Audit 8.7 · Audit of $\mathcal M_I$ for the group (A4-7)

Audit of $\mathcal M_I$ for the group (A4-7)

1. Item 1. Strata of the twisted product are \(\mathrm{Fl}_u\widetilde\times\mathrm{Fl}_v\widetilde\times\mathrm{Fl}_{u'}\), \(u\le x,v\le w_0,u'\le y\); by \(IsI\cdot IvI\subseteq IsvI\cup IvI\), a stratum maps into cells of length \(\le\ell(u)+\ell(v)+\ell(u')\). So only the open stratum meets \(m^{-1}(\mathrm{Fl}_{xw_0y})\), where \(m\) is an isomorphism (Bezrukavnikov–Yun Lemma 3.2.2, text checked), and \(M|_{\mathrm{Fl}_{xw_0y}}=\underline k[\ell(xw_0y)]\). Comparing with \(\bigoplus\mathrm{IC}_z[-n]\otimes E_{z,n}\), \(z\le xw_0y\), gives \(E_{xw_0y,n}=k\) for \(n=0\) and \(0\) otherwise.
2. Item 2. \(\kappa(T_x)=(-1)^{\ell_{S'}(x)}q^{\ell_{S''}(x)}\), \(\kappa(T_\omega)=1\). Needed: \(\Omega\) preserves \(S'\) and \(S''\). True: \(\widetilde W\) acts on affine roots \(\alpha+n\) with gradient action through \(W=\prod W_i\), so it preserves the affine roots of each simple factor, hence each component; the authors assert this without proof. The relations check as written; \(\kappa(\mathbf 1_K)=0\) since each component contains a finite simple reflection; the descent argument and Grothendieck–Lefschetz (\(\kappa(T_y)=q^{\ell(y)}=|\mathrm{Fl}_y(\mathbb F_q)|\) when \(\ell_{S'}(y)=0\), by the subword property) are correct. The eigenvalue of \(A_{z,n}\otimes\mathrm{Fr}|IH^j_z\) has weight \(w+n+j\), so the sign \((-1)^{n+j}\) is fixed by the weight, and \(IH^{-\ell(z)}_z=H^0(\mathrm{Fl}_z,k)\ne0\). For \(H\) simple, \(S'=S_{\rm aff}\) and item 2 is Audit 8.5.
3. Item 3, recomputed. In \(\widetilde A_1\) every \(z\in W_{\rm aff}\setminus\{e,s_0\}\) has an alternating reduced word through \(s_1=w_0\), so item 1 applies; \(\Omega\)-translates via \(M_{x,y\omega}=M_{x,y}\ast\Delta_\omega\). For \(s_0\): the Bott–Samelson map for \(s_0s_1s_0\) has fibres a point over \(\mathrm{Fl}_{s_0s_1s_0},\mathrm{Fl}_{s_0s_1},\mathrm{Fl}_{s_1s_0},\mathrm{Fl}_{s_1}\) and \(\mathbb P^1\) over \(\mathrm{Fl}_{s_0},\mathrm{Fl}_e\). Fibre sizes are the coefficients of \((1+T_0)(1+T_1)(1+T_0)=(1+q)+(1+q)T_0+T_1+T_{01}+T_{10}+T_{010}\). It is semismall with relevant strata \(\mathrm{Fl}_{s_0s_1s_0}\), \(\mathrm{Fl}_{s_0}\), so \(\pi_*k[3]=\mathrm{IC}_{s_0s_1s_0}\oplus\mathrm{IC}_{s_0}\) (de Cataldo–Migliorini Thm 4.2.7). Kazhdan–Lusztig computation: \(\underline H_0\underline H_1\underline H_0=\underline H_{010}+\underline H_0\) and \(\underline H_{01}\underline H_1\underline H_{10}=(v^{-2}+2+v^2)(\underline H_{010}+\underline H_0)\), confirming the authors' \([\pm2],[0]^2\) multiplicities.
4. Item 4, what is cited. (a) the KL trace dictionary and (b) positivity of the structure constants (Bezrukavnikov–Yun Prop. 3.2.5) make “\(z\in\mathcal M_I\iff\underline H_z\) occurs in some \(\underline H_x\underline H_{w_0}\underline H_y\iff z\le_{LR}w_0\)” correct: with non-negative coefficients the chain defining \(\le_{LR}\) collapses to one product. (c) Lusztig and Shi, “the lowest two-sided cell \(c_0=\{a=\nu\}=\{x\cdot w_J\cdot y\}\)”, is not in the local sources, and its theorem numbers are unchecked; the authors label it cited. The identification \(\{z\le_{LR}w_0\}=c_0\) (since \(w_0\in c_0\), the lowest cell) is correct given (c).
5. Computation for item 4 (\(\widetilde A_2\)). All \(\underline H_x\underline H_{w_0}\underline H_y\) with \(\ell(x),\ell(y)\le3\) were expanded in the KL basis (all coefficients came out in \(\mathbb N[v^{\pm1}]\)). The elements \(z\) occurring with \(\ell(z)\le6\) are exactly Shi's \(\{x\cdot w_J\cdot y\text{ reduced}:|J|=2\}\) of that length: \(0,0,0,3,6,9,12\) elements in lengths \(0,\dots,6\). No \(s_i\) and no element of length \(\le2\) occurs. This is evidence for item 4, not a proof: longer \(x,y\) could a priori add elements.