Proposition X1.4 · L(GL₂ₘ/Sp₂ₘ): codim = ⟨ρ,μ−ν⟩ in every component, char ≠ 2; σ reaches only m | s

1. Components. \(\pi_0(LX)\cong\mathbb Z\) through \(s=\operatorname{val}\operatorname{Pf}\) (Lemma 3.1, item 3). Since \(\operatorname{Pf}(t\omega)=t^m\operatorname{Pf}\omega\) and \(t(g\omega g^{\mathsf T})=g(t\omega)g^{\mathsf T}\), \(\sigma\) is an \(LG\)-equivariant automorphism of the ind-scheme \(LX\) with \(\sigma^j(LX_s)=LX_{s+jm}\); its orbits on \(\pi_0\) are the residue classes mod \(m\). So the reduction of Objection 7.11 (move to component \(0\), which is the \(\mathrm{SL}_{2m}/\mathrm{Sp}_{2m}\) case of Lemma 3.7, where \(LX\) is connected and Prop. 64 applies) is valid exactly for \(m\mid s\). (This confirms Correction 10.2, step 1.)

2. Reduction to \(a_m\ge0\). If \(b\le a\) in one component, the partial sums give \(b_m\ge a_m\). Applying \(\sigma^{-a_m}\), which preserves closures and codimensions and leaves \(\mu_a-\mu_b\) unchanged, we may assume \(a_m\ge0\); then every \(e\le a\) satisfies \(0\le e_m\le e_1\le a_1\le s=\sum a_j\).

3. Finite-type model (Correction 10.2, step 3, checked). Fix \(N>s\) and \(\pi:\Lambda^2\mathcal O^{2m}\to\Lambda^2(\mathcal O/t^N)^{2m}\). The integral forms in \(LX_s\) form \(Z_s=\pi^{-1}(Z_s^{(N)})\), \(Z_s^{(N)}=\{\operatorname{val}\operatorname{Pf}=s\}\) open. If \(\omega'\in Z_s\) and \(\omega'\equiv D_e\bmod t^N\), then \(\omega'=D_e\big(1+D_e^{-1}(\omega'-D_e)\big)\) with the second factor in \(\mathrm{GL}_{2m}(\mathcal O)\) (entries of \(D_e^{-1}(\omega'-D_e)\) have valuation \(\ge N-e_1\ge1\)); so \(\omega'\) has the Smith invariants of \(D_e\), and alternating matrices over \(\mathcal O\) with equal Smith invariants are congruent. Hence \(\mathcal O_e=\pi^{-1}(\mathcal O_e^{(N)})\) and \(\overline{\mathcal O_a}=\pi^{-1}\big(\overline{\mathcal O_a^{(N)}}\cap Z_s^{(N)}\big)\) (integral forms are closed in \(LX\), components are open and closed, and \(\pi\) is a product projection). By Chen–Yi, App. A.4(3), the dimension function of the pulled-back embedding \(\mathcal O_b\hookrightarrow\overline{\mathcal O_a}\) is that of \(\mathcal O_b^{(N)}\hookrightarrow\overline{\mathcal O_a^{(N)}}\); as \(G_N\) is connected, orbits are irreducible and \(\operatorname{codim}=\dim\mathcal O_a^{(N)}-\dim\mathcal O_b^{(N)}\).

4. Lie algebra. \(\operatorname{Lie}S_a=\{X\in\mathfrak{gl}_{2m}(\mathcal O/t^N):XD_a+D_aX^{\mathsf T}=0\}\). In \(2\times2\) blocks, using \(YJ_2+J_2Y^{\mathsf T}=\operatorname{tr}(Y)J_2\): a diagonal block is constrained only by \(t^{a_j}\operatorname{tr}X_{jj}=0\), leaving \(3N+a_j\) dimensions; for \(i<j\), \(X_{ij}J_2\equiv-t^{a_i-a_j}J_2X_{ji}^{\mathsf T}\bmod t^{N-a_j}\) with \(X_{ji}\) free, leaving \(4N+4a_j\); the \((j,i)\) equation is the transpose of the \((i,j)\) one. Total \((2m^2+m)N+\sum_j(4j-3)a_j\), in every characteristic. Recomputed by linear algebra over \(\mathbb F_3,\mathbb F_5,\mathbb F_7\) for \(m\le3\), \(N\le4\) (\(N\le3\) for \(m=3\)), all labels with \(a_1<N\).

5. Smoothness in characteristic \(\ne2\) (new; Correction 10.2 argued in characteristic \(0\) and reached \(\overline{\mathbb F}_p\) only for almost all \(p\)). For \(X\in\operatorname{Lie}S_a\) with \(1-X\) invertible put \(g=(1-X)^{-1}(1+X)\). Then

$$(1+X)D_a(1+X)^{\mathsf T}-(1-X)D_a(1-X)^{\mathsf T}=2\,(XD_a+D_aX^{\mathsf T})=0,$$

so \(gD_ag^{\mathsf T}=D_a\). As \(g+1=2(1-X)^{-1}\) is invertible, \(X=(g-1)(g+1)^{-1}\): the Cayley map is an injective morphism from the nonempty open subset \(\{\det(1-X)\in(\mathcal O/t^N)^\times\}\) of the affine space \(\operatorname{Lie}S_a\) into \(S_a\). Hence \(\dim S_a\ge\dim\operatorname{Lie}S_a\ge\dim S_a\), and the group scheme \(S_a\) is smooth. (Checked on random elements of \(\operatorname{Lie}S_a\) over \(\mathbb F_3,\mathbb F_5\).)

6. The formula. \(\dim\mathcal O_a^{(N)}=4m^2N-\dim S_a=m(2m-1)N-\sum_j(4j-3)a_j\). With \(\sum a_j=\sum b_j\),

$$\operatorname{codim}=\sum_j(4j-3)(b_j-a_j)=\sum_j4j(b_j-a_j)=\sum_j(2m+2-4j)(a_j-b_j)=\langle\rho,\mu_a-\mu_b\rangle,$$

since \(\langle\rho,e_{2j-1}+e_{2j}\rangle=2m+2-4j\). For \(\mathrm{GL}_4\), \(N=3\): dimensions \(16\) and \(12\), the degrees of the point counts \(q^{10}(q-1)(q^3-1)(q^2+1)\) and \(q^8(q-1)(q^3-1)\) of Lemma 3.1. Where Prop. 64 applies (\(m\mid s\)) the two formulas agree.

7. What Theorem 3.6 uses. \(c_\nu=D-\dim\mathcal O_\nu^{(N)}\) needs only orbit dimensions in the component \(k\). The finite-type model \(Y\) of \(\mathrm{Gr}^{\omega_k}\tilde\times L^+X\) is irreducible, so its image under the (proper) convolution map is the closure of the unique orbit of largest dimension among \(\nu\in\{0,1,2\}^m\), \(|\nu|=k\); by step 6 this is the unique minimizer of \(\sum_j(4j-3)\nu_j\), namely \(\nu^{top}=\varphi(\omega_k)\), with fibre \(\mathrm{LG}(\epsilon,2\epsilon)\) of dimension \(\epsilon=k\bmod2\). So \(D=\dim\mathcal O^{(N)}_{\nu^{top}}+\epsilon\), \(c_\nu=\epsilon+\langle\rho,\nu^{top}-\nu\rangle\), and \(\operatorname{def}(\nu)=n_1(n_1+1)-c_\nu=n_1(\nu)\) (identity checked for \(m\le8\) and all \(k\) from step 6 alone). No closure relation and no Prop. 64 enters.

8. Corrected point 2 of Objection 7.11. "The shift \(\sigma\) identifies the component \(s\) with component \(0\) only when \(m\mid s\), where Prop. 64 applies through Lemma 3.7. In every component, and over every algebraically closed field of characteristic \(\ne2\), the codimension formula follows from the finite-type stabilizer count of Lemma 3.1, item 5, completed by the smoothness of the stabilizer (Cayley transform)." Severity: cosmetic; no conclusion of the run changes, since Lemma 3.1's count was available throughout. Correction 10.2 is confirmed and extended from characteristic \(0\) to every odd characteristic, which is the setting (\(\overline{\mathbb F}_q\), \(q\) odd) in which Proposition 3.3 and Theorem 3.6 are applied.