1]\), because \(\Phi(\mathcal F)\) is a finite direct sum of shifts of the \(E_\omega\). So \(\Phi(\mathcal F)\simeq\Phi(\mathcal F)[1]\): a bounded complex with finite-dimensional cohomology isomorphic to its own shift is zero.
   • \(\Phi(\mathcal A\ast\mathcal F)=\mathcal A\ast\Phi(\mathcal F)=0\) for every \(\mathcal A\). The kernel of an exact functor is closed under shifts, cofibres and retracts, so it contains the thick span of these objects, which is \(\mathcal C_I^{\mathrm{Iwa}}\) by step 2.
6. Conclusion \(\Delta_e\) is not Iwahori-generated; what is missing is the sign part.

   $$\Phi(\Delta_e)=E_e\ne0\ \Longrightarrow\ \Delta_e\notin\mathcal C_I^{\mathrm{Iwa}};\qquad\Phi\ \text{factors through}\ \mathcal C_I/\mathcal C_I^{\mathrm{Iwa}}\twoheadrightarrow\mathrm{Vect}^{\Omega}.$$

   Proof

   • Steps 3 and 5. The quotient \(\mathcal C_I/\mathcal C_I^{\mathrm{Iwa}}\) (Verdier quotient by a thick subcategory) receives \(\Phi\), which is essentially surjective; so the Iwahori-generated part misses at least the sign-type module \(\mathrm{Vect}^{\Omega}\), on which every \(\Delta_s\) acts by \([-1]\).
   • Dictionary: \(\Phi\) is the categorical form of the sign character \(\varepsilon\) of step 1 (\(T_s\mapsto-1\) becomes \(\Delta_s\mapsto[-1]\), \(T_\omega\mapsto1\) becomes a permutation of the components), that is, of the Iwahori-fixed line of the Steinberg representation; \(\ker\Phi\) is the categorical form of \(\ker\varepsilon\). For functions, \(\ker\varepsilon\) is strictly larger than the ideal when there are other Iwahori-spherical representations without \(K\)-fixed vectors; the same is expected for \(\ker\Phi\), and nothing here identifies \(\mathcal C_I^{\mathrm{Iwa}}\) with \(\ker\Phi\).