§8 · Agent 8 · E₆/F₄: sheaf level · 2026-10-17T03:36:00Z

Lemma 8.7 · Detection lemma: a stalk class in degree \(\ge-\delta(\mu)\) forces \(\IC_\mu\) to split off

proved audit: correct

Second audit: correct.

If a decomposed complex is supported below \(\mu'\), the interval \([\mu,\mu']\) has two elements, and \(\IC_{\mu'}\) occurs only unshifted, then its stalk at \(x_\mu\) in degrees \(\ge-\delta(\mu)\) is exactly the multiplicity space of \(\IC_\mu\). So one nonzero stalk class in that range makes \(\IC_\mu\) a retract.

Lemma 8.7.

Let \(M\simeq\bigoplus_{\nu\in\Lambda_S^+}\bigoplus_{n\in\Z}\IC_\nu[-n]\otimes E_{\nu,n}\) be a finite direct sum with constant coefficients, normalized by \(\IC_\nu|_{\cO_\nu}=\underline k[\delta(\nu)]\). Let \(\mu<\mu'\) in \(\Lambda_S^+\) satisfy

  1. \(E_{\nu,n}=0\) unless \(\nu\le\mu'\);
  2. \(\{\nu:\mu\le\nu\le\mu'\}=\{\mu,\mu'\}\);
  3. \(E_{\mu',n}=0\) for \(n\ne0\).

Then for every \(j\ge-\delta(\mu)\) $$H^j(M_{x_\mu})\ \simeq\ E_{\mu,\,j+\delta(\mu)} .$$ In particular, if \(H^j(M_{x_\mu})\ne0\) for some \(j\ge-\delta(\mu)\), then \(\IC_\mu\) is a direct summand of \(M\): precisely, \(\IC_\mu[-n]\) is a summand for \(n=j+\delta(\mu)\), so \(\IC_\mu\) is a retract of \(M[n]\) and lies in \(\Thick\{M\}\).

  1. The stalk of \(\IC_\nu\) at \(x_\mu\) vanishes unless \(x_\mu\in\overline{\cO_\nu}\), i.e. \(\mu\le\nu\) (Chen–Yi, Thm. 18). With hypotheses 1 and 2, only \(\nu\in\{\mu,\mu'\}\) contribute.
  2. For \(\nu=\mu\): \((\IC_\mu[-n]\otimes E_{\mu,n})_{x_\mu}\) is \(E_{\mu,n}\) placed in degree \(-\delta(\mu)+n\).
  3. For \(\nu=\mu'\): by hypothesis 3 only \(n=0\) occurs, and the support condition of the intermediate extension along the stratum \(\cO_\mu\subset\overline{\cO_{\mu'}}\), of codimension \(\delta(\mu')-\delta(\mu)\), gives \(H^j(\IC_{\mu'})_{x_\mu}=0\) for \(j\ge-\delta(\mu)\). In Chen–Yi's notation this is Cor. 14(ii): \(\deg P_{\mu;\mu'}\le\tfrac12(\delta(\mu')-\delta(\mu)-1)\), where \(P_{\mu;\mu'}=\sum_i[H^{i-\delta(\mu')}(\IC_{\mu'})|_{\cO_\mu}]q^{i/2}\); equivalently BBD 2.1.9 in a finite-type model.
  4. Taking \(H^j\) of the direct sum for \(j\ge-\delta(\mu)\) leaves only the term of step 2 with \(n=j+\delta(\mu)\). If it is nonzero, \(\IC_\mu[-n]\) is a summand of \(M\), so \(\IC_\mu\) is a retract of \(M[n]\).

Why hypothesis 3 matters. Without it, a summand \(\IC_{\mu'}[-n]\) with \(n>0\) could place stalk classes at \(x_\mu\) in degrees up to \(-\delta(\mu)-1+n\). Weights cannot replace it: every summand \(\IC_\nu[-n]\otimes E_{\nu,n}\) of a pure complex of weight \(0\) is pointwise of weight at most its degree, so weights alone do not separate different \(n\).

Sanity check against \(\mathrm{GL}_4/\mathrm{Sp}_4\) (Theorem 4.9). There the fibre over the covering orbit is a point, the special fibre is the Lagrangian Grassmannian \(\mathrm{LG}(2,4)\cong Q^3\), and \(\delta(\mu')-\delta(\mu)=4\). The lemma gives \(E_{\mu,n}=H^{n+4}(Q^3)\) for \(n\ge0\), so \(E_{\mu,0},E_{\mu,2}\ne0\), and Verdier symmetry adds \(E_{\mu,-2}\ne0\). Thus \(E_\mu\) is three-dimensional, in degrees \(-2,0,2\), and the stalk of \(\IC_{\mu'}\) at \(x_\mu\) is one class in degree \(-4\): the slice is rationally smooth.

Builds on

  • Lemma 4.5 · \(\Csat=\cC_K\) if and only if every \(\IC_\mu\) is a direct summand of a Satake object

Built on by

  • Theorem 8.8 · \(\IC_{3k\omega_1^\vee}\) is a direct summand of the Satake object \(V_{\lambda_k}\ast\IC_0\)
  • Audit 7.30 · Audit of the detection lemma and the summand theorem: both hold, independently of the normalization
Original record, as the agent wrote it at 03:36 UTC

Detection lemma: a stalk class in high enough degree forces the IC of the orbit to be a summand

Lemma. Let \(M\simeq\bigoplus_{\nu\in\Lambda_S^+}\bigoplus_{n\in\mathbb Z}\mathrm{IC}_\nu[-n]\otimes E_{\nu,n}\) be a finite direct sum with constant coefficients, \(\mathrm{IC}_\nu|_{\mathcal O_\nu}=\underline k[\delta(\nu)]\). Let \(\mu<\mu'\) in \(\Lambda_S^+\) with (i) \(E_{\nu,n}=0\) unless \(\nu\le\mu'\); (ii) \(\{\nu:\mu\le\nu\le\mu'\}=\{\mu,\mu'\}\); (iii) \(E_{\mu',n}=0\) for \(n\ne0\). Then for every \(j\ge-\delta(\mu)\): $$H^j(M_{x_\mu})\ \simeq\ E_{\mu,\,j+\delta(\mu)} .$$ In particular, if \(H^j(M_{x_\mu})\ne0\) for some \(j\ge-\delta(\mu)\), then \(\mathrm{IC}_\mu\) is a direct summand of \(M\) (a retract, hence in \(\operatorname{Thick}\{M\}\)).

Proof. 1. The stalk of \(\mathrm{IC}_\nu\) at \(x_\mu\) vanishes unless \(x_\mu\in\overline{\mathcal O_\nu}\), i.e. \(\mu\le\nu\) (Chen–Yi Thm 18, H1(a)). With (i),(ii) only \(\nu\in\{\mu,\mu'\}\) contribute.

  1. \(\nu=\mu\): \((\mathrm{IC}_\mu[-n]\otimes E_{\mu,n})_{x_\mu}=E_{\mu,n}\) placed in degree \(-\delta(\mu)+n\).
  2. \(\nu=\mu'\): by (iii) only \(n=0\), and the support condition for the intermediate extension along the stratum \(\mathcal O_\mu\subset\overline{\mathcal O_{\mu'}}\) of codimension \(\delta(\mu')-\delta(\mu)\) gives \(H^{j}(\mathrm{IC}_{\mu'})_{x_\mu}=0\) for \(j\ge-\delta(\mu)\) (Chen–Yi Cor. 14(ii): \(\deg P_{\mu;\mu'}\le\tfrac12(\delta(\mu')-\delta(\mu)-1)\) with \(P_{\mu;\mu'}=\sum_i[H^{i-\delta(\mu')}(\mathrm{IC}_{\mu'})|_{\mathcal O_\mu}]q^{i/2}\); equivalently BBD 2.1.9 in the finite-type model).
  3. Taking \(H^j\) of the direct sum for \(j\ge-\delta(\mu)\) leaves only the term of step 2 with \(n=j+\delta(\mu)\). If it is nonzero, \(\mathrm{IC}_\mu[-n]\) is a summand of \(M\), so \(\mathrm{IC}_\mu\) is a retract of \(M[n]\). \(\square\)

Why (iii) matters: without it a summand \(\mathrm{IC}_{\mu'}[-n]\), \(n>0\), could place stalk classes at \(x_\mu\) in degrees up to \(-\delta(\mu)-1+n\); the brief's step 4 as literally stated needs this hypothesis, which A8-5 supplies. Degree-versus-weight bookkeeping cannot replace it: all summands \(\mathrm{IC}_\nu[-n]\otimes E_{\nu,n}\) of a pure weight-0 complex are pointwise of weight ≤ degree, so weights alone do not separate different \(n\). Sanity check against (F2): \(\mathrm{GL}_4/\mathrm{Sp}_4\), \(F'\) a point, \(F=\mathrm{LG}(2,4)\), \(\delta(\mu')-\delta(\mu)=4\): the lemma gives \(E_{\mu,n}=H^{n+4}(Q^3)\) for \(n\ge0\), i.e. \(E_{\mu,0},E_{\mu,2}\ne0\), and by Verdier symmetry \(E_{\mu,-2}\ne0\): \(E_\mu\) is 3-dimensional in degrees \(-2,0,2\), and the IC stalk of the bigger closure at \(x_\mu\) is one class in degree \(-4\) — the slice is rationally smooth.

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