§3 · Agent 3 · Symplectic family: fibres · 2026-10-17T02:07:17Z

Lemma 3.1 · The loop space of \(\GL_{2m}/\Sp_{2m}\): orbits, closure order, scalar shift and codimensions

proved audit: minor defect

Minor defect (second audit). Item 5 cites Chen–Yi Prop. 64, which is proved in their §7.2 under the standing assumption that LX is connected; for GL_{2m}, pi_0(LX) = Z, so the citation does not apply as written. The formula is nevertheless proved inside the node by the finite-type stabilizer count, which (after a sigma-shift to a_m >= 0) covers every component; re-verified: stabilizer dimensions by linear algebra over F_3 and F_101, the codimension identity for 539 pairs of strata in components s <= 2m+1, m <= 4 (g2_orbits_lie.py). Point counts confirmed by brute force (35840, 1792 at q = 2) and by a closed-form count valid for every q. Transversality of the slice (tangent space of the orbit = t Lambda^2) is a routine detail left implicit. Objection 7.11 flagged the citation; its proposed repair is incomplete (see Objection 7.11 and correction Correction 10.2), but the node's own count is the valid repair.

Identifies \(L(\GL_{2m}/\Sp_{2m})\) with nondegenerate alternating forms over \(k(\!(t)\!)\), labels the arc-group orbits by symplectic elementary divisors, and records the closure order, the scalar shift \(\omega\mapsto t\omega\) and the orbit codimensions. For \(\GL_4/\Sp_4\) the uncovered orbit \((1,1)\) is the closed orbit of its component, of codimension 4 in the closure of \((2,0)\), with a transversal slice that is the cone over a smooth quadric threefold.

Lemma 3.1.

Let \(X=\GL_{2m}/\Sp_{2m}\) with \(\theta(g)=Jg^{-T}J^{-1}\), \(J=\operatorname{diag}(J_2,\dots,J_2)\), and put \(\omega_x:=\tau(x)J\) for the Cartan embedding \(\tau(gG^\theta)=g\theta(g)^{-1}\). Write \(\mathcal O=k[\![t]\!]\), \(F=k(\!(t)\!)\), \(\Lambda_0=\mathcal O^{2m}\).

  1. Model. \(x\mapsto\omega_x\) identifies \(LX=X(F)\) with the set of nondegenerate alternating forms on \(F^{2m}\), with action \(g\cdot\omega=g\omega g^{T}\), and \(L^+X\) with the forms that are perfect on \(\Lambda_0\).
  2. Orbits. The \(L^+G\)-orbits are the classes \(\omega\sim\bigoplus_i t^{a_i}J_2\), \(a=(a_1\ge\dots\ge a_m)\in\Z^m\). The label \(a\) corresponds to \(\mu=(a_1,a_1,\dots,a_m,a_m)\in\Lambda_S^+\) through \(\omega_{x_\mu}=t^\mu J\); write \(\cO_a\).
  3. Components and order. \(\pi_0(LX)=\Lambda_S/(\Lambda_S\cap Q)\cong\Z\) via \(s=\sum_ia_i=\val\Pf\omega\). Within a component, \(\nu\le\mu\) (dominance in \(\Z^{2m}\)) if and only if the label of \(\nu\) is majorized by that of \(\mu\). For \(\GL_4\) and \(s=2\) the orbits form the chain \((1,1)<(2,0)<(3,-1)<\cdots\); so \(\cO_{(1,1)}\) is the closed orbit of its component and \(\overline{\cO_{(2,0)}}=\cO_{(2,0)}\sqcup\cO_{(1,1)}=\{\omega\text{ integral},\ \val\Pf\omega=2\}\).
  4. Scalar shift. \(\sigma(\omega)=t\omega\) is an \(LG\)-equivariant automorphism of \(LX\) with \(\sigma(\cO_a)=\cO_{a+(1^m)}\). Hence \(\cO_{(1^m)}=\sigma(L^+X)\) is closed, \(\IC_{(1^m)}=\Delta_{(1^m)}=\sigma_*\IC_0\), and \(\sigma^2\) is translation by the central element \(t\in LG\), so \(\sigma^2_*=V_{(1^{2m})}\ast(-)\).
  5. Codimension. \(\operatorname{codim}_{\overline{\cO_\mu}}\cO_\nu=\langle\rho,\mu-\nu\rangle\) (cited from Chen–Yi, Prop. 64, as quoted in their Example 88); for \(\GL_4\), \(\cO_{(1,1)}\) has codimension \(\langle\rho,(1,1,-1,-1)\rangle=4\) in \(\overline{\cO_{(2,0)}}\). In the truncation modulo \(t^N\) (\(a_m\ge0\), \(N>a_1\)),
$$\dim\cO^{(N)}_a=m(2m-1)N-\sum_{j=1}^m(4j-3)\,a_j ,$$

so for \(\GL_4\), \(\dim\cO^{(N)}_{(a_1,a_2)}=6N-a_1-5a_2\) (16 and 12 at \(N=3\)). Over \(\F_q\): \(\#\cO^{(3)}_{(2,0)}=q^{10}(q-1)(q^3-1)(q^2+1)\) and \(\#\cO^{(3)}_{(1,1)}=q^{8}(q-1)(q^3-1)\).

  1. Slice. At \(t\omega_0\in\cO_{(1,1)}\) (\(\GL_4\)), the affine space \(\{t\omega_0+\eta:\eta\in\Lambda^2k^4\}\) meets \(\overline{\cO_{(2,0)}}\) in \(\{\eta:\Pf(\eta)=0,\ B(\eta,\omega_0)=0\}\), where \(B\) is the polar form of \(\Pf\): the affine cone over the smooth quadric threefold \(Q^3\cong LG(2,4)\), of dimension \(4\), equal to the codimension.
  1. Model. From \(\tau(gx)=g\tau(x)\theta(g)^{-1}\) and \(\theta(g)^{-1}=Jg^{T}J^{-1}\) one gets \(\omega_{gx}=g\,\omega_x\,g^{T}\), and \(\omega_{x_0}=J\). Over a field all nondegenerate alternating forms of rank \(2m\) are equivalent (equivalently \(H^1(F,\Sp_{2m})=0\)), so \(LX\) is the set of all such forms on \(F^{2m}\); an \(\mathcal O\)-point is a form perfect on \(\Lambda_0\).
  1. Orbits. By the elementary-divisor theorem for alternating matrices over the principal ideal domain \(\mathcal O\), every \(\omega\) equals \(g(\bigoplus_it^{a_i}J_2)g^{T}\) with \(g\in\GL_{2m}(\mathcal O)\) and \(a\) unique up to order. Since \(t^\mu J=\bigoplus_it^{a_i}J_2\) and \(\tau(x_\mu)=t^\mu\), this is the parametrization of Chen–Yi, Prop. 17.
  1. Components and order. The coroot lattice is \(Q=\{x\in\Z^{2m}:\sum x_i=0\}\), so \(\Lambda_S/(\Lambda_S\cap Q)\cong\Z\) via \(\sum a_i\), and \(\val\Pf\) is locally constant on \(LX\). For doubled vectors the partial sums of \(\mu-\nu\) are \(2S_j\) at position \(2j\) and \(S_{j-1}+S_j\) at position \(2j-1\), where \(S_j\) are the partial sums of the difference of labels; so dominance is majorization. For \(\GL_4\), an integral form with \(\val\Pf=2\) has type \((2,0)\) or \((1,1)\), and Chen–Yi, Thm. 18 gives the closure.
  1. Scalar shift. \(\sigma(g\omega g^T)=g\,\sigma(\omega)\,g^T\); \(t\) is central with \(\theta(t\tau(x))=(t\tau(x))^{-1}\); the central element \(t\cdot1\) acts by \(\omega\mapsto t^2\omega\). The intersection complex of a closed orbit is its constant sheaf.
  1. Codimension. The formula is quoted from Chen–Yi, Prop. 64. Independent count: for \(D_a=\bigoplus_jt^{a_j}J_2\), the stabilizer Lie algebra \(\{X\in\mathfrak{gl}_{2m}(\mathcal O/t^N):XD_a+D_aX^T\equiv0\}\) is computed in \(2\times2\) blocks, using \(YJ_2+J_2Y^T=\tr(Y)J_2\). A diagonal block \(X_{jj}\) is constrained only by \(\tr X_{jj}\in t^{N-a_j}\mathcal O\), giving \(3N+a_j\) dimensions; a pair of blocks \((X_{ij},X_{ji})\), \(i<j\), gives \(4N+4a_j\). The total is \((2m^2+m)N+\sum_j(4j-3)a_j\), whence the displayed orbit dimension; for \(\GL_4\) the stabilizer dimensions are \(10N+a_1+5a_2\). The difference \(16-12=4\) agrees with Prop. 64, and in general the count agrees with \(\langle\rho,\mu-\nu\rangle\) because \(\langle\rho,e_{2j-1}+e_{2j}\rangle=2m+2-4j\). The two point counts were confirmed by brute force at \(q=2\) (35840 and 1792).
  1. Slice. \(\Pf(t\omega_0+\eta)=\Pf(\eta)+t\,B(\eta,\omega_0)+t^2\Pf(\omega_0)\), so \(\val\Pf=2\) forces \(\Pf(\eta)=0\) and \(B(\eta,\omega_0)=0\). On \(\Lambda^2k^4\cong k^6\) the Pfaffian is a nondegenerate quadratic form (the Klein quadric) and \(\omega_0\) is anisotropic, so the hyperplane \(B(\cdot,\omega_0)=0\) is nondegenerate and the intersection is the affine cone over a smooth quadric threefold, which is the Lagrangian Grassmannian of \(\omega_0\). The tangent space of \(\cO_{(1,1)}\) at \(t\omega_0\) is \(t\Lambda^2\), so the constant forms \(\eta\) span a complementary affine subspace and the slice is transversal.

Verified by computation: g2_orbits_lie.py

Builds on

  • —

Built on by

  • Computation 3.2 · Convolution fibres over \(x_{(2,0)}\) and \(x_{(1,1)}\) for \(\lambda=\omega_2\) and \(\lambda=(2,0,0,0)\)
  • Proposition 3.3 · Decomposition of \(V_{\omega_2}\ast\IC_0\) for \(\GL_4/\Sp_4\): \(\IC_{(1,1)}\) is a retract with multiplicity \(k[2]\oplus k\oplus k[-2]\)
  • 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
  • Objection 7.11 · Objection to the codimension citation: Chen–Yi's Prop. 64 assumes \(LX\) connected; a one-line reduction is proposed
  • Correction 10.2 · Codimensions of orbits in \(L(\GL_{2m}/\Sp_{2m})\) in every component, without connectedness of \(LX\)
  • Proposition X1.4 · L(GL₂ₘ/Sp₂ₘ): codim = ⟨ρ,μ−ν⟩ in every component, char ≠ 2; σ reaches only m | s
Original record, as the agent wrote it at 02:07 UTC

Forms model, closure order and codimensions for GL_4/Sp_4 (and GL_2m/Sp_2m)

Model. \(X=\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(\theta(g)=J g^{-T}J^{-1}\). Through \(x\mapsto\omega_x:=\tau(x)J\) (\(\tau(gG^\theta)=g\theta(g)^{-1}\)), \(LX=X(F)\) is the space of nondegenerate alternating forms \(\omega\) on \(F^{2m}\) (all are \(G(F)\)-conjugate, \(H^1(F,\mathrm{Sp})=0\)), with \(\omega_{gx}=g\,\omega_x\,g^{T}\), \(L^+X=\{\omega\text{ perfect on }\Lambda_0=\mathcal O^{2m}\}\).

  1. (Orbits.) By the elementary-divisor theorem for alternating forms over \(\mathcal O\), \(L^+G\)-orbits are the types \((a_1,a_1,\dots,a_m,a_m)\), \(a_1\ge\dots\ge a_m\): \(\omega\sim\bigoplus_i t^{a_i}J_2\). This matches Chen–Yi Prop. 17 through \(\tau(x_\mu)=t^\mu\), \(\omega_{x_\mu}=t^\mu J\). I write labels as \(m\)-tuples \(a=(a_1,\dots,a_m)\).
  2. (Components.) \(\pi_0(LX)=\Lambda_S/(\Lambda_S\cap Q)\cong\mathbb Z\) via \(s=\sum a_i=v(\mathrm{Pf}\,\omega)\).
  3. (Order.) \(\nu\le\mu\) (dominance in \(\mathbb Z^{2m}\)) \(\iff\) same \(s\) and \(\nu\) majorized by \(\mu\). For \(\mathrm{GL}_4\), \(s=2\): the chain \((1,1)<(2,0)<(3,-1)<\cdots\). So \(\mathcal O_{(1,1)}\) is the closed (minimal) orbit of its component and \(\overline{\mathcal O_{(2,0)}}=\mathcal O_{(2,0)}\sqcup\mathcal O_{(1,1)}=\{\omega\text{ integral},\ v(\mathrm{Pf}\,\omega)=2\}\) (H1(a), Chen–Yi Thm. 18).
  4. (Scalar shift.) \(\sigma(\omega)=t\omega\) is an \(LG\)-equivariant automorphism of \(LX\) (\(t\) central: \(\theta(t\tau(x))=(t\tau(x))^{-1}\)), \(\sigma(\mathcal O_a)=\mathcal O_{a+(1^m)}\). Hence \(\mathcal O_{(1^m)}=\sigma(L^+X)\cong L^+X\) as \(L^+G\)-spaces, \(\mathrm{IC}_{(1^m)}=\Delta_{(1^m)}=\sigma_*\mathrm{IC}_0\), and \(\sigma^2=\) translation by the central element \(t\in LG\), so \(\sigma^2_*=V_{(1^{2m})}\ast(-)\).
  5. (Codimension.) Chen–Yi Prop. 64 (quoted in Example 88): \(\operatorname{codim}_{\overline{\mathcal O_{\mu}}}\mathcal O_\nu=\langle\rho,\mu-\nu\rangle\). For \(\mathrm{GL}_4\): \(\langle\rho,(1,1,-1,-1)\rangle=4\). Independent check: in the finite-type model \(Z^{(N)}=\{\bar\omega\in\Lambda^2(\mathcal O/t^N)^{4*}:v(\mathrm{Pf})=2\}\) (\(N\ge3\)) the stabilizer Lie algebra of \(\omega=\mathrm{diag}(t^aJ_2,t^bJ_2)\) has dimension \(10N+a+5b\), so \(\dim\mathcal O^{(N)}_{(a,b)}=6N-a-5b\): \(16\) vs \(12\), codimension \(4\). Point counts over \(\mathbb F_q\): \(\#\mathcal O^{(3)}_{(2,0)}=q^{10}(q-1)(q^3-1)(q^2+1)\), \(\#\mathcal O^{(3)}_{(1,1)}=q^{8}(q-1)(q^3-1)\); verified by brute force at \(q=2\) (35840, 1792).
  6. (Slice.) At \(t\omega_0\in\mathcal O_{(1,1)}\), \(\{t\omega_0+\eta:\eta\in\Lambda^2k^4\}\cap Z=\{\eta:\mathrm{Pf}(\eta)=0,\ B(\eta,\omega_0)=0\}\) (from \(\mathrm{Pf}(t\omega_0+\eta)=\mathrm{Pf}(\eta)+tB(\eta,\omega_0)+t^2\mathrm{Pf}(\omega_0)\)): the affine cone over the smooth quadric threefold \(Q^3\cong LG(2,4)\), dimension 4 = codimension.

For \(\mathrm{GL}_{2m}\): \(\dim\mathcal O^{(N)}_{a}=m(2m-1)N-\sum_j(4j-3)a_j\) (same Lie-algebra computation, \(a_m\ge0\), \(N>a_1\)), consistent with Prop. 64 since \(\langle\rho,e_{2j-1}+e_{2j}\rangle=2m+2-4j\).

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