§3 · Agent 3 · Split varieties · 2026-10-17T22:32:49Z

Proposition 3.8 · Each \(m_\lambda\) is semismall; multiplicities are dual branching numbers

proved audit: minor defect ★ key step

The defect of the convolution map at every stratum is \(-2\sum_i(u_i-w_i)_+\le0\), so all convolution maps are semismall; a label is relevant exactly when a Clebsch–Gordan triangle closes, the number of top components equals the branching multiplicity for Sakellaridis's \(\iota:\check G_X\to\check G\), and every label — covered or not — is relevant for an explicit \(\lambda\).

Proposition 3.8.

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 Lemma 3.3(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.

Proof.

  1. Dimensions. By Computation 3.7(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 \(a=d+\max(u,w)\), \(b=u_++w_++\tfrac\sigma2-\max(u,w)\), \(D=d+2u\), \(\rho'-\rho=2\min(w,0)-2\min(u,0)\):
$$\Phi=2u+2\min(w,0)-2\min(u,0)+2w_+-2u_+-2\max(u,w)=-2(u-w)_+\le0 .$$
  1. Both sides. \(\delta(\nu)=3d_1+3d_2+e\) (Proposition 3.6(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.
  2. 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).
  3. 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).
  4. (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.

Builds on

  • Proposition 3.6 · Orbits \((d_1,d_2,e)\), connected stabilizers, \(\delta=3d_1+3d_2+e\)
  • Computation 3.7 · Convolution fibres and the counts \((T_\lambda\ast\varphi_0)(x_\nu)\)

Built on by

  • Corollary 3.9 · Given D1–D2 for \(X\): \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\), \(m^{(n)}=D_ncD_n'\)
  • Remark 3.11 · No trace-zero shape for \(X\); the failure is \(q\)- and \(n\)-independent
  • Audit 5.8 · Audit of A3-7, A3-8: position formula, semismallness, dual branching
  • Theorem 6.3 · Verdict: sheaf generation over \(\overline{\mathbb F}_q\) does not imply spherical generation over \(\mathbb F_q(\!(t)\!)\)

Sources