Lemma 3.3 · A model of \(X=\mathrm{SL}_2^3\backslash\mathrm{Sp}_4^2\) and its fusion divisor

Proof.

1. Transitivity. By Witt's theorem \(\mathrm{Sp}(V_i)(E)\) is transitive on nondegenerate planes; for fixed \(W_1,W_2\) the isometries \(W_1\to W_2\) form a torsor under \(\mathrm{Sp}(W_1)(E)\), and \(\mathrm{Sp}(W_1)\times\mathrm{Sp}(W_1^\perp)\subseteq\operatorname{Stab}(W_1)\). So \(G(E)\) is transitive on triples, which gives \(X(E)=G(E)x_0=G(E)/H(E)\) (also \(H^1(E,\mathrm{SL}_2^3)=1\)).
2. Stabilizer. \((g_1,g_2)\) fixes \(x_0\) iff \(g_i\) preserves \(W_i^0\) (hence \(W_i^{0\perp}\)) and \(g_2|_{W_2^0}=\varphi_0\,g_1|_{W_1^0}\,\varphi_0^{-1}\). This is (a). \(H\) is connected, reductive and split, so \(X\) is affine and smooth.
3. Models. \((e_1,f_1,e_2,f_2)\mapsto(\langle e_1,f_1\rangle,\langle e_2,f_2\rangle,\ e_1\mapsto e_2,\ f_1\mapsto f_2)\) is constant exactly on the orbits of the diagonal right action of \(\mathrm{SL}_2\) on pairs of frames, giving \(X=(Z_1\times Z_2)/\mathrm{SL}_2\); \(Z_i\cong\mathrm{Sp}_4/\mathrm{Sp}_2\) since \(\mathrm{Sp}(V_i)\) is transitive on \(\{\omega_i(e,f)=1\}\) with stabilizer \(\mathrm{Sp}(\langle e,f\rangle^\perp)\). The graph \(\Gamma_\varphi\subset V_1\oplus V_2\) is a 2-plane isotropic for \(\omega_1-\omega_2\); conversely an isotropic 2-plane \(\Gamma\) with \(\omega_1|_{\mathrm{pr}_1\Gamma}\ne0\) has both projections injective onto nondegenerate planes and is a graph. The condition is the non-vanishing of a section of an ample line bundle, so the open set is affine. \(\dim X=20-9=11\).
4. Dual data. (c) is Sakellaridis, Example 8.0.1 (sak.txt, §8; arXiv:0905.4244v3, p. 64), which also states that \(W_X\)-nonconjugate points become \(W\)-conjugate.
5. Fusion. On maximal tori, \(\iota(a,b,c)=\big((ab,a/b),(bc,b/c)\big)\) in the coordinates \((x,y)\mapsto\operatorname{diag}(x,y,1,y^{-1},x^{-1})\) of \(\mathrm{SO}_5\) (eigenvalues of \([g_1,g_2]\) on \(k^2\otimes k^2\) are \(ab,a/b,b/a,1/ab\)). \(W(B_2)\) acts by all signed permutations of \((\log x,\log y)\); \(y\mapsto y^{-1}\) in both factors sends \(\iota(a,b,a)=((ab,a/b),(ab,b/a))\) to \(((ab,b/a),(ab,a/b))=\iota(b,a,b)\). \(W_X=\{\pm1\}^3\) acts by \(a\mapsto a^{\pm1},b\mapsto b^{\pm1},c\mapsto c^{\pm1}\), so \((a,b,a)\sim_{W_X}(b,a,b)\) forces \(b=a^{\pm1}\). \(\square\)

Remark. The fused locus is the divisor \(\{c=a\}\) of the 3-torus, folded by \((a,b,a)\leftrightarrow(b,a,b)\); since \(\delta(X)\) is trivial, no power of \(q\) enters. This is the mechanism of Proposition 3.2(b) and the shape the task asks for.