Audit 5.9 · Audit of A3-9, A3-2(c): given D1–D2, \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\)

Re-derivation.

1. Perversity. The support condition: on \(\mathrm{Gr}^{\lambda'}\tilde\times L^+X\) the source complex sits in degrees \(\le-\langle2\rho,\lambda'\rangle\). Hence the stalk at \(x_\nu\) vanishes above degree \(\max_{\lambda'}(2\dim F^{\lambda'}_\nu-\langle2\rho,\lambda'\rangle)\le-\delta(\nu)\) (Audit 5.8). The cosupport condition follows from Verdier self-duality of \(V_\lambda\tilde\boxtimes\mathrm{IC}_0\), since \(m_\lambda\) is proper; this needs Verdier duality in the formalism (part of D1).
2. Multiplicities. Constant coefficients come from connected stabilizers (H3). In degree \(-\delta(\nu)\) at \(x_\nu\):

• \(\mathrm{IC}_{\nu'}\) for \(\nu'>\nu\) contributes nothing, by the IC support condition;
• the strata \(\lambda'<\lambda\) contribute nothing, because \(\mathrm{IC}_{\mathrm{Gr}_{\le\lambda}}|_{\mathrm{Gr}^{\lambda'}}\) lives in degrees \(\le-\langle2\rho,\lambda'\rangle-2\) and \(\dim F^{\lambda'}_\nu\le f'\).

So \(E_\nu=H^{2f}_c(F^\lambda_\nu)\), of dimension \(c_{\nu\lambda}\).

3. Generation. Every label is relevant for some \(\lambda\) (Audit 5.8, step 6). So every \(\mathrm{IC}_\nu\) is a direct summand, and Proposition I gives \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\).
4. Traces.

• Top components are defined over \(\mathbb F_q\) and geometrically irreducible (Lang–Weil, Audit 5.8 step 5), so \(A_{\nu,0}(\lambda)=q^f\cdot\mathrm{id}\) and \(m^{(n)}=D_n\,c\,D_n'\) with invertible diagonal \(D_n,D_n'\).
• IIb(iii) needs the IC traces to form a basis of \(C_{K_n}\otimes\overline{\mathbb Q}_\ell\). This holds because orbit closures are finite unions of orbits (Audit 5.6) and one \(K_n\)-orbit of rational points lies in each \(L^+G\)-orbit.
• IIa (\(\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)=\operatorname{tr}V_\lambda\ast\varphi_0\)) needs D1 and Lang's theorem for \(L^+G\).

5. Fusion. Columns of \(c\) are restrictions of \(\mathrm{SO}_5^2\)-characters. The class function \(\psi(\chi)=\chi(a,b,a)-\chi(b,a,b)\), at generic \((a,b)\) (that is, \(a\ne\pm b^{\pm1}\), Audit 5.2), vanishes on every column. On \(V_{(0,2,0)}\) it equals \(b^2+b^{-2}-a^2-a^{-2}\ne0\). Numerically, the restricted characters agree at \((a,b,a)\) and \((b,a,b)\) to \(10^{-14}\) for all \(a_i\le3\).
6. A3-2(c)(iii) and (b).

• (b): every irreducible of a closed reductive \(\check G_X\subseteq\check G\) is a summand of some restricted irreducible, and characters separate semisimple classes of the connected \(\check G_X\).
• (c)(iii): the hypotheses are exactly the conclusions of (a)–(c) above.

Status. This is a correct conditional statement. Its unconditional consequence, the failure of \((\mathrm{SG})_n\), is already proved at the function level (Audit 5.5). The only open content is the existence of the formalism; see Objection 5.10.