§8 · Agent 8 · E₆/F₄: sheaf level · 2026-10-17T03:15:00Z

Lemma 8.4 · Both convolution fibres lie in the open cell \(\Gr^{\lambda_k}\)

proved audit: correct ★ key step

Second audit: correct.

Every point of the convolution fibres over \(x_{\mu_k}\) and \(x_{\mu_k'}\) lies in the open cell of \(\Gr^{\le\lambda_k}\). So the Satake sheaf is constant there, and the two stalks of \(M_{\lambda_k}\) are the shifted cohomology of projective varieties; no intersection cohomology of the singular Schubert variety is needed.

Lemma 8.4.

Let \(m:\Gr^{\le\lambda_k}\mathbin{\widetilde\times}L^+X\to LX\), \([g,y]\mapsto gy\), be the convolution map, and for \(x\in LX\) let \(F_x:=m^{-1}(x)\). The first projection identifies \(F_x\) with the closed subset \(\{gL^+G\in\Gr^{\le\lambda_k}:g^{-1}x\in L^+X\}\) of the projective variety \(\Gr^{\le\lambda_k}\) (Proposition III, step 3). Put \(F_k:=F_{x_{\mu_k}}\), \(F_k':=F_{x_{\mu_k'}}\) and \(D_k:=\dim\Gr^{\lambda_k}=\delta(\mu_k')\) (Lemma 8.3).

Then \(F_k\) and \(F_k'\) are contained in the open cell \(\Gr^{\lambda_k}\). Consequently both are projective varieties, \(V_{\lambda_k}\mathbin{\widetilde\boxtimes}\IC_0\) restricts to \(\underline k[D_k]\) on each, and by proper base change $$(M_{\lambda_k})_{x_{\mu_k}}\simeq R\Gamma(F_k,\underline k)[D_k],\qquad(M_{\lambda_k})_{x_{\mu_k'}}\simeq R\Gamma(F_k',\underline k)[D_k].$$

  1. Reduction to labels. Let \(gL^+G\in\Gr^{\lambda''}\) with \(\lambda''\in\Lambda_T^+\), \(\lambda''\le\lambda_k\), and suppose \(g^{-1}x\in L^+X\) for some \(x\in\{x_{\mu_k},x_{\mu_k'}\}\). Then \(x\in L^+G\,t^{\lambda''}L^+X\). The support bound of Proposition III, step 2 (proved there from the convolution bound on \(\Gr_G\) and the projection \(p=\tfrac12(1-\theta)\)) says that \(\cO_\nu\cap L^+Gt^{\lambda''}L^+X\ne\varnothing\) implies \(\nu\le\varphi(\lambda'')\); the same step gives \(\varphi(\lambda'')\le\varphi(\lambda_k)=\mu_k'\). Hence \(\mu_k\le\varphi(\lambda'')\le\mu_k'\), and \(\varphi(\lambda'')\in\{\mu_k,\mu_k'\}\) by Lemma 8.2 (3).
  2. The lower label is excluded. \(\varphi(\lambda'')=\mu_k\) is impossible, because \(\mu_k=3k\omega_1^\vee\) has no dominant lift (family page, section 5).
  3. The upper label forces \(\lambda''=\lambda_k\). Suppose \(\varphi(\lambda'')=\varphi(\lambda_k)\) with \(\lambda''<\lambda_k\). Then \(\kappa:=\lambda_k-\lambda''\in Q^+\setminus0\) and \(\varphi(\kappa)=0\). The kernel of \(\varphi=1-\theta\) is spanned by the black coroots, so \(\kappa=\sum_{j=2}^5c_j\alpha_j^\vee\) with \(c_j\ge0\). Since \(\langle\alpha_j,\lambda_k\rangle=0\) for \(j=2,\dots,5\), dominance of \(\lambda''\) forces \(\langle\alpha_j,\kappa\rangle\le0\) for every black \(j\). But \(0<(\kappa,\kappa)=\sum_jc_j(\alpha_j^\vee,\kappa)\) forces \(\langle\alpha_j,\kappa\rangle>0\) for some black \(j\) with \(c_j>0\). Contradiction.
  4. Stalks. So \(\lambda''=\lambda_k\), and both fibres lie in \(\Gr^{\lambda_k}\). This cell is smooth and \(V_{\lambda_k}|_{\Gr^{\lambda_k}}=\underline k[D_k]\) (perverse normalization of the IC sheaf). The twisted product is locally \(V_{\lambda_k}\boxtimes\IC_0\) with \(\IC_0=\underline k_{L^+X}\), so it restricts to \(\underline k[D_k]\) on each fibre. Each fibre is closed in \(\Gr^{\le\lambda_k}\) and contained in \(\Gr^{\lambda_k}\), hence projective. Proper base change for \(m\), which is proper on \(\Gr^{\le\lambda_k}\mathbin{\widetilde\times}Z\) for every closed finite union of orbits \(Z\) (Proposition III, step 3), gives the stalk formula.

A finite search agrees: in the box \(c_j\le6\) there is no dominant \(\lambda''\in\lambda_k-\big(\sum_{j=2}^5\Z_{\ge0}\alpha_j^\vee\setminus0\big)\) for \(k=1,3,5,7\).

Verified by computation: e6_fibre.py

Builds on

  • Lemma 8.2 · Orbit labels, the dimension function, and the two-element interval \([\mu_k,\mu_k']\)
  • Lemma 8.3 · The Hecke label \(\lambda_k\): a dominant lift of \(\mu_k'\) whose cell has dimension \(\delta(\mu_k')\)

Built on by

  • Lemma 8.5 · The fibre over the covering orbit is finite, so \(\IC_{\mu_k'}\) occurs only unshifted
  • Computation 8.6 · Torus-fixed points of \(F_k\) and a seven-dimensional orbit of \(Z_{G^\theta}(\mu_k)^\circ\)
  • Theorem 8.8 · \(\IC_{3k\omega_1^\vee}\) is a direct summand of the Satake object \(V_{\lambda_k}\ast\IC_0\)
  • Audit 7.27 · Audit of the open-cell lemma: support bound, monotonicity and the black-cone step hold
Original record, as the agent wrote it at 03:15 UTC

Both convolution fibres lie in the open Schubert cell

Notation: \(F_x:=m^{-1}(x)\subseteq\mathrm{Gr}^{\le\lambda_k}\widetilde\times L^+X\); the first projection identifies it with the closed subset \(\{gL^+G\in\mathrm{Gr}^{\le\lambda_k}:g^{-1}x\in L^+X\}\) of the projective \(\mathrm{Gr}^{\le\lambda_k}\) (criterion.html Step 3: \([g,y]\mapsto(gL^+G,gy)\) is a closed embedding; A4-3 step 1). Put \(F_k:=F_{x_{\mu_k}}\), \(F_k':=F_{x_{\mu_k'}}\).

Lemma. \(F_k\) and \(F_k'\) are contained in the open cell \(\mathrm{Gr}^{\lambda_k}\). Consequently they are projective varieties, and \(V_{\lambda_k}\widetilde\boxtimes\mathrm{IC}_0\) restricted to either fibre is the constant sheaf \(\underline k[D_k]\), \(D_k:=\dim\mathrm{Gr}^{\lambda_k}=\delta(\mu_k')\); by proper base change $$(M_{\lambda_k})_{x_{\mu_k}}\simeq R\Gamma(F_k,\underline k)[D_k],\qquad(M_{\lambda_k})_{x_{\mu_k'}}\simeq R\Gamma(F_k',\underline k)[D_k].$$

Proof. 1. Let \(gL^+G\in\mathrm{Gr}^{\lambda''}\) with \(\lambda''\in\Lambda_T^+\), \(\lambda''\le\lambda_k\), and suppose \(g^{-1}x\in L^+X\) for \(x\in\{x_{\mu_k},x_{\mu_k'}\}\). Then \(x\in L^+G\,t^{\lambda''}L^+X\), and the support bound of criterion.html Step 2 (bullets Bound/Projection, proved there from the convolution bound on \(\mathrm{Gr}_G\)) gives \(\mathcal O_\nu\cap L^+Gt^{\lambda''}L^+X\ne\varnothing\Rightarrow\nu\le\varphi(\lambda'')\); also \(\varphi(\lambda'')\le\varphi(\lambda_k)=\mu_k'\) (same step). Hence \(\mu_k\le\varphi(\lambda'')\le\mu_k'\), so by A8-2(iii) \(\varphi(\lambda'')\in\{\mu_k,\mu_k'\}\).

  1. \(\varphi(\lambda'')=\mu_k\) is impossible: \(\mu_k=3k\omega_1^\vee\) has no dominant lift (page §5 Lemma; F3).
  2. \(\varphi(\lambda'')=\mu_k'=\varphi(\lambda_k)\) with \(\lambda''<\lambda_k\) is impossible: \(\kappa:=\lambda_k-\lambda''\in Q^+\setminus0\) and \(\varphi(\kappa)=0\), so \(\kappa\in\ker\varphi\cap Q^+=\sum_{j=2}^5\mathbb Z_{\ge0}\alpha_j^\vee\) (the kernel of \(\varphi=1-\theta\) is the span of the black coroots, page §2). Since \(\langle\alpha_j,\lambda_k\rangle=0\) for \(j=2,\dots,5\), dominance of \(\lambda''\) requires \(\langle\alpha_j,\kappa\rangle\le0\) for all black \(j\); but \(0<(\kappa,\kappa)=\sum_jc_j(\alpha_j^\vee,\kappa)\) with \(c_j\ge0\) forces some \(\langle\alpha_j,\kappa\rangle>0\). Contradiction.
  3. So \(\lambda''=\lambda_k\): both fibres lie in \(\mathrm{Gr}^{\lambda_k}\), which is smooth with \(V_{\lambda_k}|_{\mathrm{Gr}^{\lambda_k}}=\underline k[D_k]\) (perverse normalization of the IC sheaf). The twisted product is locally \(V_{\lambda_k}\boxtimes\mathrm{IC}_0\) and \(\mathrm{IC}_0=\underline k_{L^+X}\), so its restriction to the fibre is \(\underline k[D_k]\). Closedness: \(F_x\) is closed in \(\mathrm{Gr}^{\le\lambda_k}\) and contained in \(\mathrm{Gr}^{\lambda_k}\). Proper base change for the proper map \(m\) on \(\mathrm{Gr}^{\le\lambda_k}\widetilde\times Z\) (criterion Step 3) gives the stalk formula. \(\square\)

Script check: in the box \(c_j\le6\) there is no dominant \(\lambda''\in\lambda_k-\sum_{j=2}^5\mathbb Z_{\ge0}\alpha_j^\vee\setminus0\) for \(k=1,3,5,7\), as the proof predicts.

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