# Claims to audit (verbatim from the agents' returns)

Each block gives the step id, its title, the claimed statement and the argument as written. Audit each one independently.


## A3-1 · (SG) forces \(\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\le1\) over every local field

**Kind/status as reported:** lemma / proved

**Builds on:** —

### Statement

Let \(F\) be a non-archimedean local field (either characteristic), \(G\) reductive over \(\mathcal O\), \(K=G(\mathcal O)\), and \(X=G/H\) with \(H\) reductive, such that \(X(F)=G(F)x_0\) is one orbit. Then:

(a) If \(\mathcal H_K\ast\varphi_0=C_K\), then \(\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\le1\) for every character \(\chi:\mathcal H_K\to\mathbb C\), and every \(\mathcal H_K\)-endomorphism of \(C_K\) is \(h_0\ast(-)\) for some \(h_0\in\mathcal H_K\).

(b) For an irreducible unramified \(\pi\) with \(\pi^K=\mathbb Cv_0\), \(\ell\mapsto\Psi_\ell\), \(\Psi_\ell(gx_0)=\ell(\pi(g)^{-1}v_0)\), embeds \(\operatorname{Hom}_{H(F)}(\pi,\mathbb C)\) into the \(K\)-invariant functions with \(f\ast\Psi=\chi_\pi(f)\Psi\), and \(\Psi\mapsto\Lambda_\Psi=\int_X(-)\Psi\,dx\) identifies the latter with \(\operatorname{Hom}_{\mathcal H_K}(C_K,\chi_\pi\circ\vee)\), \(f^\vee(y)=f(y^{-1})\).

(c) Hence (SG) implies \(\dim\operatorname{Hom}_{H(F)}(\pi,\mathbb C)\le1\) for every irreducible unramified \(\pi\); and if \(\dim\operatorname{Hom}_{H(F)}(\pi,\mathbb C)\ge2\) for one such \(\pi\), then (SG) fails over \(F\).

### Argument

**Proof.**

1. *Cyclicity.* (SG) says that \(h\mapsto h\ast\varphi_0\) maps \(\mathcal H_K\) onto \(C_K\); so \(C_K\cong\mathcal H_K/I\), \(I=\operatorname{Ann}(\varphi_0)\). The algebra \(\mathcal H_K\) is commutative (Satake isomorphism).
2. *Eigenfunctionals and endomorphisms.* If \(\Lambda\in\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\), then \(\Lambda(h\ast\varphi_0)=\chi(h)\Lambda(\varphi_0)\), so \(\Lambda\) is determined by the number \(\Lambda(\varphi_0)\). An \(\mathcal H_K\)-endomorphism \(D\) with \(D\varphi_0=h_0\ast\varphi_0\) satisfies \(D(h\ast\varphi_0)=hh_0\ast\varphi_0\). This is (a).
3. *Periods give eigenfunctions.* \(\Psi_\ell\) is well defined because \(\ell\) is \(H(F)\)-invariant, and \(K\)-invariant because \(v_0\) is. With \(\pi(f)=\int_Gf(y)\pi(y)\,dy\) (acting on \(\pi^K\) by \(\chi_\pi(f)\)),
$$(f\ast\Psi_\ell)(gx_0)=\sum_{yK}f(y)\,\ell\big(\pi(g)^{-1}\pi(y)v_0\big)=\ell\big(\pi(g)^{-1}\pi(f)v_0\big)=\chi_\pi(f)\,\Psi_\ell(gx_0).$$
If \(\Psi_\ell=0\), then \(\ell\) vanishes on \(\pi(G)v_0\), which spans \(\pi\) because \(\pi\) is irreducible; so \(\ell=0\).
4. *Eigenfunctions give eigenfunctionals.* \(G(F)\) and \(H(F)\) are unimodular, so \(X(F)=G(F)/H(F)\) carries an invariant measure \(dx\). Substituting \(y\mapsto y^{-1}\) (unimodularity, \(\operatorname{vol}K=1\)) gives \(\Lambda_\Psi(f\ast\varphi)=\Lambda_{f^\vee\ast\Psi}(\varphi)\). Since \(C_K\) has the basis \(\mathbf 1_{Kx}\) and \(\Lambda_\Psi(\mathbf 1_{Kx})=\operatorname{vol}(Kx)\Psi(x)\) with \(\operatorname{vol}(Kx)>0\) (orbits are compact open), \(\Psi\mapsto\Lambda_\Psi\) is a bijection from \(K\)-invariant functions onto \(\operatorname{Hom}_{\mathbb C}(C_K,\mathbb C)\); it carries \(\chi_\pi\)-eigenfunctions onto \(\chi_\pi\circ\vee\)-eigenfunctionals. This is (b).
5. (a) and (b) give (c). \(\square\)

**What dimension two needs.** Two linearly independent \(H(F)\)-invariant functionals on a single irreducible unramified \(\pi\), equivalently two independent \(K\)-invariant eigenfunctions with one eigencharacter. When \(H\supseteq H_1\times M\) with \(M\) acting on \(G/H_1\) from the right, a natural source is a \(\pi\) with two non-isomorphic irreducible \(M\)-quotients of its \(H_1\)-coinvariants ([[A3-4]], [[A3-5]]). Equivalently, by (a), an \(\mathcal H_K\)-endomorphism of \(C_K\) that is not multiplication by an element of \(\mathcal H_K\).

**Scope.** Only Frobenius reciprocity-type identities and the invariant measure are used: the lemma holds over \(\mathbb F_q(\!(t)\!)\), its unramified extensions \(F_n\), and \(p\)-adic fields alike. Sakellaridis's Corollary 8.0.4(2) (sak.txt, §8) states the same implication under his assumptions; here it is unconditional.

**Sources cited:** sakellaridis-v3.pdf, Cor. 8.0.4; trace-pages/_definitions.html (Hecke action, vol K = 1)


## A3-2 · Thick closure versus span: twisted traces, restriction of class functions, fusion

**Kind/status as reported:** lemma / proved

**Builds on:** A3-1

### Statement

(a) Under D0–D3 (or their analogues for a spherical \(X\)), for \(\lambda\in\Lambda_T^+\) and \(e\in\operatorname{End}(V_\lambda\ast\mathrm{IC}_0)\) put \(t_{\lambda,e}=\operatorname{tr}(\Phi_\lambda\circ\mathrm{Fr}^*e)\in C_K\otimes\overline{\mathbb Q}_\ell\). Then
$$\operatorname{span}_{\lambda,e}t_{\lambda,e}\ \supseteq\ \operatorname{span}\{\operatorname{tr}\mathrm{IC}_\nu:\nu\in\mathcal M\},\qquad \mathcal H_K\ast\varphi_0\otimes\overline{\mathbb Q}_\ell=\operatorname{span}_\lambda t_{\lambda,\mathrm{id}} .$$
So \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) makes the twisted traces span \(C_K\), while (SG) asks the same of the untwisted ones.

(b) Let \(\check G_X\subseteq\check G\) be a closed reductive subgroup over \(\overline{\mathbb Q}_\ell\). Every irreducible representation of \(\check G_X\) is a direct summand of \(\operatorname{Res}V\) for some irreducible \(V\) of \(\check G\); the map induced on \(K_0\otimes\overline{\mathbb Q}_\ell\) is restriction of class functions \(\overline{\mathbb Q}_\ell[\check G]^{\check G}\to\overline{\mathbb Q}_\ell[\check G_X]^{\check G_X}\), and it is not surjective as soon as two semisimple elements of \(\check G_X\) are \(\check G\)-conjugate but not \(\check G_X\)-conjugate (fusion).

(c) Each of the following, together with \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\), forces (SG) to fail: (i) a nonzero \(\psi:\Lambda_S^+\to\overline{\mathbb Q}_\ell\) with \(\sum_\nu\psi(\nu)m^{(1)}_{\nu\lambda}=0\) for all \(\lambda\) (for instance two proportional rows); (ii) a character \(\chi\) with \(\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\ge2\); (iii) every \(V_\lambda\ast\mathrm{IC}_0\) perverse, every top-dimensional component of every convolution fibre defined over \(\mathbb F_q\), and the matrix \(c_{\nu\lambda}\) of top-component counts equal to the branching matrix of a subgroup \(\check G_X\subseteq\check G\) with fusion; in case (iii), \((\mathrm{SG})_n\) fails for every \(n\) and every \(q\).

### Argument

**Proof of (a).**

1. By D2, over \(k\), \(V:=V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_{\nu,i}\mathrm{IC}_\nu[-i]\otimes E_{\nu,i}(\lambda)\). By D1 the Weil structure \(\Phi\) preserves the perverse truncations, so in this decomposition it is block-triangular for the perverse degree, and on \({}^p\mathcal H^i(V)\) it is \(\bigoplus_\nu\Phi^{\mathrm{can}}_\nu\otimes A_{\nu,i}(\lambda)\) (IIb, step 4).
2. Fix \(\nu\in\mathcal M\) and \(i\) with \(E=E_{\nu,i}(\lambda)\ne0\), and let \(e=\mathrm{id}_{\mathrm{IC}_\nu[-i]}\otimes\varepsilon\) on that summand and \(0\) on the others, \(\varepsilon\in\operatorname{End}(E)\). For \(x\in X(F)\) the point is Frobenius-fixed, so \((\mathrm{Fr}^*e)_x=e_x\) and \(t_{\lambda,e}(x)=\sum_j(-1)^j\operatorname{tr}(e_x\Phi_x\mid\mathcal H^j)\) by cyclicity of the trace.
3. \(e_x\Phi_x\) is the compression of \(\Phi_x\) to one summand; by step 1 only the diagonal block \((\Phi^{\mathrm{can}}_\nu)_x\otimes A_{\nu,i}(\lambda)\) survives. Hence \(t_{\lambda,e}=(-1)^i\operatorname{tr}(\varepsilon A_{\nu,i}(\lambda))\operatorname{tr}\mathrm{IC}_\nu\). With \(\varepsilon=A_{\nu,i}(\lambda)^{-1}E_{11}\) the coefficient is \(\pm1\), so \(\operatorname{tr}\mathrm{IC}_\nu\) lies in the span.
4. \(t_{\lambda,\mathrm{id}}=\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)=\operatorname{tr}(V_\lambda)\ast\varphi_0\), and these span \(\mathcal H_K\ast\varphi_0\otimes\overline{\mathbb Q}_\ell\) (IIa). \(\square\)

**Proof of (b).** \(\overline{\mathbb Q}_\ell[\check G]\to\overline{\mathbb Q}_\ell[\check G_X]\) is onto (closed subgroup), and \(\overline{\mathbb Q}_\ell[\check G_X]\) contains every irreducible \(W\) of \(\check G_X\) (matrix coefficients). So \(W\) is a quotient, hence (complete reducibility) a summand, of the restriction of a finite-dimensional \(\check G\)-subrepresentation of \(\overline{\mathbb Q}_\ell[\check G]\). On \(K_0\otimes\overline{\mathbb Q}_\ell\), classes are characters and \(\operatorname{Res}\) is restriction of functions. If \(s,s'\in\check G_X\) are semisimple, \(\check G\)-conjugate and not \(\check G_X\)-conjugate, then every restricted character takes equal values at \(s,s'\), while characters of \(\check G_X\) separate semisimple classes; so some \([W]\) is not in the image. \(\square\)

**Proof of (c).** (i) By IIb (iii), (SG) is the spanning of the columns of \(m^{(1)}\); a nonzero \(\psi\) vanishing on all columns prevents it. (ii) is [[A3-1]](a). (iii) In the perverse case only \(i=0\) occurs and \(E_{\nu,0}(\lambda)\) is the top cohomology of the fibre over \(x_\nu\), on which \(\mathrm{Fr}^n\) acts by \(q^{nf_{\nu\lambda}}\) on each rational component, \(f_{\nu\lambda}=\tfrac12(\langle2\rho,\lambda\rangle-\delta(\nu))\). So \(m^{(n)}=D_n\,c\,D'_n\) with invertible diagonal \(D_n=\operatorname{diag}(q^{-n\delta(\nu)/2})\), \(D_n'=\operatorname{diag}(q^{n\langle\rho,\lambda\rangle})\), and the columns of \(m^{(n)}\) span iff those of \(c\) do; by (b) they do not. Every row of a branching matrix of a subgroup is nonzero, so \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) by Proposition I. \(\square\)

**The mechanism in words.** \(\operatorname{Tr}(\mathrm{Fr},-)\) is functorial for the module functor \(V\mapsto V\ast\mathrm{IC}_0\) and turns it into \(f\mapsto f\ast\varphi_0\) (IIa). Thick generation adds retracts; the trace of a Karoubi-generating functor need not be onto, because the trace of the target receives Frobenius-twisted traces of idempotents \(e\) of \(V_\lambda\ast\mathrm{IC}_0\) that do not come from \(\operatorname{End}(V_\lambda)=\overline{\mathbb Q}_\ell\). Laumon's injectivity (Thm 1.1.2) concerns classes over all \(n\) and does not help: in case (iii) the failure is the same for every \(n\). [[A3-9]] realizes case (iii) for Sakellaridis's \(X\).

**Sources cited:** trace-pages/trace-b.html (IIb steps 4-7); trace-pages/trace-a.html (IIa); trace-pages/_definitions.html (Proposition I); laumon-1987.pdf, Thm 1.1.2


## A3-3 · A model of \(X=\mathrm{SL}_2^3\backslash\mathrm{Sp}_4^2\) and its fusion divisor

**Kind/status as reported:** lemma / proved

**Builds on:** —

### Statement

Let \(V_1,V_2\) be symplectic spaces of dimension 4 over \(\mathbb F_q\), \(q\) odd, \(G=\mathrm{Sp}(V_1)\times\mathrm{Sp}(V_2)\) (split). Fix nondegenerate planes \(W_i^0\subset V_i\) and an isometry \(\varphi_0:W_1^0\to W_2^0\), and put \(x_0=(W_1^0,W_2^0,\varphi_0)\).

(a) \(H:=\operatorname{Stab}_G(x_0)=\mathrm{Sp}(W_1^{0\perp})\times\mathrm{Sp}(W_2^{0\perp})\times\{(g,\varphi_0g\varphi_0^{-1}):g\in\mathrm{Sp}(W_1^0)\}\cong\mathrm{SL}_2^3\): the first two factors are \(\mathrm{Sp}_2\) inside each \(\mathrm{Sp}_4\), the third is diagonal in both — Sakellaridis's \(H\).

(b) For every field \(E\supseteq\mathbb F_q\), \(X(E)=G(E)/H(E)=\{(W_1,W_2,\varphi)\}\), \(W_i\subset V_i\otimes E\) nondegenerate planes, \(\varphi\) an isometry. Equivalently \(X=(Z_1\times Z_2)/\mathrm{SL}_2\) with \(Z_i=\{(e,f)\in V_i^2:\omega_i(e,f)=1\}\cong\mathrm{Sp}_4/\mathrm{Sp}_2\), and \(X\cong\{\Gamma\in\mathrm{IG}(2,V_1\oplus V_2;\omega_1-\omega_2):\omega_1|_{\mathrm{pr}_1\Gamma}\ne0\}\), an affine open subset (\(\Gamma\) = graph of \(\varphi\)). \(\dim X=11\).

(c) (Cited.) \(X\) is spherical of rank 3 with \(\delta(X)\) trivial, and its dual group is the image of \(\iota:\mathrm{SL}_2\times\mathrm{SL}_2^{\mathrm{diag}}\times\mathrm{SL}_2\to\mathrm{SO}_4\times\mathrm{SO}_4\subset\mathrm{SO}_5\times\mathrm{SO}_5=\check G\), \((g_1,g_2,g_3)\mapsto([g_1,g_2],[g_2,g_3])\), with \(\ker\iota=\{\pm(1,1,1)\}\).

(d) For \(a,b\in\overline{\mathbb Q}_\ell^\times\) with \(a\ne b^{\pm1}\), the torus points \((a,b,a)\) and \((b,a,b)\) of \(\check G_X\) are \(\check G\)-conjugate but not \(\check G_X\)-conjugate: fusion along a divisor, with no \(q\) in it.

### Argument

**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 [[A3-2]](b) and the shape the task asks for.

**Sources cited:** sakellaridis-v3.pdf, Example 8.0.1 (p. 64); sakellaridis-v3.pdf, Theorem 8.0.2


## A3-4 · Generic \(I(\chi)\) of \(\mathrm{Sp}_4\) has two \(\mathrm{SL}_2\)-quotients

**Kind/status as reported:** lemma / proved

**Builds on:** A3-3

### Statement

Let \(F\) be any non-archimedean local field, \(V=P\oplus P^\perp\) a 4-dimensional symplectic space over \(F\) with \(P\) a nondegenerate plane, \(H_1=\mathrm{Sp}(P)\times\mathrm{Sp}(P^\perp)\cong\mathrm{SL}_2\times\mathrm{SL}_2\). Let \(B\subset\mathrm{Sp}(V)\) be the stabilizer of an isotropic flag \(\ell\subset L\), \(\chi=(\chi_1,\chi_2)\) an unramified character of \(B/U\) (acting on \(\ell\), \(L/\ell\)), \(I(\chi)\) the normalized induction; for a character \(c\) of \(F^\times\), \(\sigma_c\) is the normalized principal series of \(\mathrm{SL}_2(F)\) induced from \(\operatorname{diag}(t,t^{-1})\mapsto c(t)\).

There are constants \(s_1,s_2\in\tfrac12\mathbb Z\), depending only on the groups, such that for \((\chi_1(\varpi),\chi_2(\varpi))\) outside a proper Zariski-closed subset of \((\mathbb C^\times)^2\), with \(c_1=\chi_1\chi_2^{-1}|\cdot|^{s_1}\) and \(c_2=\chi_1\chi_2|\cdot|^{s_2}\):
$$\operatorname{Hom}_{H_1}\big(I(\chi),\mathbf 1\boxtimes\sigma_{c_j}\big)\ne0\quad(j=1,2),\qquad \sigma_{c_1}\not\cong\sigma_{c_2}\ \text{irreducible},\qquad I(\chi)\ \text{irreducible}.$$

### Argument

**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 \(\iota\), [[A3-3]]), 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 \(\{a,b\}\).

**Sources cited:** Bernstein–Zelevinsky, geometric lemma / Mackey theory (standard); sakellaridis-v3.pdf, Example 8.0.1


## A3-5 · (SG) fails for Sakellaridis's \(X\) over \(\mathbb F_q(\!(t)\!)\), every \(q\)

**Kind/status as reported:** lemma / proved

**Builds on:** A3-1, A3-3, A3-4

### Statement

Let \(X=G/H\) be as in [[A3-3]] and \(F\) any non-archimedean local field with ring of integers \(\mathcal O\) — in particular \(F=\mathbb F_{q^n}(\!(t)\!)\) for any \(q\) and \(n\). For \(\chi\) outside a proper Zariski-closed subset of the unramified characters of the torus of \(\mathrm{Sp}_4\), let \(\pi=I(\chi)\) and \(\Pi=\pi\boxtimes\pi\). Then
$$\dim\operatorname{Hom}_{H(F)}(\Pi,\mathbb C)\ge2,\qquad\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi_\Pi\circ\vee)\ge2,\qquad \mathcal H_K\ast\varphi_0\subsetneq C_K .$$
So \((\mathrm{SG})\) and \((\mathrm{SG})_n\) fail for \(X\) for every \(q\) and every \(n\).

### Argument

**Proof.**

1. *Setup.* \(H(F)=\mathrm{Sp}(P_1)\times\mathrm{Sp}(P_2)\times\mathrm{SL}_2^{(3)}\) with \(P_i=W_i^{0\perp}\) and \(\mathrm{SL}_2^{(3)}=\{(g,\varphi_0g\varphi_0^{-1})\}\subset\mathrm{Sp}(W_1^0)\times\mathrm{Sp}(W_2^0)\) ([[A3-3]](a)); \(X(F)=G(F)/H(F)\) ([[A3-3]](b)).
2. *Two quotients on each side.* Apply [[A3-4]] to \(V_i=P_i\oplus W_i^0\): for generic \(\chi\) there are nonzero maps \(f_j:\pi\to\sigma_{c_j}\) (\(j=1,2\)), invariant under \(\mathrm{Sp}(P_1)\) and equivariant for \(\mathrm{Sp}(W_1^0)\), and likewise \(f_j':\pi\to\sigma_{c_j}\) for the second copy, with \(\mathrm{Sp}(W_2^0)\) identified with \(\mathrm{Sp}(W_1^0)\) through \(\varphi_0\).
3. *Functionals.* \(\sigma_{c_j}\) is irreducible and \(\sigma_{c_j}^\vee\cong\sigma_{c_j^{-1}}\cong\sigma_{c_j}\), so there is a nonzero \(\mathrm{SL}_2(F)\)-invariant pairing \(\beta_j\) on \(\sigma_{c_j}\). Put \(\ell_j(v\otimes w)=\beta_j(f_j(v),f_j'(w))\). For \((h_1,h_2,g)\in H(F)\): \(h_1,h_2\) act trivially after \(f_j,f_j'\), and \(g\) acts diagonally, preserved by \(\beta_j\). So \(\ell_j\in\operatorname{Hom}_{H(F)}(\Pi,\mathbb C)\).
4. *Independence.* \(F_\bullet=(f_1,f_2):\pi\to\sigma_{c_1}\oplus\sigma_{c_2}\) is onto: its image is an \(\mathrm{SL}_2\)-subrepresentation projecting onto both factors, which are irreducible and non-isomorphic ([[A3-4]]). Likewise \(F'_\bullet\). So \(\Pi\twoheadrightarrow(\sigma_{c_1}\oplus\sigma_{c_2})^{\otimes2}\), and \(\ell_1,\ell_2\) are pullbacks of \(\beta_1\circ\mathrm{pr}_{11}\) and \(\beta_2\circ\mathrm{pr}_{22}\), which are nonzero on different direct summands. Pullback along a surjection preserves independence.
5. *Conclusion.* \(\Pi\) is irreducible and unramified for \(G(F)\) (outer tensor product of irreducible unramified representations). By [[A3-1]](b), \(\ell_1,\ell_2\) give two independent \(K\)-invariant \(\chi_\Pi\)-eigenfunctions and two independent eigenfunctionals on \(C_K\); by [[A3-1]](c) \(C_K\) is not cyclic, so \(\mathcal H_K\ast\varphi_0\ne C_K\). The same argument runs over every \(F_n\). \(\square\)

**Remarks.** (i) This is the multiplicity two predicted by Sakellaridis's Example 8.0.1 and his remark after Corollary 8.0.4: the \(X\)-parameters \((a,b,a)\) and \((b,a,b)\) of [[A3-3]](d) correspond to \(\sigma_{c_1}\) and \(\sigma_{c_2}\). (ii) The failure holds for every residue characteristic, so it lives in the same characteristic as the sheaf side over \(\overline{\mathbb F}_q\). (iii) It occurs on a 2-parameter family of eigencharacters (the fused divisor), not at isolated \(q\)-dependent points as for \(E_6/F_4\).

**Sources cited:** sakellaridis-v3.pdf, Example 8.0.1 and remark after Cor. 8.0.4


## A3-6 · Orbits \((d_1,d_2,e)\), connected stabilizers, \(\delta=3d_1+3d_2+e\)

**Kind/status as reported:** lemma / proved

**Builds on:** A3-3

### Statement

Let \(\kappa\) be \(\mathbb F_{q^n}\) or \(k=\overline{\mathbb F}_q\), \(L=L_1\oplus L_2\) the standard self-dual \(\kappa[\![t]\!]\)-lattices, \(K_\kappa=G(\kappa[\![t]\!])\). For \(x=(W_1,W_2,\varphi)\in X(\kappa(\!(t)\!))\) let \(N_i=W_i\cap L_i\), let \(d_i\ge0\) be the valuation of \(\omega\) on \(N_i\), and let \((t^{m_1},t^{m_2})\), \(m_1\ge m_2\), be the elementary divisors of \(\varphi(N_1)\) relative to \(N_2\) (so \(m_1+m_2=d_1-d_2\)); put \(e=m_1-m_2\).

(a) \(x\mapsto(d_1,d_2,e)\) is a bijection \(K_\kappa\backslash X(\kappa(\!(t)\!))\to\Lambda_X^+:=\{(d_1,d_2,e)\in\mathbb Z_{\ge0}^3:e\equiv d_1-d_2\ (2)\}\); \(X(\kappa[\![t]\!])\) is the orbit \((0,0,0)\).

(b) For \(x\in LX(k)\), \(\operatorname{Stab}_{L^+G}(x)\) is connected; hence \(\mathcal O_\nu(\mathbb F_q)\) is a single \(K\)-orbit (Lang).

(c) \(\delta(\nu):=\dim\mathcal O_\nu-\dim\mathcal O_{(0,0,0)}=3d_1+3d_2+e\).

(d) Every \(L^+G\)-orbit closure in \(LX\) contains finitely many orbits.

(e) For \(\lambda=((a_1,b_1),(a_2,b_2))\in\Lambda_T^+\) put \(\varphi(\lambda)=(a_1+b_1,\ a_2+b_2,\ (a_1-b_1)+(a_2-b_2))\); then \(\delta(\varphi(\lambda))=\langle2\rho,\lambda\rangle\). A label \(\nu\) lies in \(\varphi(\Lambda_T^+)\) iff \(e\le d_1+d_2\) and not (\(d_1,d_2\) odd and \(e=0\)). Uncovered examples: \((0,0,2k)\), \(k\ge1\); \((1,1,0)\); \((1,0,3)\).

### Argument

**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 [[A3-7]].
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 [[A3-7]], 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 ([[A3-8]]). \(\nu=\varphi(\lambda)\) 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.

**Sources cited:** sakellaridis-v3.pdf, Example 8.0.1 (spherical roots); chen-yi-v4.pdf, Prop. 17 and Thm 18 (the symmetric analogue)


## A3-7 · Convolution fibres and the counts \((T_\lambda\ast\varphi_0)(x_\nu)\)

**Kind/status as reported:** computation / proved

**Builds on:** A3-6

### Statement

Fix \(x=x_\nu\), \(\nu=(d_1,d_2,e)\), and \(\lambda=(\lambda^1,\lambda^2)\in\Lambda_T^+\).

(a) \(x\) is integral for \(L'=(L_1',L_2')\) iff \(L_1'=C\oplus Q_1\), \(L_2'=\varphi(C)\oplus Q_2\) with \(C\subset W_1\), \(Q_i\subset W_i^\perp\) self-dual lattices. So \(F^\lambda_\nu:=\{L'\in\mathrm{Gr}^\lambda: x\ \text{integral for}\ L'\}\cong\{(C,Q_1,Q_2):\operatorname{pos}_{L_1}(C\oplus Q_1)=\lambda^1,\ \operatorname{pos}_{L_2}(\varphi C\oplus Q_2)=\lambda^2\}\).

(b) (Position formula.) For a self-dual \(L\) with \(N=W\cap L\) of type \(t^d\), and self-dual \(C\subset W\), \(Q\subset W^\perp\): write \(D_C=d+2u\) for the tree distance from \(C\) to the class of \(N\), \(D_Q=d+2w\) for that from \(Q\) to the class of \(N'\), and \(\sigma\) for the distance between the projections of \(C\) and \(\varepsilon^{-1}Q\) to the ball of radius \(d\). Then
$$\operatorname{pos}_L(C\oplus Q)=\big(d+\max(u,w),\ u_++w_++\tfrac\sigma2-\max(u,w)\big).$$

(c) (Counts.) With \(P_1=[N_1]\), \(P_2=[\varphi^{-1}N_2]\) at distance \(e\) in the tree of \(\mathrm{SL}(W_1)\), \(D^i=\operatorname{dist}(C,P_i)\):
$$(T_\lambda\ast\varphi_0)(x_\nu)=\#F^\lambda_\nu(\mathbb F_q)=\sum_{D^1,D^2}h_e(D^1,D^2)\,g_{d_1}(D^1,\lambda^1)\,g_{d_2}(D^2,\lambda^2),$$
where \(h_e(D^1,D^2)\) counts vertices at distances \(D^1,D^2\) from two vertices at distance \(e\) (\(1\), \((q-1)q^{h-1}\), \(q^h\) or \((q+1)q^{h-1}\), \(h=\tfrac12(D^1+D^2-e)\)) and \(g_d(D,\lambda)\) counts the \(Q\) of (b) with position \(\lambda\).

### Argument

**Proof.**

1. *(a).* Integrality for \(L'\) means label \((0,0,0)\) relative to \(L'\) ([[A3-6]](a)): \(W_i\cap L_i'\) self-dual — which forces \(L_i'=(W_i\cap L_i')\oplus(W_i^\perp\cap L_i')\), the gluing being trivial — and \(\varphi(W_1\cap L_1')=W_2\cap L_2'\).
2. *\(\lambda_1\) in (b).* For self-dual \(L,L'\) at position \((\lambda_1,\lambda_2)\), \(\lambda_1=\min\{n:t^nL'\subseteq L\}\) and \(\lambda_1+\lambda_2=\operatorname{len}(L'/L\cap L')\). Since \(L\cap W=N\) and \(L\cap W^\perp=N'\), \(t^n(C\oplus Q)\subseteq L\) iff \(t^nC\subseteq N\) and \(t^nQ\subseteq N'\). In a common adapted basis \(C=\langle t^\alpha e',t^\beta f'\rangle\), \(N=\langle e',f'\rangle\), \(\alpha+\beta=-d\), \(|\alpha-\beta|=D_C\); so \(\min\{n:t^nC\subseteq N\}=\tfrac12(d+D_C)=d+u\). Same for \(Q\).
3. *\(\lambda_1+\lambda_2\) in (b).* As \(L\subseteq N^\#\oplus N'^\#\) is the preimage of the graph of \(\varepsilon\),
$$\operatorname{len}\frac{C\oplus Q}{L\cap(C\oplus Q)}=\operatorname{len}\frac{C}{C\cap N}+\operatorname{len}\frac{Q}{Q\cap N'}-\operatorname{len}(\bar A_C\cap\varepsilon^{-1}\bar A_Q),$$
with \(\bar A_C\) the image of \(C\cap N^\#\) in \(N^\#/N\). Here \(\operatorname{len}(C/C\cap N)=(-\alpha)_++(-\beta)_+=d+u_+\); \(\bar A_C\) is Lagrangian, i.e. \(C':=(C\cap N^\#)+N\) is the self-dual lattice nearest to \(C\) in the ball of radius \(d\); and for Lagrangians \(\operatorname{len}(\bar A_C\cap\varepsilon^{-1}\bar A_Q)=d-\tfrac\sigma2\). Summing gives \(d+u_++w_++\tfrac\sigma2\).
4. *(c).* Lang's theorem identifies \(\mathrm{Gr}^\lambda(\mathbb F_q)\) with \(Kt^\lambda K/K\), so \(\#F_\nu^\lambda(\mathbb F_q)=\#\{gK\subseteq Kt^\lambda K:g^{-1}x_\nu\in X(\mathcal O)\}=(T_\lambda\ast\varphi_0)(x_\nu)\). The count of \(C\) is a tree count; for fixed \(C\), the number of \(Q_i\) depends only on \(D^i\) (the stabilizer of the centre is transitive on vertices at a given distance), which gives the product formula. \(g_d\) is a sum over \((D_Q,\sigma)\) of a tree count \(h_\rho(\rho',\sigma)\), \(\rho=\min(D_C,d)\), \(\rho'=\min(D_Q,d)\), times the outward extension \((q\ \text{or}\ q+1)q^{D_Q-d-1}\) when \(D_Q>d\). \(\square\)

**Check (computation).** The script fibres.py implements (b)–(c) with polynomials in \(q\); mass.py tests the identity \(\sum_\nu(T_\lambda\ast\varphi_0)(x_\nu)\operatorname{vol}(Kx_\nu)/\operatorname{vol}(Kx_0)=|Kt^\lambda K/K|\) (integration of \(T_\lambda\ast\varphi_0\) over \(X(F)\)), with the volumes of [[A3-6]] step 4 and \(|Kt^{(a,b)}K/K|=q^{4a+2b}W(q^{-1})/W_\lambda(q^{-1})\) for \(\mathrm{Sp}_4\). It holds exactly for \(\lambda=((1,0),0),((1,1),0),((1,0),(1,0)),((2,1),0),((1,1),(1,0)),((2,0),(1,1)),((2,1),(1,0))\) at \(q=3,5\) (e.g. \(14400\) for \(((1,0),(1,0))\), \(q=3\)). This tests the position formula, the orbit labels and the volumes simultaneously; it is a check, not part of the proof.

**Sources cited:** scratchpad/solve/A3/fibres.py; scratchpad/solve/A3/mass.py; zhu-satake-v2.pdf, §5.6 (trace of convolution, Lang)


## A3-8 · Each \(m_\lambda\) is semismall; multiplicities are dual branching numbers

**Kind/status as reported:** lemma / proved

**Builds on:** A3-6, A3-7

### Statement

For \(\lambda\in\Lambda_T^+\) and \(\nu\in\Lambda_X^+\) let \(r_\lambda(\nu)=2\dim F^\lambda_\nu-(\langle2\rho,\lambda\rangle-\delta(\nu))\).

(a) \(r_\lambda(\nu)\le0\) for all \(\lambda,\nu\) (with \(F^\lambda_\nu\ne\varnothing\)).

(b) For \(d\ge0\) and \((a,b)\) dominant put \(Z_d(a,b)=\{D\equiv d\ (2):|2b-d|\le D\le\min(2b+d,\,2a-d)\}\). Then \(r_\lambda(\nu)=0\) iff there are \(D^i\in Z_{d_i}(\lambda^i)\) with \(|D^1-D^2|\le e\le D^1+D^2\), and the number \(c_{\nu\lambda}\) of top-dimensional irreducible components of \(F^\lambda_\nu\) (all geometrically irreducible and defined over \(\mathbb F_q\)) is the number of such pairs.

(c) \(c_{\nu\lambda}=\big[\operatorname{Res}^{\mathrm{SO}_5\times\mathrm{SO}_5}_{\iota(\check G_X)}V_\lambda:V^{\check G_X}_{(d_1,e,d_2)}\big]\), with \(\iota\) as in [[A3-3]](c) and \(V_{(n_1,n_2,n_3)}\) the irreducible \(\mathrm{SL}_2^3\)-module (outer, middle, outer).

(d) Every \(\nu\) is relevant for some \(\lambda\): \(\nu=\varphi(\lambda)\) has \(F^\lambda_\nu=\) point; if \(e>d_1+d_2\), take \(k=\tfrac12(e-d_1-d_2)\) and \(\lambda=((k+d_1,k),(d_2,0))\); if \(d_1,d_2\) are odd and \(e=0\), take \(\lambda=((d_1,0),(\tfrac{d_1+d_2}2,\tfrac{|d_1-d_2|}2))\). E.g. \((0,0,2)\) is relevant for \(((1,1),0)\), with \(F\cong\mathrm{Gr}^{(1,-1)}_{\mathrm{SL}_2}\) of dimension 2 and codimension 4.

### Argument

**Proof.**

1. *Dimensions.* By [[A3-7]](c) \(F^\lambda_\nu\) is a finite disjoint union of locally closed pieces indexed by \((D^1,D^2)\) and, on each side, \((D_{Q_i},\sigma_i)\); each piece is a tower of \(\mathbb A^1\)-, \(\mathbb G_m\)- or \(\mathbb P^1\)-fibrations over \(\mathbb F_q\), of dimension \(h+\dim_1+\dim_2\), \(h=\tfrac12(D^1+D^2-e)\), \(\dim_i=\tfrac12(\rho_i'+\sigma_i-\rho_i)+2(w_i)_+\) (height of \(Q_i'\) over the geodesic, plus outward extension).
2. *One side.* Put \(\Phi=D+2\dim_Q-(4a+2b-3d)\) for a configuration with position \((a,b)\). Substituting \(a=d+\max(u,w)\), \(b=u_++w_++\tfrac\sigma2-\max(u,w)\), \(D=d+2u\), \(\rho'-\rho=2\min(w,0)-2\min(u,0)\):
$$\Phi=2u+2\min(w,0)-2\min(u,0)+2w_+-2u_+-2\max(u,w)=-2(u-w)_+\le0 .$$
3. *Both sides.* \(\delta(\nu)=3d_1+3d_2+e\) ([[A3-6]](c)) gives \(2(h+\dim_1+\dim_2)-\langle2\rho,\lambda\rangle+\delta(\nu)=\Phi_1+\Phi_2\): the \(e\)'s cancel. So (a), and \(r_\lambda(\nu)=0\) iff some piece has \(w_i\ge u_i\) on both sides.
4. *Zero set.* With \(w=a-d\), the condition \(w\ge u\) is \(D\le2a-d\); \(\sigma/2=b-u_++\min(w,0)\) must satisfy the triangle bounds \(|\rho-\rho'|\le\sigma\le\rho+\rho'\), which in the cases \(u\gtrless0\), \(w\gtrless0\) all reduce to \(|2b-d|\le D\le2b+d\). For given \(D\in Z_d(a,b)\) the top configuration \((w,\sigma)\) is unique and its count has leading coefficient 1. With the \(C\)-count (leading coefficient 1 for \(h\ge0\), nonempty iff \(|D^1-D^2|\le e\le D^1+D^2\)) this gives (b).
5. *Branching.* \(\mathrm{SO}_5\downarrow\mathrm{SO}_4\) is multiplicity free with \(a\ge p_1\ge b\ge|p_2|\) (Gelfand–Tsetlin); \(\mathrm{SO}_4\)-weight \((p_1,p_2)\) is \(V_m\boxtimes V_n\), \((m,n)=(p_1+p_2,p_1-p_2)\). The conditions read \(\tfrac{m+n}2\le a\), \(\tfrac{m+n}2\ge b\ge\tfrac{|m-n|}2\), i.e. \(n\in Z_m(a,b)\) (symmetric in \(m,n\)). Restricting further along the diagonal middle \(\mathrm{SL}_2\), \(V_{D^1}\otimes V_{D^2}\ni V_e\) iff \(|D^1-D^2|\le e\le D^1+D^2\), \(e\equiv D^1+D^2\) (Clebsch–Gordan). Summing over \((D^1,D^2)\) gives (c).
6. *(d).* For \(\nu=\varphi(\lambda)\): \(Z_{a_i+b_i}(a_i,b_i)=\{a_i-b_i\}\) and \(e=D^1+D^2\) force \(h=0\), \(w_i=u_i\), \(\sigma_i=0\): one point. For \(e>d_1+d_2\): \(D^1=2k+d_1\in Z_{d_1}(k+d_1,k)\), \(D^2=d_2\in Z_{d_2}(d_2,0)\), and \(|D^1-D^2|\le e=D^1+D^2\). For \(d_1,d_2\) odd, \(e=0\): \(D^1=D^2=d_1\) lies in both zero sets. For \(\nu=(0,0,2)\), \(\lambda=((1,1),0)\): \(C=\varphi^{-1}N_2\), \(Q_2=N_2^\perp\), \(Q_1\) at distance 2 from \(N_1^\perp\): \(\dim F=2\), \(\langle2\rho,\lambda\rangle-\delta(\nu)=6-2=4\). \(\square\)

**Check (computation).** branch.py compares, for all \(\lambda\) with \(a_i\le3\) and all \(\nu\) with \(d_i\le4\), \(e\le8\) (4484 nonempty pairs, 913 of multiplicity \(\ge2\)): no positive defect, and the leading coefficient of \((T_\lambda\ast\varphi_0)(x_\nu)\) at relevant pairs equals the branching multiplicity in every case. fusion.py checks that restricted characters agree at \((a,b,a)\) and \((b,a,b)\). Both are consistency checks of the general proof above.

**Sources cited:** scratchpad/solve/A3/branch.py; scratchpad/solve/A3/rank1.py; scratchpad/solve/A3/fusion.py; de Cataldo–Migliorini (decataldo-migliorini-v2.pdf): semismall maps and relevant strata


## A3-9 · Given D1–D2 for \(X\): \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\), \(m^{(n)}=D_ncD_n'\)

**Kind/status as reported:** lemma / proved

**Builds on:** A3-2, A3-6, A3-8

### Statement

Assume for \(X\) the inputs D1 and D2 of the pages: a Frobenius-compatible formalism \(\mathcal C_k\) on \(L^+G\backslash LX\) with perverse t-structure whose simple objects are the \(\mathrm{IC}_\nu\) (\(\operatorname{End}=\overline{\mathbb Q}_\ell\)), proper base change for \(m_\lambda:\mathrm{Gr}_{\le\lambda}\tilde\times L^+X\to LX\), and the decomposition theorem for \(m_\lambda\). Then, over \(k=\overline{\mathbb F}_q\), \(q\) odd:

(a) \(V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_\nu\mathrm{IC}_\nu\otimes\overline{\mathbb Q}_\ell^{\,c_{\nu\lambda}}\) is perverse, with \(c\) the branching matrix of [[A3-8]](c).

(b) \(\mathcal M=\Lambda_X^+\), hence \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\).

(c) \(m^{(n)}_{\nu\lambda}=c_{\nu\lambda}\,q^{n(\langle2\rho,\lambda\rangle-\delta(\nu))/2}\) for every \(n\).

(d) \((\mathrm{SG})_n\) holds iff \(R(\mathrm{SO}_5\times\mathrm{SO}_5)\otimes\mathbb Q\to R(\iota\check G_X)\otimes\mathbb Q\) is onto; it is not: \(\operatorname{tr}\mathrm{IC}_{(0,0,2)}\notin\mathcal H_{K_n}\ast\varphi_0\otimes\overline{\mathbb Q}_\ell\) for every \(n\) and \(q\).

### Argument

**Proof (given D1–D2).**

1. *Perversity.* [[A3-8]](a) applied to every \(\lambda'\le\lambda\) is stratified semismallness of \(m_\lambda\) for the strata \(\mathrm{Gr}^{\lambda'}\tilde\times L^+X\) and \(\mathcal O_\nu\) (with \(\delta\) from [[A3-6]](c)); hence \(m_{\lambda*}(\mathrm{IC}_{\mathrm{Gr}_{\le\lambda}}\tilde\boxtimes\mathrm{IC}_0)\) is perverse (the argument of Mirković–Vilonen for convolution; de Cataldo–Migliorini for semismall maps).
2. *Semisimplicity and multiplicities.* By D2 it is a sum of \(\mathrm{IC}_\nu\otimes E_\nu\); local systems are constant since stabilizers are connected ([[A3-6]](b)). \(E_\nu\) is the stalk at \(x_\nu\) in degree \(-\delta(\nu)\), i.e. \(H^{2f}(F^\lambda_\nu)\), \(f=\tfrac12(\langle2\rho,\lambda\rangle-\delta(\nu))\): lower strata \(\mathrm{Gr}^{\lambda'}\) contribute only in degrees \(<-\delta(\nu)\), since \(\mathrm{IC}_{\mathrm{Gr}_{\le\lambda}}|_{\mathrm{Gr}^{\lambda'}}\) lives in degrees \(\le-\langle2\rho,\lambda'\rangle-2\). So \(\dim E_\nu=c_{\nu\lambda}\) ([[A3-8]](b)): (a).
3. *(b).* Every \(\nu\) is relevant for some \(\lambda\) ([[A3-8]](d)), so \(\mathrm{IC}_\nu\) is a summand and \(\mathcal M=\Lambda_X^+\). Proposition I applies: H1 from [[A3-6]](a),(d), H2 from D1, H3 from [[A3-6]](b).
4. *(c).* The top components are geometrically irreducible and defined over \(\mathbb F_q\) ([[A3-8]](b)), so \(\mathrm{Fr}^n\) acts on \(H^{2f}\) by \(q^{nf}\); only \(i=0\) occurs.
5. *(d).* IIb (iii) holds verbatim for \(X\) given D1–D3 (D3 by [[A3-6]](b)): \((\mathrm{SG})_n\) iff the columns of \(m^{(n)}\) span. By (c), \(m^{(n)}=D_n\,c\,D_n'\) with invertible diagonal matrices, so iff the columns of \(c\) span, i.e. iff restriction of characters is onto. The class function \(\chi\mapsto\chi(a,b,a)-\chi(b,a,b)\) kills every restricted character ([[A3-3]](d)) and takes \(b^2-a^2+b^{-2}-a^{-2}\ne0\) on \(V_{(0,2,0)}\), the label \(\nu=(0,0,2)\). \(\square\)

**Status.** All geometric and combinatorial inputs are proved in [[A3-6]]–[[A3-8]]; the corollary is conditional only on D1–D2 for \(X\) ([[A3-10]]). Its conclusion (d) is consistent with, and independent of, the unconditional [[A3-5]]. It also realizes condition (iii) of [[A3-2]](c): sheaves generate because \(\iota\check G_X\) is a subgroup, functions fail because of fusion.

**Sources cited:** trace-pages/trace-b.html (IIb); trace-pages/_definitions.html (Proposition I); sun-decomposition.pdf, Thm 1.2; bbd-perverse.pdf (decomposition theorem); decataldo-migliorini-v2.pdf


## A3-10 · What blocks an unconditional \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) for \(X\)

**Kind/status as reported:** blocked / blocked

**Builds on:** A3-6, A3-9

### Statement

**Missing lemma (placidity).** For every \(\nu\in\Lambda_X^+\), the reduced orbit closure \(\overline{\mathcal O_\nu}\subset LX\) (ind-topology) is a placid scheme on which \(L^+G\) acts through finite-type quotients on a cofinal system of finite-type models compatible with the convolution maps \(m_\lambda\).

Given it, Chen–Yi's App. A.5 construction gives \(\mathcal C_k\) with a Frobenius-compatible perverse t-structure (D1), and BBD/Sun's Thm 1.2 on the finite-type models give D2; [[A3-9]] then makes \(X\) a counterexample to IId with both levels in characteristic \(p\).

### Argument

**Why it is not immediate.**

1. *No embedding into \(G\).* Chen–Yi's proof (§5) uses the symmetric-space embedding \(\tau:X\hookrightarrow G\), \(gH\mapsto g\theta(g)^{-1}\), and the geometry of \(L^+G\)-orbits in \(LG\). \(X\) is not symmetric; no such \(\tau\) is available.
2. *The compactification gives the wrong topology.* \(X\) is open in the projective \(\mathrm{IG}(2,V_1\oplus V_2)\) ([[A3-3]](b)), so \(LX(k)\) is an open subset of the placid arc space \(L^+\mathrm{IG}\). But the induced topology is not the ind-topology of \(LX\): already for \(\mathbb G_m\subset\mathbb P^1\), the components \(\{\operatorname{val}=n\}\) of \(L\mathbb G_m\) are open and closed in the ind-topology but not closed in \(L^+\mathbb P^1\). Orbit closures must be taken in the ind-topology (where \(\mathcal C_\lambda\) of [[A3-6]] step 5 is closed).
3. *The torsor over \(Y_1\times Y_2\).* \(X\to Y_1\times Y_2\), \(Y_i=\mathrm{Sp}_4/(\mathrm{Sp}_2\times\mathrm{Sp}_2)\) symmetric, is an \(\mathrm{SL}_2\)-torsor, Zariski-locally trivial (\(\mathrm{SL}_2\) is special), and Chen–Yi apply to the \(Y_i\). But loops do not respect Zariski covers, so local triviality of \(LX\to LY_1\times LY_2\) over orbit closures needs an argument. Over a single pair of \(Y\)-orbits the fibres are \(\mathrm{SL}(N_1)\backslash L\mathrm{SL}_2/\mathrm{SL}(N_2)\)-type sets, i.e. pieces of \(\mathrm{Gr}_{\mathrm{SL}_2}\), placid; what is open is the gluing across \((d_1,d_2)\).
4. *What does hold.* The source of \(m_\lambda\) is finite type modulo a pro-unipotent gerbe: \(L^+G\backslash(\mathrm{Gr}_{\le\lambda}\tilde\times L^+X)\cong L^+H\backslash\mathrm{Gr}_{\le\lambda^*}\), and \(L^+H\) acts on \(\mathrm{Gr}_{\le\lambda^*}\) through a finite-type quotient. Proper base change and Grothendieck–Lefschetz on the fibres hold (they are the varieties of [[A3-7]]). The orbit set, stabilizers, dimensions and finiteness of closures are proved ([[A3-6]]). Only the target side (the stack \(L^+G\backslash LX\cong LH\backslash\mathrm{Gr}_G\) near finitely many orbits) lacks a reference.

**Theorem-strength lemma needed:** the placidity statement above (an analogue of Chen–Yi §5 for affine spherical varieties with connected stabilizers, or at least for this \(X\)).

**Sources cited:** chen-yi-v4.pdf, §5 (placidness of orbit closures) and App. A.5; sun-decomposition.pdf, Thm 1.2; trace-pages/_hypotheses.html (D1, D2)


## A3-11 · No trace-zero shape for \(X\); the failure is \(q\)- and \(n\)-independent

**Kind/status as reported:** check / proved

**Builds on:** A3-5, A3-8

### Statement

For Sakellaridis's \(X\) (split, as in [[A3-3]]): (a) every top-dimensional irreducible component of every \(F^\lambda_\nu\) is geometrically irreducible and defined over \(\mathbb F_q\), and \(\#F^\lambda_\nu(\mathbb F_{q^n})=c_{\nu\lambda}q^{nf_{\nu\lambda}}+O(q^{n(f_{\nu\lambda}-1)})\); (b) consequently no Frobenius eigenvalue cancellation of the form \(\operatorname{tr}A=0\) at \(n=1\) occurs in the relevant multiplicity spaces; (c) the function-level failure ([[A3-5]]) holds for every \(F_n\), and in the conditional dictionary ([[A3-9]](c)) the obstruction is the same integer matrix \(c\) for every \(n\).

### Argument

**Proof.**

1. The pieces of [[A3-7]] are towers of fibrations with fibres \(\mathbb A^1\), \(\mathbb A^1\smallsetminus\{\mathrm{pt}\}\) or \(\mathbb P^1\), all defined over \(\mathbb F_q\) (the tree of \(\mathrm{SL}_2(\mathbb F_q(\!(t)\!))\) and its spheres are), hence geometrically irreducible and rational; the top pieces are counted with leading coefficient 1 each ([[A3-8]] step 4). This gives (a).
2. Frobenius therefore acts on \(H^{2f}(F^\lambda_\nu)\) by the scalar \(q^f\); a permutation of components with a fixed-point-free part, as for a non-split even-dimensional quadric, does not occur. This gives (b).
3. (c): [[A3-5]] is stated over every non-archimedean \(F\), including \(F_n\); and \(m^{(n)}=D_ncD'_n\) with \(c\) independent of \(n\). \(\square\)

**Remark.** The task's alternative shape — a multiplicity space with trace \(0\) at \(n=1\) — would make a single row of \(m^{(1)}\) vanish, contradicting nothing in IIc but producing a failure that disappears at even \(n\). In split examples with rational pavings of the convolution fibres (as here) it cannot occur; a candidate would need fibres with Frobenius-permuted components over a split base point, which (R) and (C) do not exclude but which no example on the pages exhibits.

**Sources cited:** trace-pages/trace-dictionary.html §4 (why n varies); scratchpad/solve/A3/fibres.py