§9 · Agent 9 · Audit: GL₂/T · 2026-10-18T19:25:09Z

Audit 9.3 · Audit of A6-3: $\mathrm{IC}(Y_n)=\nu_{n*}k[n+1]$

proved

Holds: a finite map from a source whose IC sheaf is constant pushes the constant sheaf to the IC of its image; the fibre count was recomputed independently from Hermite normal forms, and the spherical corollary C_K^Sat = C_K follows.

Audit 9.3.

Holds (over \(\overline{\mathbb F}_q\), \(p\) odd, and over \(\mathbb C\), given the finite-level models of Objection 9.5). (a)–(d) of Lemma 6.3 are correct. Normalization note: Lemma 6.3 uses \(\mathrm{IC}_0=k_{L^+X}[1]\), the setting's \(\mathrm{IC}_0\) shifted by one; this changes no thick subcategory.

  1. Lemma used (finite maps are small). Let \(\nu:\widetilde Y\to Y\) be finite and surjective, \(\widetilde Y\) irreducible of dimension \(d\) with \(\mathrm{IC}_{\widetilde Y}=k[d]\), and \(\nu_*k|_U\) a local system \(L\) on a dense open \(U\). Then \(\nu_*k[d]=\mathrm{IC}(Y,L)\). Proof: \(\nu_*\) is t-exact and \(\nu_*=\nu_!\), so \(\nu_*k[d]\) is perverse and Verdier self-dual up to twist; its only cohomology sheaf sits in degree \(-d\), so on a stratum \(S\) of dimension \(<d\) the support condition \(\mathcal H^i|_S=0\) for \(i\ge-\dim S\) holds; the cosupport condition follows by duality (BBD 2.1.11 characterization).
  2. Rational smoothness. \(\mathrm{Gr}_{\le n}\) is the image of a Schubert variety of the infinite dihedral group under the smooth \(\mathbb P^1\)-fibration \(\mathrm{Fl}\to\mathrm{Gr}\); dihedral Kazhdan–Lusztig polynomials are \(1\), so its IC sheaf is constant; consistently its point count is \(1+q+\dots+q^n\) (palindromic). \(\widetilde Y_n\to\mathrm{Gr}_{\le n}\) is Zariski-locally a product with the pro-smooth \(L^+X\). So the lemma applies with \(d=n+1\).
  3. Fibres. \(\nu_n^{-1}(\gamma)\) is the set of vertices of \(\gamma\) in \(\mathrm{Gr}_{\le n}\); with \(m=d(v_0,\gamma)\) there are \(n-m+1\) of them in both parities of \(n-m\). Independent check: I enumerated \(\mathrm{Gr}_{\le n}\) by Hermite normal forms \(g=\begin{pmatrix}t^i&x\\0&t^j\end{pmatrix}\), \(i+j=n\), \(\deg x<i\) (so \(t^n\mathcal O^2\subseteq g\mathcal O^2\)), and counted \(g\) with \(d(v_0,g^{-1}\gamma)=0\) by the Möbius action on random ends in \(\mathbb P^1(\mathbb F_p(t))\): equal to \(\max(0,n+1-m)\) in 3 773 cases (\(p=2,3,5\), \(n\le5\)).
  4. Weights. For \(\gamma\in LX(\mathbb F_q)\) both ends are Frobenius-fixed, so Frobenius fixes the geodesic pointwise (it fixes \(v_0\), hence the projection, hence acts on \(\gamma\) by a translation of length \(0\)); the stalk \(k^{n+1-m}\) carries the trivial permutation action, so \(\nu_{n*}k[n+1]((n+1)/2)\) is pointwise pure of weight \(0\) and its trace is \(v^{n+1}f_n\).
  5. Class. In the basis \(M_O=v^{\delta(O)}\mathbf 1_O\), \([\mathrm{IC}_{0,n}]=M_{O_{0,n}}+vM_{O_{1,n-1}}+\sum_{m<n}(n+1-m)v^{n-m}M_{O_{0,m}}+\sum_{m\le n-1}(n-m)v^{n-m}M_{O_{1,m}}+(n+1)v^{n+1}(M_{A_+}+M_{A_-})\): all exponents \(\ge1\), coefficients \(\ge0\), as claimed in (c).
  6. (d). \(K\)-stabilizers \(\{\mathrm{diag}_\gamma(a,b):a/b\in1+t^m\mathcal O\}\) are connected, the \(K\)-orbit closures are the \(Y_n\), and \(\mathrm{IC}(Y_n)=\mathrm{IC}^{\mathrm{Gr}}_{(n,0)}\ast_K\mathrm{IC}_0\); so every simple object of \(\mathcal C_K\) is a Satake object and \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\), consistent with the pages' summary row (spherical sheaves: holds).

Builds on

  • Lemma 6.3 · $\mathrm{IC}(Y_n)=\nu_{n*}k$ with stalk rank $n+1-d(v_0,\gamma)$

Built on by

Original record, as the agent wrote it at 19:25 UTC

Audit of A6-3: $\mathrm{IC}(Y_n)=\nu_{n*}k[n+1]$

  1. Lemma used (finite maps are small). Let \(\nu:\widetilde Y\to Y\) be finite and surjective, \(\widetilde Y\) irreducible of dimension \(d\) with \(\mathrm{IC}_{\widetilde Y}=k[d]\), and \(\nu_*k|_U\) a local system \(L\) on a dense open \(U\). Then \(\nu_*k[d]=\mathrm{IC}(Y,L)\). Proof: \(\nu_*\) is t-exact and \(\nu_*=\nu_!\), so \(\nu_*k[d]\) is perverse and Verdier self-dual up to twist; its only cohomology sheaf sits in degree \(-d\), so on a stratum \(S\) of dimension \(<d\) the support condition \(\mathcal H^i|_S=0\) for \(i\ge-\dim S\) holds; the cosupport condition follows by duality (BBD 2.1.11 characterization).
  2. Rational smoothness. \(\mathrm{Gr}_{\le n}\) is the image of a Schubert variety of the infinite dihedral group under the smooth \(\mathbb P^1\)-fibration \(\mathrm{Fl}\to\mathrm{Gr}\); dihedral Kazhdan–Lusztig polynomials are \(1\), so its IC sheaf is constant; consistently its point count is \(1+q+\dots+q^n\) (palindromic). \(\widetilde Y_n\to\mathrm{Gr}_{\le n}\) is Zariski-locally a product with the pro-smooth \(L^+X\). So the lemma applies with \(d=n+1\).
  3. Fibres. \(\nu_n^{-1}(\gamma)\) is the set of vertices of \(\gamma\) in \(\mathrm{Gr}_{\le n}\); with \(m=d(v_0,\gamma)\) there are \(n-m+1\) of them in both parities of \(n-m\). Independent check: I enumerated \(\mathrm{Gr}_{\le n}\) by Hermite normal forms \(g=\begin{pmatrix}t^i&x\\0&t^j\end{pmatrix}\), \(i+j=n\), \(\deg x<i\) (so \(t^n\mathcal O^2\subseteq g\mathcal O^2\)), and counted \(g\) with \(d(v_0,g^{-1}\gamma)=0\) by the Möbius action on random ends in \(\mathbb P^1(\mathbb F_p(t))\): equal to \(\max(0,n+1-m)\) in 3 773 cases (\(p=2,3,5\), \(n\le5\)).
  4. Weights. For \(\gamma\in LX(\mathbb F_q)\) both ends are Frobenius-fixed, so Frobenius fixes the geodesic pointwise (it fixes \(v_0\), hence the projection, hence acts on \(\gamma\) by a translation of length \(0\)); the stalk \(k^{n+1-m}\) carries the trivial permutation action, so \(\nu_{n*}k[n+1]((n+1)/2)\) is pointwise pure of weight \(0\) and its trace is \(v^{n+1}f_n\).
  5. Class. In the basis \(M_O=v^{\delta(O)}\mathbf 1_O\), \([\mathrm{IC}_{0,n}]=M_{O_{0,n}}+vM_{O_{1,n-1}}+\sum_{m<n}(n+1-m)v^{n-m}M_{O_{0,m}}+\sum_{m\le n-1}(n-m)v^{n-m}M_{O_{1,m}}+(n+1)v^{n+1}(M_{A_+}+M_{A_-})\): all exponents \(\ge1\), coefficients \(\ge0\), as claimed in (c).
  6. (d). \(K\)-stabilizers \(\{\mathrm{diag}_\gamma(a,b):a/b\in1+t^m\mathcal O\}\) are connected, the \(K\)-orbit closures are the \(Y_n\), and \(\mathrm{IC}(Y_n)=\mathrm{IC}^{\mathrm{Gr}}_{(n,0)}\ast_K\mathrm{IC}_0\); so every simple object of \(\mathcal C_K\) is a Satake object and \(\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\), consistent with the pages' summary row (spherical sheaves: holds).

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