§3 · Agent 3 · Symplectic family: fibres · 2026-10-17T02:48:26Z

Lemma 3.5 · Maximal-defect lemma: at a stratum of maximal defect the top stalk forces an IC summand

proved audit: correct

Second audit: correct. Lemma and proof verified step by step: perverse amplitude from fibre dimensions, Verdier duality, decomposition with constant coefficients, and the strict support condition at a stratum of maximal defect ('ties allowed' is right). The closing remark computes c_nu from Chen-Yi Prop. 64, which presumes LX connected; for GL_{2m}/Sp_{2m} the codimensions come from the dimension count of Lemma 3.1 (written out in correction Correction 10.2). The lemma itself is unaffected.

A general detection criterion for summands of a proper pushforward from a smooth space. At a stratum where twice the fibre dimension minus the codimension is largest, the top cohomology of the fibre can only come from that stratum's intersection complex with the extreme shift. This is the tool that produces retracts from non-semismall convolutions.

Lemma 3.5.

Let \(m:Y\to Z\) be a proper \(G\)-equivariant morphism of varieties of finite type, with \(Y\) smooth and irreducible of dimension \(D\) and \(Z=\bigsqcup_\nu\cO_\nu\) a union of finitely many \(G\)-orbits with connected stabilizers. Put $$f_\nu:=\dim m^{-1}(x_\nu),\qquad c_\nu:=D-\dim\cO_\nu,\qquad \operatorname{def}(\nu):=2f_\nu-c_\nu,\qquad r:=\max_\nu\operatorname{def}(\nu).$$ If \(\operatorname{def}(\mu)=r\), then \(\IC_\mu[-r]\) is a direct summand of \(K:=m_!k_Y[D]\). Other strata may also attain \(r\).

In the loop setting the lemma is applied to the finite-type models of Proposition 3.3. It applies equally to a chain of minuscule convolutions \(Y=\Gr^{\omega_{k_1}}\mathbin{\tilde\times}\cdots\mathbin{\tilde\times}\Gr^{\omega_{k_r}}\mathbin{\tilde\times}L^+X\), which is smooth and whose \(m_!k_Y[D]=(V_{\omega_{k_1}}\ast\cdots\ast V_{\omega_{k_r}})\ast\IC_0\) is a sum of Satake objects, hence lies in \(\Csat\).

  1. Perverse amplitude. \(\mathcal H^i(K)_x=H^{i+D}(m^{-1}x)\), which vanishes unless \(i+D\le2f_\nu\) for \(x\in\cO_\nu\). So \(\supp\mathcal H^i(K)\subseteq\bigcup\{\cO_\nu:2f_\nu\ge i+D\}\), and for each such \(\nu\), \(\dim\cO_\nu=D-c_\nu\le r-i\) because \(r\ge2f_\nu-c_\nu\ge i+D-c_\nu\). Hence \(K\in{}^pD^{\le r}\). Since \(Y\) is smooth and \(m\) is proper, \(\mathbb DK\simeq K\), which gives \(K\in{}^pD^{\ge-r}\).
  1. Decomposition. By the decomposition theorem for \(Y/G\to Z/G\) (Sun, Thm. 1.2; BBD 5.4.5–5.4.6) and connected stabilizers, \(K\simeq\bigoplus_{\nu,j}\IC_\nu[-j]^{\,n_{\nu,j}}\), and step 1 gives \(n_{\nu,j}=0\) for \(|j|>r\).
  1. Stalks at \(x_\mu\). For \(\nu\ne\mu\) with \(x_\mu\in\overline{\cO_\nu}\), the support condition puts \(\IC_\nu[-j]\) at \(x_\mu\) in degrees \(\le-\dim\cO_\mu-1+j\le-D+c_\mu-1+r=-D+2f_\mu-1\). The summand \(\IC_\mu[-j]\) sits in the single degree \(-D+c_\mu+j\le-D+2f_\mu\), with equality exactly when \(j=r\). But \(\mathcal H^{-D+2f_\mu}(K)_{x_\mu}=H^{2f_\mu}(m^{-1}x_\mu)\ne0\), the top cohomology of a proper variety of dimension \(f_\mu\). Hence \(n_{\mu,r}\ge1\). \(\square\)
  1. Application. In the loop setting \(c_\nu=\dim Y-\delta(\cO_\nu)\) for Chen–Yi's dimension theory (Prop. 41), computed from the codimension formula (the node cites Chen–Yi, Prop. 64) and the generic fibre dimension. A chain of minuscule convolutions is an iterated Grassmannian bundle over \(L^+X\), hence smooth, and a convolution of Satake sheaves is a sum of Satake sheaves.

Builds on

Built on by

  • Theorem 3.6 · \(\GL_{2m}/\Sp_{2m}\): the labels \((1^k0^{m-k})\), \((2^{k-m}1^{2m-k})\) and their shifts lie in \(\mathcal M\); \(\GL_4/\Sp_4\) is complete
  • Audit 7.13 · Check of the summand criterion, the maximal-defect lemma, the minuscule fibre lemma and the \(\SL\)/\(\GL\) reduction
Original record, as the agent wrote it at 02:48 UTC

Maximal-defect lemma: the top stalk degree at a maximal-defect stratum forces an IC summand

Lemma. Let \(m:Y\to Z\) be a proper \(G\)-equivariant map of finite type, \(Y\) smooth irreducible of dimension \(D\), \(Z=\bigsqcup_\nu\mathcal O_\nu\) finitely many \(G\)-orbits with connected stabilizers, \(f_\nu:=\dim m^{-1}(x_\nu)\), \(c_\nu:=D-\dim\mathcal O_\nu\), \(\mathrm{def}(\nu):=2f_\nu-c_\nu\), \(r:=\max_\nu\mathrm{def}(\nu)\). If \(\mathrm{def}(\mu)=r\), then \(\mathrm{IC}_\mu[-r]\) is a direct summand of \(K:=m_!\underline k_Y[D]\) (ties with other strata allowed).

Proof. 1. \(K\in{}^pD^{[-r,r]}\): \(\mathcal H^i(K)_x=H^{i+D}(m^{-1}x)\) is supported on \(\bigcup\{\mathcal O_\nu:2f_\nu\ge i+D\}\), of dimension \(\max(D-c_\nu)\le r-i\) since \(r\ge 2f_\nu-c_\nu\ge i+D-c_\nu\); the lower bound by \(\mathbb DK\simeq K\).

  1. Decomposition theorem (Sun Thm. 1.2 on \(Y/G\to Z/G\); BBD 5.4.5–6): \(K\simeq\bigoplus_{\nu,j}\mathrm{IC}_\nu[-j]^{n_{\nu,j}}\), \(n_{\nu,j}=0\) for \(|j|>r\) by step 1.
  2. Stalks at \(x_\mu\): \(\mathrm{IC}_\nu[-j]\) with \(\nu\ne\mu\) lives in degrees \(\le-\dim\mathcal O_\mu-1+j\le-D+c_\mu-1+r=-D+2f_\mu-1\); \(\mathrm{IC}_\mu[-j]\) in degree \(-D+c_\mu+j\le-D+2f_\mu\), with equality iff \(j=r\). But \(\mathcal H^{-D+2f_\mu}(K)_{x_\mu}=H^{2f_\mu}(m^{-1}x_\mu)\ne0\). Hence \(n_{\mu,r}\ge1\). \(\square\)

In the loop setting the lemma is applied to the finite-type model of A3-3 (placid presentation of a closed finite union of orbits); \(c_\nu\) is then \(\dim Y-\delta(\mathcal O_\nu)\) with Chen–Yi's dimension theory (Prop. 41), computed from Prop. 64 and the generic fibre dimension. The same statement holds for a chain of minuscule convolutions \(Y=\mathrm{Gr}^{\omega_{k_1}}\widetilde\times\cdots\widetilde\times\mathrm{Gr}^{\omega_{k_r}}\widetilde\times L^+X\) (smooth), whose \(m_!\underline k[D]=(V_{\omega_{k_1}}\ast\cdots\ast V_{\omega_{k_r}})\ast\mathrm{IC}_0\in\mathcal C^{\mathrm{Sat}}\) is a sum of Satake objects.

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