§4 · Agent 4 · The group: sign and weights · 2026-10-18T18:48:50Z

Proposition 4.7 · $\mathcal M_I$ for the group: top cells in, $\Omega$ out, exact in rank one

proved

Determines which IC_z are Iwahori-generated: every reduced x·w_0·y is (top cell), no length-zero element is (key theorem), nor any z with no letter from a whole simple factor (mixed sign/trivial character); in semisimple rank one this is everything, and in general the exact answer is the lowest two-sided cell, which needs Kazhdan–Lusztig cell theory.

Proposition 4.7.

Over \(\overline{\mathbb F}_q\), \(W\ne1\), with \(\mathcal M_I^\flat\) as in Proposition 4.2 (\(=\mathcal M_I\) under the equivariant splitting):

  1. (proved) If \(\ell(xw_0y)=\ell(x)+\ell(w_0)+\ell(y)\), then \(\mathrm{IC}_{xw_0y}\) occurs in \(M_{x,y}\) exactly once, in \({}^pH^0\). So \(\{xw_0y\ \text{reduced}\}\subseteq\mathcal M_I^\flat\).
  2. (proved) \(\mathcal M_I^\flat\cap\Omega=\emptyset\) (Theorem 4.5). More generally, if \(S_{\rm aff}=S'\sqcup S''\) with \(S'\neq\emptyset\) a union of connected components of the affine Dynkin diagram, then \(\mathcal M_I^\flat\cap\{z:\ \ell_{S'}(z)=0\}=\emptyset\), \(\ell_{S'}\) the number of letters from \(S'\) in a reduced word.
  3. (proved) If the root system of \(H\) is of type \(A_1\) (\(\mathrm{SL}_2,\mathrm{PGL}_2,\mathrm{GL}_2\), times tori): \(\mathcal M_I^\flat=\widetilde W\setminus\Omega\), so \(\mathcal C_I^{\mathrm{Iwa}}=\mathcal D_{\widetilde W\setminus\Omega}=\operatorname{Thick}\{\mathrm{IC}_z:\ell(z)>0\}\) (equivariant splitting).
  4. (cited) In general \(\mathcal M_I=\{z:\ z\le_{LR}w_0\}=c_0\Omega\), \(c_0=\{x\cdot w_J\cdot y\ \text{reduced}:\ J\subsetneq S_{\rm aff},\ \ell(w_J)=\nu:=\ell(w_0)\}\) the lowest two-sided cell. For \(H\) simple of rank \(\ge2\), \(\mathrm{IC}_s\notin\mathcal C_I^{\mathrm{Iwa}}\) for every simple affine reflection \(s\), so \(\mathcal C_I^{\mathrm{Iwa}}\subsetneq\mathcal D_{\widetilde W\setminus\Omega}\).
  5. The description \(\{x\cdot w_0\cdot y\}\) of the brief is too small: for \(\mathrm{SL}_2\), \(s_0\in\mathcal M_I\) although \(s_0\) has no reduced factorisation through \(w_0=s_1\).
  1. Top cell. Products of cells satisfy \(IuI\cdot IvI\subseteq\bigcup_{\ell(z)\le\ell(u)+\ell(v)}IzI\) (BN-pair rule \(IsI\cdot IvI\subseteq IsvI\cup IvI\)). So \(m\) maps every stratum \(\mathrm{Fl}_u\widetilde\times\mathrm{Fl}_v\widetilde\times\mathrm{Fl}_{u'}\ne\mathrm{Fl}_x\widetilde\times\mathrm{Fl}_{w_0}\widetilde\times\mathrm{Fl}_y\) of the source into cells of dimension \(<\ell(xw_0y)\), and \(\mathrm{supp}\,M_{x,y}\subseteq\overline{\mathrm{Fl}_{xw_0y}}\). Over the open cell, \(m\) is the isomorphism \(\mathrm{Fl}_x\widetilde\times\mathrm{Fl}_{w_0}\widetilde\times\mathrm{Fl}_y\cong\mathrm{Fl}_{xw_0y}\) (lengths add; bezrukavnikov-yun-v2.pdf Lemma 3.2.2, (3.8)), so \(M_{x,y}|_{\mathrm{Fl}_{xw_0y}}=\underline k[\ell(xw_0y)]\). In \(\bigoplus\mathrm{IC}_z[-n]\otimes E_{z,n}\) only \(z\le xw_0y\) occur, and the restriction to the open cell is \(\bigoplus_n\underline k[\ell(xw_0y)-n]\otimes E_{xw_0y,n}\); hence \(E_{xw_0y,n}=k\) for \(n=0\) and \(0\) otherwise.
  2. A whole factor missing. Define \(\kappa(T_x)=(-1)^{\ell_{S'}(x)}q^{\ell_{S''}(x)}\), \(\kappa(T_\omega)=1\) (\(\Omega\) permutes each of \(S',S''\)). The relations of Lemma 4.3 step 1 hold: for \(s\in S''\), \(xs<x\): \(q\cdot q^{-1}\kappa(T_x)+(q-1)\kappa(T_x)=q\kappa(T_x)\). \(W=W'\times W''\), so \(\kappa(\mathbf 1_{K_H})=\big(\sum_{W'}(-1)^{\ell}\big)\big(\sum_{W''}q^{\ell}\big)=0\) (\(W'\ne1\): each affine component contains a finite simple reflection). If \(\ell_{S'}(z)>0\), \(z\) has a right descent \(s\in S'\) and the descent argument of Lemma 4.3 step 5 with \(\kappa(\operatorname{tr}\mathrm{IC}_s)=-(1-1)=0\) gives \(\kappa(\operatorname{tr}_m\mathrm{IC}_z)=0\). If \(\ell_{S'}(z)=0\), every \(y\le z\) has \(\kappa(T_y)=q^{\ell(y)}=|\mathrm{Fl}_y(\mathbb F_q)|\), so by Grothendieck–Lefschetz \(\kappa(\operatorname{tr}_m\mathrm{IC}_z)=\sum_j(-1)^j\operatorname{tr}(\mathrm{Fr}^m\mid IH^j_z)\), \(IH^j_z:=H^j(\overline{\mathrm{Fl}_z},\mathrm{IC}_z)\), pure of weight \(\ell(z)+j\) (proper pushforward) and nonzero for \(j=-\ell(z)\) (\(H^{-\ell(z)}=H^0(\mathrm{Fl}_z,\underline k)\)). As in Theorem 4.5: \(0=\kappa(\operatorname{tr}_mM_{x,y})=\sum_{\ell_{S'}(z)=0}\sum_{n,j}(-1)^{n+j}\operatorname{tr}(A_{z,n}^m)\operatorname{tr}(\mathrm{Fr}^m|IH^j_z)\); an eigenvalue product \(\alpha\beta\) has weight \((w+n-\ell(z))+(\ell(z)+j)=w+n+j\), so the sign \((-1)^{n+j}\) is fixed by the weight, nothing cancels, and \(E_{z,n}\otimes IH^*_z=0\); since \(IH^*_z\ne0\), \(E_{z,n}=0\).
  3. Rank one. \(W_{\rm aff}=\langle s_0,s_1\rangle\) infinite dihedral, \(w_0=s_1\). Every \(z\in W_{\rm aff}\setminus\{e,s_0\}\) has a reduced word containing \(s_1\), so step 1 applies; \(\Omega\)-translates follow from \(M_{x,y\omega}=M_{x,y}\ast\Delta_\omega\). For \(s_0\): \(M_{s_0s_1,s_1s_0}=\mathrm{IC}_{s_0s_1}\ast\mathrm{IC}_{s_1}\ast\mathrm{IC}_{s_1s_0}\). Geometrically \(\mathrm{IC}_{s_0s_1}\ast\mathrm{IC}_{s_1}\simeq\mathrm{IC}_{s_0s_1}[1]\oplus\mathrm{IC}_{s_0s_1}[-1]\) (Lemma 4.3 step 5, \(\pi_{s*}\overline{\mathbb Q}_\ell\) split), \(\mathrm{IC}_{s_0s_1}=\mathrm{IC}_{s_0}\ast\mathrm{IC}_{s_1}\), \(\mathrm{IC}_{s_1s_0}=\mathrm{IC}_{s_1}\ast\mathrm{IC}_{s_0}\) (Bott–Samelson maps bijective onto length-2 Schubert varieties). So \(M_{s_0s_1,s_1s_0}\) contains \((\mathrm{IC}_{s_0}\ast\mathrm{IC}_{s_1}\ast\mathrm{IC}_{s_0})[\pm2],[0]\), and \(\mathrm{IC}_{s_0}\ast\mathrm{IC}_{s_1}\ast\mathrm{IC}_{s_0}=\pi_*\underline k[3]\) for \(\pi:P_{s_0}\times^IP_{s_1}\times^IP_{s_0}/I\to\overline{\mathrm{Fl}_{s_0s_1s_0}}\). Fibres: a point over \(\mathrm{Fl}_{s_0s_1s_0},\mathrm{Fl}_{s_1}\), finite over the 2-dimensional cells, \(\mathbb P^1\) over \(\mathrm{Fl}_{s_0}\) and \(\mathrm{Fl}_e\) (if \(g_3\in P_{s_0}\) then \(g_2\in P_{s_0}\) and \(g_1^{-1}g_2\in P_{s_0}\cap P_{s_1}=I\)). So \(\pi\) is semismall with relevant strata \(\mathrm{Fl}_{s_0s_1s_0}\) and \(\mathrm{Fl}_{s_0}\) (\(2\cdot1=3-1\)), and \(\pi_*\underline k[3]\simeq\mathrm{IC}_{s_0s_1s_0}\oplus\mathrm{IC}_{s_0}\) (decataldo-migliorini-v2.pdf Thm 4.2.7; summand.tex Prop. I(e) with \(r=0\)). Hence \(s_0\in\mathcal M_I^\flat\), and with step 2 (\(\Omega\) excluded) \(\mathcal M_I^\flat=\widetilde W\setminus\Omega\). Proposition 4.2 steps 3–4 give the thick span.
  4. General case (cited, numbers not checked against local copies). (a) \(\operatorname{tr}_m\mathrm{IC}_z=(-1)^{\ell(z)}\sum_yP_{y,z}(q^m)T_y\) (Kazhdan–Lusztig, Schubert varieties and Poincaré duality, 1980, Thm 4.2 and the affine case §5); (b) positivity: \(M_{x,y}\) is a sum of \(\mathrm{IC}_z[n](n/2)\) (bezrukavnikov-yun-v2.pdf Prop. 3.2.5), so the KL structure constants are \(\sum_n\dim E_{z,n}v^{\pm n}\in\mathbb N[v^{\pm1}]\) and \(z\in\mathcal M_I\iff C'_z\) occurs in some \(C'_xC'_{w_0}C'_y\iff z\le_{LR}w_0\) (positivity makes the chain condition defining \(\le_{LR}\) collapse to one product); (c) Lusztig (Cells in affine Weyl groups, 1985) and Shi (A two-sided cell in an affine Weyl group, I–II, 1987–88): \(\{a=\nu\}=c_0\) is the lowest two-sided cell with the stated description. Then \(s\notin c_0\) since \(\ell(s)=1<\nu\) when \(\nu\ge2\), i.e. rank \(\ge2\) for simple \(H\). This item is not used in Theorem 4.5.
  5. The brief's description. Step 3 shows \(s_0\in\mathcal M_I\) for \(\mathrm{SL}_2\); \(J=\{s_0\}\) has \(\ell(w_J)=1=\nu\). The computation Computation 4.8 confirms this and \(\mathcal M_I\cap\{\ell\le\nu+2\}=c_0\cap\{\ell\le\nu+2\}\) for \(\widetilde A_2\).

Builds on

  • Proposition 4.2 · Summand criterion for the group: generator constituents decide $\mathcal C_I^{\rm Iwa}$
  • Lemma 4.3 · Sign character on IC traces: $\varepsilon(\operatorname{tr}\mathrm{IC}_z)=[z\in\Omega]$
  • Lemma 4.4 · Trace expansion of $M_{x,y}$ with pure multiplicity Frobenii
  • Theorem 4.5 · No $\Delta_\omega$ in any generator $M_{x,y}$; so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$

Built on by

  • Computation 4.8 · KL expansions of $C_xC_{w_0}C_y$ in types $\widetilde A_1$, $\widetilde A_2$
  • Audit 4.9 · Transfer to $\mathbb C$ and checklist audit for the group
  • Proposition 4.10 · Pure sign-type functionals force sheaf failure: obligations for other rows
  • Summary 4.11 · Verdict for $X=H$: both levels fail, no divergence
  • Audit 8.7 · Audit of $\mathcal M_I$ for the group (A4-7)
  • Summary 10.2 · Verdict: the criterion and the bridge hold; the decided rows of Table 1 agree; GL₂/T fails at both levels
Original record, as the agent wrote it at 18:48 UTC

$\mathcal M_I$ for the group: top cells in, $\Omega$ out, exact in rank one

  1. Top cell. Products of cells satisfy \(IuI\cdot IvI\subseteq\bigcup_{\ell(z)\le\ell(u)+\ell(v)}IzI\) (BN-pair rule \(IsI\cdot IvI\subseteq IsvI\cup IvI\)). So \(m\) maps every stratum \(\mathrm{Fl}_u\widetilde\times\mathrm{Fl}_v\widetilde\times\mathrm{Fl}_{u'}\ne\mathrm{Fl}_x\widetilde\times\mathrm{Fl}_{w_0}\widetilde\times\mathrm{Fl}_y\) of the source into cells of dimension \(<\ell(xw_0y)\), and \(\mathrm{supp}\,M_{x,y}\subseteq\overline{\mathrm{Fl}_{xw_0y}}\). Over the open cell, \(m\) is the isomorphism \(\mathrm{Fl}_x\widetilde\times\mathrm{Fl}_{w_0}\widetilde\times\mathrm{Fl}_y\cong\mathrm{Fl}_{xw_0y}\) (lengths add; bezrukavnikov-yun-v2.pdf Lemma 3.2.2, (3.8)), so \(M_{x,y}|_{\mathrm{Fl}_{xw_0y}}=\underline k[\ell(xw_0y)]\). In \(\bigoplus\mathrm{IC}_z[-n]\otimes E_{z,n}\) only \(z\le xw_0y\) occur, and the restriction to the open cell is \(\bigoplus_n\underline k[\ell(xw_0y)-n]\otimes E_{xw_0y,n}\); hence \(E_{xw_0y,n}=k\) for \(n=0\) and \(0\) otherwise.
  2. A whole factor missing. Define \(\kappa(T_x)=(-1)^{\ell_{S'}(x)}q^{\ell_{S''}(x)}\), \(\kappa(T_\omega)=1\) (\(\Omega\) permutes each of \(S',S''\)). The relations of Lemma 4.3 step 1 hold: for \(s\in S''\), \(xs<x\): \(q\cdot q^{-1}\kappa(T_x)+(q-1)\kappa(T_x)=q\kappa(T_x)\). \(W=W'\times W''\), so \(\kappa(\mathbf 1_{K_H})=\big(\sum_{W'}(-1)^{\ell}\big)\big(\sum_{W''}q^{\ell}\big)=0\) (\(W'\ne1\): each affine component contains a finite simple reflection). If \(\ell_{S'}(z)>0\), \(z\) has a right descent \(s\in S'\) and the descent argument of Lemma 4.3 step 5 with \(\kappa(\operatorname{tr}\mathrm{IC}_s)=-(1-1)=0\) gives \(\kappa(\operatorname{tr}_m\mathrm{IC}_z)=0\). If \(\ell_{S'}(z)=0\), every \(y\le z\) has \(\kappa(T_y)=q^{\ell(y)}=|\mathrm{Fl}_y(\mathbb F_q)|\), so by Grothendieck–Lefschetz \(\kappa(\operatorname{tr}_m\mathrm{IC}_z)=\sum_j(-1)^j\operatorname{tr}(\mathrm{Fr}^m\mid IH^j_z)\), \(IH^j_z:=H^j(\overline{\mathrm{Fl}_z},\mathrm{IC}_z)\), pure of weight \(\ell(z)+j\) (proper pushforward) and nonzero for \(j=-\ell(z)\) (\(H^{-\ell(z)}=H^0(\mathrm{Fl}_z,\underline k)\)). As in Theorem 4.5: \(0=\kappa(\operatorname{tr}_mM_{x,y})=\sum_{\ell_{S'}(z)=0}\sum_{n,j}(-1)^{n+j}\operatorname{tr}(A_{z,n}^m)\operatorname{tr}(\mathrm{Fr}^m|IH^j_z)\); an eigenvalue product \(\alpha\beta\) has weight \((w+n-\ell(z))+(\ell(z)+j)=w+n+j\), so the sign \((-1)^{n+j}\) is fixed by the weight, nothing cancels, and \(E_{z,n}\otimes IH^*_z=0\); since \(IH^*_z\ne0\), \(E_{z,n}=0\).
  3. Rank one. \(W_{\rm aff}=\langle s_0,s_1\rangle\) infinite dihedral, \(w_0=s_1\). Every \(z\in W_{\rm aff}\setminus\{e,s_0\}\) has a reduced word containing \(s_1\), so step 1 applies; \(\Omega\)-translates follow from \(M_{x,y\omega}=M_{x,y}\ast\Delta_\omega\). For \(s_0\): \(M_{s_0s_1,s_1s_0}=\mathrm{IC}_{s_0s_1}\ast\mathrm{IC}_{s_1}\ast\mathrm{IC}_{s_1s_0}\). Geometrically \(\mathrm{IC}_{s_0s_1}\ast\mathrm{IC}_{s_1}\simeq\mathrm{IC}_{s_0s_1}[1]\oplus\mathrm{IC}_{s_0s_1}[-1]\) (Lemma 4.3 step 5, \(\pi_{s*}\overline{\mathbb Q}_\ell\) split), \(\mathrm{IC}_{s_0s_1}=\mathrm{IC}_{s_0}\ast\mathrm{IC}_{s_1}\), \(\mathrm{IC}_{s_1s_0}=\mathrm{IC}_{s_1}\ast\mathrm{IC}_{s_0}\) (Bott–Samelson maps bijective onto length-2 Schubert varieties). So \(M_{s_0s_1,s_1s_0}\) contains \((\mathrm{IC}_{s_0}\ast\mathrm{IC}_{s_1}\ast\mathrm{IC}_{s_0})[\pm2],[0]\), and \(\mathrm{IC}_{s_0}\ast\mathrm{IC}_{s_1}\ast\mathrm{IC}_{s_0}=\pi_*\underline k[3]\) for \(\pi:P_{s_0}\times^IP_{s_1}\times^IP_{s_0}/I\to\overline{\mathrm{Fl}_{s_0s_1s_0}}\). Fibres: a point over \(\mathrm{Fl}_{s_0s_1s_0},\mathrm{Fl}_{s_1}\), finite over the 2-dimensional cells, \(\mathbb P^1\) over \(\mathrm{Fl}_{s_0}\) and \(\mathrm{Fl}_e\) (if \(g_3\in P_{s_0}\) then \(g_2\in P_{s_0}\) and \(g_1^{-1}g_2\in P_{s_0}\cap P_{s_1}=I\)). So \(\pi\) is semismall with relevant strata \(\mathrm{Fl}_{s_0s_1s_0}\) and \(\mathrm{Fl}_{s_0}\) (\(2\cdot1=3-1\)), and \(\pi_*\underline k[3]\simeq\mathrm{IC}_{s_0s_1s_0}\oplus\mathrm{IC}_{s_0}\) (decataldo-migliorini-v2.pdf Thm 4.2.7; summand.tex Prop. I(e) with \(r=0\)). Hence \(s_0\in\mathcal M_I^\flat\), and with step 2 (\(\Omega\) excluded) \(\mathcal M_I^\flat=\widetilde W\setminus\Omega\). Proposition 4.2 steps 3–4 give the thick span.
  4. General case (cited, numbers not checked against local copies). (a) \(\operatorname{tr}_m\mathrm{IC}_z=(-1)^{\ell(z)}\sum_yP_{y,z}(q^m)T_y\) (Kazhdan–Lusztig, Schubert varieties and Poincaré duality, 1980, Thm 4.2 and the affine case §5); (b) positivity: \(M_{x,y}\) is a sum of \(\mathrm{IC}_z[n](n/2)\) (bezrukavnikov-yun-v2.pdf Prop. 3.2.5), so the KL structure constants are \(\sum_n\dim E_{z,n}v^{\pm n}\in\mathbb N[v^{\pm1}]\) and \(z\in\mathcal M_I\iff C'_z\) occurs in some \(C'_xC'_{w_0}C'_y\iff z\le_{LR}w_0\) (positivity makes the chain condition defining \(\le_{LR}\) collapse to one product); (c) Lusztig (Cells in affine Weyl groups, 1985) and Shi (A two-sided cell in an affine Weyl group, I–II, 1987–88): \(\{a=\nu\}=c_0\) is the lowest two-sided cell with the stated description. Then \(s\notin c_0\) since \(\ell(s)=1<\nu\) when \(\nu\ge2\), i.e. rank \(\ge2\) for simple \(H\). This item is not used in Theorem 4.5.
  5. The brief's description. Step 3 shows \(s_0\in\mathcal M_I\) for \(\mathrm{SL}_2\); \(J=\{s_0\}\) has \(\ell(w_J)=1=\nu\). The computation Computation 4.8 confirms this and \(\mathcal M_I\cap\{\ell\le\nu+2\}=c_0\cap\{\ell\le\nu+2\}\) for \(\widetilde A_2\).

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