Audit 5.3 · Audit of A3-4: the Mackey filtration and the two SL₂-quotients of I(χ)

1. Orbits. A Lagrangian \(L\subset P\oplus P^\perp\) either meets \(P\), and then \(L=\ell_1\oplus\ell_2\) with \(\ell_1\subset P\), \(\ell_2\subset P^\perp\), or it is the graph of an anti-isometry \(P\to P^\perp\). Graphs form one torsor under \(\mathrm{Sp}(P^\perp)(F)\). This gives exactly the four orbits \(O_0,O_c,O_a,O_b\) of A3-4, each a single \(H_1(F)\)-orbit on \(F\)-points.

• For \(O_c\), the torus element \((t,t')\in T_P\times T_{P^\perp}\) rescales the parameter of \(\ell=\langle v_1+cv_2\rangle\) by \(t'/t\).
• The scheme-theoretic stabilizers are smooth in every characteristic: the graph of a homomorphism, a Borel, a product of Borels, and \(\{t=t'\}\).

2. Characters. Normalize \(I(\chi)=\{f(bg)=\chi\delta_B^{1/2}(b)f(g)\}\), with \(\delta_B^{1/2}(t_1,t_2)=|t_1|^2|t_2|\). The piece on the orbit through \(x\) is \(\operatorname{c-ind}_{S}^{H_1}\mu\), where \(S=H_1\cap x^{-1}Bx\) and \(\mu(s)=\chi\delta_B^{1/2}(xsx^{-1})\):

• \(O_0\): \(\mu_0(t)=\chi_1\chi_2^{-1}(t)|t|\), since \(T^\Delta\) acts on \((\ell,L/\ell)\) by \((t,t^{-1})\);
• \(O_c\): \(\mu_c(t)=\chi_1\chi_2(t)|t|^3\);
• \(O_a\): \(\mu_a(t,t')=\chi_1(t)\chi_2(t')|t|^2|t'|\);
• \(O_b\): \(\mu_b(t,t')=\chi_1(t')\chi_2(t)|t'|^2|t|\).

3. Ext formula. \(\operatorname{Ext}^i_{H_1}(\operatorname{c-ind}_S\mu,\tau)\cong\operatorname{Ext}^i_{T_S}((\tau^\vee)_{N_S},\mu^{-1}\delta_S)\), with \(\delta_S=|\det\operatorname{Ad}|_{\operatorname{Lie}S}|\). The chain of identifications:

• \(\tau\cong\tau^{\vee\vee}\) (admissible);
• \(\operatorname{Ext}^i(A,B^\vee)\cong\operatorname{Ext}^i(B,A^\vee)\), because both compute \(\operatorname{Ext}^i(A\otimes B,\mathbb C)\) and \(P\otimes B\) is projective for projective \(P\) (smooth representations of profinite groups are semisimple over \(\mathbb C\));
• \((\operatorname{c-ind}_S\mu)^\vee\cong\operatorname{Ind}_S(\mu^{-1}\delta_S)\) (Bernstein–Zelevinsky);
• Shapiro: \(\operatorname{Ind}\) is exact and right adjoint to the exact restriction, so it preserves injectives;
• inflation from \(T_S\) is right adjoint to the exact functor of \(N_S\)-coinvariants (\(N_S\) is a union of compact open subgroups).

All of this holds for closed subgroups of any \(\ell\)-group, so the non-reductive \(S\) and \(\operatorname{char}F=p\) are harmless. Ext between distinct characters of \(F^\times\) (or \((F^\times)^2\)) vanishes, by decomposing over \(\mathcal O^\times\)-types and a Koszul argument on \(\varpi^{\mathbb Z}\).

4. Exact Homs (no shifts).

• Open orbit. Transitivity gives \(\operatorname{c-ind}_{B^\Delta}^{H_1}\mu_0=\operatorname{c-ind}_\Delta^{H_1}(\operatorname{Ind}_{B^\Delta}^{\Delta}\mu_0)\), and \(\operatorname{Ind}_{B^\Delta}^\Delta\mu_0=\sigma_{\mu_0\delta^{-1/2}}=\sigma_{\chi_1\chi_2^{-1}}\). Since \(\Delta,H_1\) are unimodular, \(\operatorname{Hom}_{H_1}(\operatorname{c-ind}_\Delta\rho,\tau_c)=\operatorname{Hom}_\Delta(\tau_c^\vee|_\Delta,\rho^\vee)=\operatorname{Hom}_{\mathrm{SL}_2}(\sigma_{c^{-1}},\sigma_{c_1}^\vee)\). For irreducible \(\sigma\)'s this is nonzero iff \(c\in\{c_1^{\pm1}\}\), with \(c_1=\chi_1\chi_2^{-1}\), so \(s_1=0\).
• Middle orbit. \(\delta_{S_c}(t)=|t|^4\) and \(\mu_c^{-1}\delta_{S_c}=(\chi_1\chi_2)^{-1}|\cdot|\). The exponents of \((\sigma_c^\vee)_N\) are \(c^{\mp1}|\cdot|\). So Hom and Ext vanish unless \(c\in\{(\chi_1\chi_2)^{\pm1}\}\), with \(c_2=\chi_1\chi_2\) and \(s_2=0\).
• Closed orbits. \(T_P\) acts trivially on \((\tau^\vee)_{N}\). The \(T_P\)-part of \(\mu^{-1}\delta_S\) is \(\chi_1^{-1}\) for \(O_a\) and \(\chi_2^{-1}|\cdot|\) for \(O_b\). So all Ext vanish unless \(\chi_1=1\) or \(\chi_2=|\cdot|\).

5. Lifting.

• For \(c=c_1\notin\{c_2^{\pm1}\}\), the long exact sequences give \(\operatorname{Ext}^1(\pi/\pi_0,\tau)=0\), so \(\operatorname{Hom}(\pi,\tau)\cong\operatorname{Hom}(\pi_0,\tau)\), which is one-dimensional.
• For \(c=c_2\), the map on \(\pi_{0c}/\pi_0\) extends to \(\pi/\pi_0\), because \(\operatorname{Ext}^1(\pi/\pi_{0c},\tau)=0\).
• \(c_1=c_2^{\pm1}\) iff \(\chi_2^2=1\) or \(\chi_1^2=1\).
• The \(R\)-group reducibility points of \(I(\chi)\) (some \(\chi^{\alpha^\vee}\) quadratic) lie in the union of \(\{\chi_i^2=1\}\) and \(\{c_j^2=1\}\). So the excluded set stated above is a proper closed subset.

6. Dual reading. \(c_1(\varpi)=x/y=b^2\) and \(c_2(\varpi)=xy=a^2\) for the Satake parameter \((x,y)=(ab,a/b)\). Since \(s_1=s_2=0\), the two quotients have \(\mathrm{SL}_2\)-parameters exactly \(b\) and \(a\). This confirms A3-5's remark (i) identifying them with the \(X\)-parameters \((a,b,a)\) and \((b,a,b)\).

7. Test. The finite-field analogue is computed in Computation 5.4: it has the same orbit and character bookkeeping, with no shifts and no Ext terms. It agrees.