Proposition 3.6 · Orbits \((d_1,d_2,e)\), connected stabilizers, \(\delta=3d_1+3d_2+e\)

Proof.

1. One plane. \(N=W\cap L\) is free of rank 2 with \(\omega(N,N)\subseteq\kappa[\![t]\!]\); a basis with \(\omega(e,f)=t^d\) exists. \(N'=W^\perp\cap L\) has the same \(d\), and \(N\oplus N'\subseteq L\subseteq N^\#\oplus N'^\#\) with \(L/(N\oplus N')\) the graph of an anti-isometry \(\varepsilon\) of the discriminant forms \(N^\#/N\cong N'^\#/N'\cong(\kappa[\![t]\!]/t^d)^2\). Since \(\mathrm{SL}_2(\kappa[\![t]\!])\to\mathrm{SL}_2(\kappa[\![t]\!]/t^d)\) is onto and the latter is the automorphism group of the discriminant form, two planes with the same \(d\) are \(\mathrm{Sp}(L)\)-conjugate, and \(\operatorname{Stab}(W)\) acts on \(W\) through all of \(\mathrm{SL}(N)\).
2. The isometry. Given \((W_1,W_2)\), \(\operatorname{Stab}(W_1,W_2)\) acts on \(\varphi\) through \(\mathrm{SL}(N_2)\times\mathrm{SL}(N_1)\). In bases with \(\omega=t^{d_i}\), \(\varphi\) is a matrix of determinant exactly \(t^{d_1-d_2}\); its \(\mathrm{SL}_2(\kappa[\![t]\!])\)-double coset is its Smith form \(\operatorname{diag}(t^{m_1},t^{m_2})\) (units are absorbed because the determinant is fixed). This proves (a); label \((0,0,0)\) means \(L_i=N_i\oplus N_i'\) and \(\varphi(N_1)=N_2\), i.e. integrality.
3. Stabilizers. \(\operatorname{Stab}(x)=\{(a,b_1,b_2)\}\) with \(a\in\Gamma:=\mathrm{SL}(N_1)\cap\varphi^{-1}\mathrm{SL}(N_2)\varphi\cong\{g\in\mathrm{SL}_2(k[\![t]\!]):g_{21}\in t^e k[\![t]\!]\}\) (stabilizer of two vertices at distance \(e\) of the tree of \(\mathrm{SL}(W_1)\)), and \(b_i\in\mathrm{SL}(N_i')\) with \(b_i\bmod t^{d_i}\) determined by \(a\) through \(\varepsilon_i\). This is a fibration over the connected group \(\Gamma\) with fibres torsors under level-\(d_i\) congruence subgroups (connected); so (b). Lang's theorem for connected pro-algebraic groups gives one \(K\)-orbit of rational points per label, matching (a) over \(\mathbb F_q\).
4. Dimensions. \(\operatorname{Stab}(x_0)\cong\mathrm{SL}_2(k[\![t]\!])^3\); in \(\operatorname{Stab}(x_\nu)\), \(\Gamma\) has codimension \(e\) and the congruence conditions codimension \(3d_1+3d_2\). This gives (c). Consistently \(\operatorname{vol}(Kx_\nu)/\operatorname{vol}(Kx_0)=[(q+1)q^{e-1}]_{e\ge1}\prod_{d_i\ge1}q^{3d_i}(1-q^{-2})\), used in Computation 3.7.
5. Finiteness. For \(\lambda\in\Lambda_T^+\), \(\mathcal C_\lambda=\{x:\ x\ \text{is integral for some}\ L'\in\mathrm{Gr}_{\le\lambda}\}\) is the image of \(\mathrm{Gr}_{\le\lambda}\tilde\times L^+X\), closed because \(\mathrm{Gr}_{\le\lambda}\) is proper and \(L^+X\subset LX\) is closed. By the position formula of Computation 3.7, integrality for \(L'\) at position \(\le\lambda\) forces \(d_i\le2a_i\) and \(e\le D^1+D^2\le2(a_1+a_2)\): finitely many labels. Every \(x\) lies in some \(\mathcal C_\lambda\) because \(LX(k)=LG(k)x_0\) (\(H^1(k(\!(t)\!),\mathrm{SL}_2^3)=1\)). So \(\overline{\mathcal O_\nu}\subseteq\mathcal C_\lambda\) for suitable \(\lambda\): (d).
6. Leading labels. \(\delta(\varphi(\lambda))=3(a_1+b_1)+3(a_2+b_2)+(a_1-b_1)+(a_2-b_2)=4a_1+2b_1+4a_2+2b_2=\langle2\rho,\lambda\rangle\), and the fibre over \(x_{\varphi(\lambda)}\) in \(\mathrm{Gr}^\lambda\) is one point (Proposition 3.8). \(\nu=\varphi(\lambda)\) iff there are \(s_i=a_i-b_i\equiv d_i\ (2)\), \(0\le s_i\le d_i\), with \(s_1+s_2=e\), i.e. iff \((d_1\bmod2)+(d_2\bmod2)\le e\le d_1+d_2\). \(\square\)

Comparison with Sakellaridis. With \(\gamma_1=\varepsilon_1+\varepsilon_2\), \(\gamma_2=(\varepsilon_1-\varepsilon_2)+(\varepsilon_1'-\varepsilon_2')\), \(\gamma_3=\varepsilon_1'+\varepsilon_2'\) (his normalized spherical roots), \((d_1,d_2,e)=(\langle\gamma_1,v\rangle,\langle\gamma_3,v\rangle,\langle\gamma_2,v\rangle)\) on \(\Lambda_X=\Lambda_T/\mathbb Z(1,-1,-1,1)\): the orbit labels are \(\Lambda_X^+\) and \(\varphi\) is the projection. The uncovered labels play the role of \(3\omega_1^\vee\) for \(E_6/F_4\), but there are infinitely many families of them.