Lemma 3.4 · Generic \(I(\chi)\) of \(\mathrm{Sp}_4\) has two \(\mathrm{SL}_2\)-quotients

Proof.

1. Orbits. \(H_1\) has four orbits on the flag variety, each a single \(H_1(F)\)-orbit on \(F\)-points:
   • \(O_0\) (open, dim 4): \(L\) is the graph of an anti-isometry \(P\to P^\perp\) (these form one torsor), \(\ell\subset L\) arbitrary; stabilizer \(S_0=B^\Delta\), a Borel subgroup of the twisted diagonal \(\mathrm{SL}_2\), whose torus \(T^\Delta\) acts on \((\ell,L/\ell)\) by \((t,t^{-1})\).
   • \(O_c\) (dim 3): \(L=\ell_1\oplus\ell_2\), \(\ell_1\subset P\), \(\ell_2\subset P^\perp\), \(\ell\ne\ell_1,\ell_2\); stabilizer \(S_c=\{(b,b')\in B_P\times B_{P^\perp}:\text{equal torus parts}\}\), torus \(T^\Delta\) acting on \((\ell,L/\ell)\) by \((t,t)\).
   • \(O_a\), \(O_b\) (closed, dim 2): \(\ell=\ell_1\), resp. \(\ell=\ell_2\); stabilizer \(B_P\times B_{P^\perp}\). (Lagrangians meeting \(P\) meet \(P^\perp\); these are all cases.)
2. Mackey filtration (Bernstein–Zelevinsky): \(0\subset\pi_0\subset\pi_{0c}\subset\pi=I(\chi)|_{H_1}\), sections over \(O_0\), over \(O_0\cup O_c\), with \(\pi_0\cong\operatorname{c-ind}_{S_0}^{H_1}\mu_0\), \(\pi_{0c}/\pi_0\cong\operatorname{c-ind}_{S_c}^{H_1}\mu_c\), \(\pi/\pi_{0c}\cong\bigoplus_{\bullet=a,b}\operatorname{c-ind}_{S_\bullet}^{H_1}\mu_\bullet\), each \(\mu_\bullet\) the restriction of \(\chi\delta_B^{1/2}\) through the flag.
3. Ext formula. For \(\tau\) admissible and \(S=T_S\ltimes N_S\) closed: \(\operatorname{Ext}^i_{H_1}(\operatorname{c-ind}_S^{H_1}\mu,\tau)\cong\operatorname{Ext}^i_{H_1}(\tau^\vee,\operatorname{Ind}_S^{H_1}\mu^{-1}\Delta_S)\cong\operatorname{Ext}^i_S(\tau^\vee|_S,\mu^{-1}\Delta_S)\cong\operatorname{Ext}^i_{T_S}((\tau^\vee)_{N_S},\mu^{-1}\Delta_S)\), \(\Delta_S\) a fixed modulus character (duality \(\operatorname{Ext}^i(A,B^\vee)\cong\operatorname{Ext}^i(B,A^\vee)\); \((\operatorname{c-ind}\mu)^\vee\cong\operatorname{Ind}(\mu^{-1}\Delta_S)\); Shapiro; exactness of \(N_S\)-coinvariants). For a split torus, \(\operatorname{Ext}^*\) between distinct characters vanishes.
4. Apply to \(\tau=\mathbf 1\boxtimes\sigma_c\) (\(\sigma_c^\vee\cong\sigma_{c^{-1}}\), Jacquet module with characters \(c^{\pm1}\delta^{1/2}\)). By step 1 the relevant characters of \(T_S\) are, up to fixed powers of \(|\cdot|\): \(\chi_1\chi_2^{-1}\) on \(T^\Delta\subset S_0\); \(\chi_1\chi_2\) on \(T^\Delta\subset S_c\); \(\chi_1^{-1}\) (for \(O_a\)) or \(\chi_2^{-1}\) (for \(O_b\)) on the factor \(T_P\), on which \(\tau^\vee\) is trivial. Hence:
   • \(\operatorname{Ext}^*_{H_1}(\pi/\pi_{0c},\tau)=0\) unless \(\chi_1\) or \(\chi_2\) is a fixed power of \(|\cdot|\);
   • \(\operatorname{Ext}^*_{H_1}(\pi_{0c}/\pi_0,\tau)=0\) unless \(c\in\{c_2,c_2^{-1}\}\), and \(\operatorname{Hom}\ne0\) for \(c=c_2\);
   • \(\operatorname{Hom}_{H_1}(\pi_0,\tau)\ne0\) iff \(c\in\{c_1,c_1^{-1}\}\).
5. Lifting. For \(c=c_1\notin\{c_2^{\pm1}\}\): \(\operatorname{Ext}^1(\pi/\pi_0,\tau)=0\) by the long exact sequences, so the nonzero map on \(\pi_0\) extends to \(\pi\). For \(c=c_2\): \(\operatorname{Ext}^1(\pi/\pi_{0c},\tau)=0\), so the nonzero map on \(\pi_{0c}/\pi_0\) extends to \(\pi/\pi_0\), and \(\pi\twoheadrightarrow\pi/\pi_0\).
6. Genericity. The excluded set is a finite union of translated subtori: \(\chi_i\) a fixed power of \(|\cdot|\); \(c_1=c_2^{\pm1}\) (i.e. \(\chi_2^2\) or \(\chi_1^2\) a fixed power of \(|\cdot|\)); reducibility of \(I(\chi)\), \(\sigma_{c_1}\), \(\sigma_{c_2}\) (finitely many relations \(\chi^{\alpha^\vee}=|\cdot|^{\pm1}\), \(c_j=|\cdot|^{\pm1}\), \(c_j^2=1\)). \(\square\)

Remarks. Nothing depends on the characteristic of \(F\): Mackey theory, Frobenius reciprocity, Jacquet modules and the irreducibility criteria used are algebraic. The precise \(s_j\) are not needed, only that \(c_1\) and \(c_2\) move independently with \(\chi\). In dual terms (Sakellaridis's \(\iota\), Lemma 3.3), the parameter \((x,y)=(ab,a/b)\) of \(I(\chi)\) has \(x/y=b^2\) and \(xy=a^2\): the two quotients are the two ways of reading the unordered pair \(\{a,b\}\).