§10 · Audit · Second audit, then re-checks · 2026-10-17T07:03:35Z

Correction 10.4 · Correction: the function-level support bound for the pair model of \(\GL_n/(\GL_p\times\GL_q)\) over every local field

proved correction ★ key step

Correction. Corrects Proposition 5.8 · The support bound of H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ.(a) at function level over every \(F\), and the same-points question, Objection 7.24 · Objection: the support-bound check passes; the \(E_6/F_4\) expectation and two table cells are stale.

The orbit label of a pair \((U,V)\) relative to \(\cO^n\) is the polar part of the Smith form of the projector onto \(U\) along \(V\); bounding the minors of \(t^\lambda\pi t^{-\lambda}\) gives \(Kt^\lambda X(\cO)\subseteq\bigcup_{\nu\le\varphi(\lambda)}O_\nu\) for every dominant \(\lambda\) and every \(F\), with no involution and no Cartan embedding. This restores the conclusion of Proposition 5.8 in residue characteristic 2, where its input (a) fails.

Correction 10.4.

Let \(F\) be any non-archimedean local field, \(1\le p\le n-p\), and \(X(F)=\{(U,V):F^n=U\oplus V,\ \dim V=p\}\), the pair model of Theorem 5.4, with \(L_0=\cO^n\). For \(x=(U,V)\) let \(\pi_x\) be the projection onto \(U\) along \(V\), and let \(\ell(x)\) be the type of \(\pi_V(L_0)/(L_0\cap V)\), so that \(O_b=\{x:\ell(x)=b\}\) for \(b_1\ge\dots\ge b_p\ge0\).

  1. \(\ell(x)\) is the polar part of the Smith form of \(\pi_x\): if \(\pi_x=k_1\operatorname{diag}(\varpi^{c_1},\dots,\varpi^{c_n})k_2\) with \(k_1,k_2\in\GL_n(\cO)\) and \(c_i\in\Z\cup\{\infty\}\), then \(\ell(x)\) consists of the numbers \(-c_i\) with \(c_i<0\).
  2. For every dominant \(\lambda\in\Z^n\),
$$K\,t^\lambda X(\cO)\ \subseteq\ \bigcup_{b\le_{\mathrm{dom}}\varphi(\lambda)}O_b,\qquad\varphi(\lambda)=\big(\lambda_1-\lambda_n,\ \lambda_2-\lambda_{n-1},\ \dots,\ \lambda_p-\lambda_{n+1-p}\big),$$

and \(t^\lambda x_0\in O_{\varphi(\lambda)}\) for the base point \(x_0\) of Theorem 5.4. So function-level H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ.(a), with its top value, holds for the pair model over every \(F\).

  1. Input (a) of Proposition 5.8 fails for this model in residue characteristic 2: over \(\Q_2\) with \(n=2\), the point \(t^{(1,0)}x_0\) has label \(1\) but \(\tau(t^{(1,0)}x_0)\in K\).

Write \(L_0=\cO^n\) and, for \(x=(U,V)\), put \(A=\pi_xL_0\), \(A'=L_0\cap U\), \(B=(1-\pi_x)L_0\) and \(B'=L_0\cap V\).

  1. The label as a cokernel. \(L_0+\pi_xL_0=A\oplus B\): every \(l\in L_0\) equals \(\pi_xl+(1-\pi_x)l\), and both \(A\) and \(B=\{l-\pi_xl\}\) lie in \(L_0+\pi_xL_0\). Let \(\psi:A/A'\to B/B'\) be the Goursat map \(\pi_xl\mapsto(1-\pi_x)l\); it is well defined, because \(\pi_xl\in A'\) forces \((1-\pi_x)l=l-\pi_xl\in L_0\cap V\). The map \(A\oplus B\to B/B'\), \((a,b)\mapsto b-\psi(a)\), is onto, and its kernel is \(L_0\): if \(b-\psi(\pi_xl)\in B'\) then \(\pi_xl+b\in l+B'\subseteq L_0\), and conversely \(l\in L_0\) maps to \(0\). Hence
$$(L_0+\pi_xL_0)/L_0\ \cong\ B/B',$$

whose type is \(\ell(x)\) by definition.

  1. Smith form and minors. If \(\pi_x=k_1Dk_2\) with \(D=\operatorname{diag}(\varpi^{c_i})\), then \(L_0+\pi_xL_0=k_1(L_0+DL_0)\) and \((L_0+DL_0)/L_0\cong\bigoplus_{c_i<0}\cO/\varpi^{-c_i}\); this proves item 1. Order the exponents \(c_1\le\dots\le c_n\). By Cauchy–Binet, multiplying by elements of \(\GL_n(\cO)\) on either side does not change \(\delta_j(\pi_x):=\min\{\val\det(\pi_x)_{I,J}:|I|=|J|=j\}\), so \(\delta_j(\pi_x)=c_1+\dots+c_j\). Since \(\ell(x)\) has at most \(p\) parts (\(B/B'\) is a quotient of a lattice of rank \(p\)), for \(k\le p\)
$$\ell_1(x)+\dots+\ell_k(x)\ =\ -\min_{0\le j\le k}\delta_j(\pi_x)\qquad(\delta_0=0).$$
  1. Bounding the minors. Let \(x=kt^\lambda y\) with \(k\in K\) and \(y\in X(\cO)\). By Theorem 5.4, \(X(\cO)=Kx_0\), and \(\pi_{x_0}\in M_n(\cO)\) because \(x_0\) is split for \(L_0\); so \(\pi':=\pi_y\in M_n(\cO)\), and \(\pi_x=k\,t^\lambda\pi't^{-\lambda}\,k^{-1}\) has the Smith exponents of \(Y:=t^\lambda\pi't^{-\lambda}\), whose entries are \(\varpi^{\lambda_i-\lambda_j}\pi'_{ij}\). For index sets with \(|I|=|J|=j\),
$$\val\det Y_{I,J}\ \ge\ \sum_{i\in I}\lambda_i-\sum_{i\in J}\lambda_i\ \ge\ -\sum_{i\le j}\big(\lambda_i-\lambda_{n+1-i}\big),$$

because a sum of \(j\) entries of \(\lambda\) lies between the sum of the \(j\) smallest and the sum of the \(j\) largest. Hence \(\delta_j(\pi_x)\ge-\sum_{i\le j}\varphi(\lambda)_i\) for \(j\le p\).

  1. The bound. The numbers \(\varphi(\lambda)_i=\lambda_i-\lambda_{n+1-i}\) are nonnegative, so step 2 gives
$$\sum_{i\le k}\ell_i(x)\ \le\ \max_{j\le k}\sum_{i\le j}\varphi(\lambda)_i\ =\ \sum_{i\le k}\varphi(\lambda)_i\qquad(k\le p).$$

For labels written as \((b_1,\dots,b_p,0,\dots,0,-b_p,\dots,-b_1)\in\Lambda_S^+\), the partial sums beyond the first \(p\) coordinates reduce to these and the totals vanish, so \(\ell(x)\le_{\mathrm{dom}}\varphi(\lambda)\).

  1. The top orbit. \(t^\lambda U_0=\langle\varpi^{\lambda_i}e_i+\varpi^{\lambda_{n+1-i}}e_{n+1-i},\,e_j\rangle\) and \(t^\lambda V_0=V_0\). The projection onto \(V_0\) along \(t^\lambda U_0\) kills the \(e_j\), fixes the \(e_{n+1-i}\) and sends \(e_i\) to \(-\varpi^{-(\lambda_i-\lambda_{n+1-i})}e_{n+1-i}\), so \(\ell(t^\lambda x_0)=\varphi(\lambda)\). This proves item 2; with Lemma A(2) the top value is \(\ge1\).
  1. Item 3. The only involution with fixed group \(\Stab(x_0)=\GL(U_0)\times\GL(V_0)\) is \(\theta=\operatorname{Ad}\varepsilon_0\), with \(\varepsilon_0=1\) on \(U_0\) and \(-1\) on \(V_0\). For \(n=2\), \(U_0=\langle e_1+e_2\rangle\), \(V_0=\langle e_2\rangle\) and \(\varepsilon_0=\begin{pmatrix}1&0\\2&-1\end{pmatrix}\), which does not normalize the diagonal torus. Over \(\Q_2\), with \(t=\operatorname{diag}(2,1)\),
$$\tau(tx_0)=t\varepsilon_0t^{-1}\varepsilon_0^{-1}=\begin{pmatrix}1&0\\1&-1\end{pmatrix}\begin{pmatrix}1&0\\2&-1\end{pmatrix}=\begin{pmatrix}1&0\\-1&1\end{pmatrix}\in K,$$

whereas \(tx_0\) has label \(1\) by Theorem 5.4. More generally, for \(n=2\) the Cartan label of \(t^{(a,0)}x_0\) is \(\max(a-\val2,0)\), and in characteristic 2 there is no such involution at all. \(\square\)

Consequence for Proposition 5.8. Its conclusion holds as stated. Function-level H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ.(a) holds over every \(F\) for the spheres (pole-order bound, step 3 of Theorem 5.3, valid for every dominant \(\lambda\)), for alternating forms (inputs (a) and (b) are valid there in every characteristic, with \(\tau(\omega)=\omega J^{-1}\)), and for the pair model by item 2. Only the route through input (a) fails, and only for the pair model in residue characteristic 2; nothing downstream used that route. A machine check over \(\F_2[[t]]\) of all lattices of every dominant type with parts at most \(2\) for \(n=4\), \(p=2\) (and of smaller cases for \(n=3,5\)) confirms item 2.

Verified by computation: pairmodel.py, pair_cartan.py.

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  • Proposition X3.2 · Support bound and top orbit for the pair model, every dominant \(\lambda\), every \(F\)
  • Proposition X3.3 · Cartan label of \(\tau(x)\) is \(\mu((\ell(x)-e)^+)\), \(e=\operatorname{val}2\)
  • Summary 9.5 · Verdict after the second audit: \(E_6/F_4\) is still the one divergence