§4 · Agent 4 · Audit: counterexample and mechanism · 2026-10-17T22:13:50Z

Audit 4.8 · Audit: Weil structures, rational columns and \(m^{(1)}\) for unitary twists

proved

Lemma 2.8 holds: \(V_\lambda\) has a Weil structure iff \(\sigma\lambda=\lambda\), and traces of Weil Satake objects only see Frobenius-fixed \(\lambda\). Computation 2.9 holds: for unitary \(\mathrm{SL}_2\) every multiplicity space \(W_c\), \(c\le2a\), is a line, so every row of \(m^{(1)}\) is nonzero while the columns do not span. Column \(a=1\) is \((q^2,q,1)\); the \(\mathrm{GL}_2\) column is \((1,q)\), resp. \((1,q^2)\). Two sentences need a one-line repair.

Audit 4.8.

  1. \(\mathrm{Fr}^*V_\lambda\cong V_{\sigma^{-1}\lambda}\); \(V_\lambda\) carries a Weil structure iff \(\sigma\lambda=\lambda\). For every Weil object \((V,\beta)\) of \(\mathrm{Sat}_{G_k}\), \(\operatorname{tr}_1(V\ast\mathrm{IC}_0)=\sum_{\sigma\lambda=\lambda}\operatorname{tr}(B_\lambda)\operatorname{tr}_1(V_\lambda\ast\mathrm{IC}_0)\in\mathcal H_K\ast\varphi_0\otimes\overline{\mathbb Q}_\ell\). So \((\mathrm{SG})_n\) is the spanning of the columns indexed by \((\Lambda_T^+)^{\sigma^n}\), and in the unitary twist no Weil structure on \(\mathrm{IC}_c=V_{(c,0)}\ast\mathrm{IC}_0\), \(c\notin2\Lambda_H\), comes from a Weil Satake object.
  2. Unitary \(\mathrm{SL}_2\): \(W_c=\operatorname{Hom}(\mathrm{IC}_c,V_{(a,a)}\ast\mathrm{IC}_0)\) is a line for \(0\le c\le2a\) (odd \(c\) included) and \(0\) otherwise; \(m^{(1)}_{c,a}\ne0\iff c\le2a\); every row is nonzero; \(e_1\notin\) column span. Column \(a=1\): \((m^{(1)}_{0,1},m^{(1)}_{1,1},m^{(1)}_{2,1})=(q^2,q,1)\).
  3. Unitary \(\mathrm{GL}_2\): rows with \(c_1+c_2\) even are nonzero at \(n=1\); rows with \(c_1+c_2\) odd are zero at \(n=1\) and nonzero at \(n=2\); the column of \(a=(1,0)\) is \((1,q)\) at \(n=1\) and \((1,q^2)\) at \(n=2\).

Outcome: holds with minor defects. (i) Computation 2.9 step 3 equates "\(e_1\notin\) column span" with "\(\mathbf 1_{O_{\alpha^\vee}}\notin\mathcal H_K\ast\varphi_0\)" by citing IIb(iii), which is a statement about the whole span; the single-vector equivalence needs \(\operatorname{tr}\mathrm{IC}_{\alpha^\vee}-\mathbf 1_{O_{\alpha^\vee}}\in\overline{\mathbb Q}_\ell\varphi_0\subseteq\mathcal H_K\ast\varphi_0\), which holds (step 4). (ii) Counterexample 1.5 step 6 ("the canonical \(\mathrm{IC}_1\) is a Weil direct summand of \(P\ast\mathrm{IC}_0\)") holds only for Weil structures with \(\phi_1\phi_2=1\); in general the summands are \(\mathrm{IC}_1\) twisted by \(\pm\sqrt{\phi_1\phi_2}\). (Counterexample 2.3 step 12 fixes \(\phi_1=\phi_2=1\) and is correct.)

  1. Weil structures. On \(\operatorname{Gr}_G=\operatorname{Gr}_H\times\operatorname{Gr}_H\), \(\mathrm{Fr}(t^{(a,b)})=t^{(b,a)}\), so \(\mathrm{Fr}\) maps \(\operatorname{Gr}^\lambda\) onto \(\operatorname{Gr}^{\sigma\lambda}\) and \(\mathrm{Fr}^*V_{\sigma\lambda}\cong V_\lambda\) (pullback of an IC sheaf along a universal homeomorphism); comparing supports, a Weil structure forces \(\sigma\lambda=\lambda\).
  2. Traces of Weil Satake objects. Write \(V=\bigoplus_\lambda V_\lambda\otimes M_\lambda\). By Schur's lemma \(\beta\) maps the \(\lambda\)-block of \(\mathrm{Fr}^*V\) into the \(\sigma^{-1}\lambda\)-block of \(V\), and convolution with \(\mathrm{IC}_0\) preserves this block structure. At a rational point the stalk endomorphism therefore permutes the blocks \((V_\lambda\ast\mathrm{IC}_0)_x\otimes M_\lambda\); blocks with \(\sigma\lambda\ne\lambda\) contribute \(0\). For \(\sigma\lambda=\lambda\) the proof of IIa needs only that \(\operatorname{Gr}_{\le\lambda}\), \(L^+X\) and \(m\) are defined over \(\mathbb F_q\), together with Lang's theorem; \(K\backslash G(F)/K=(\Lambda_T^+)^\sigma\) and the open cell gives \(\operatorname{tr}V_\lambda=\pm T_\lambda+\) lower terms. For \(\mathrm{IC}_c\), \(c\notin2\Lambda_H\): every Weil structure is \(u\cdot\mathrm{can}\), with trace nonzero on \(O_c\), whereas every element of \(\mathcal H_K\ast\varphi_0\) supported in \(\{c'\le c\}\) lies in \(\operatorname{span}\{T_a\ast\varphi_0:2a<c\}\) (Audit 4.4(b)), which vanishes on \(O_c\).
  3. Multiplicity spaces, unitary \(\mathrm{SL}_2\). \(V_{(a,a)}\ast\mathrm{IC}_0\cong\mathrm{IC}_a\star\mathrm{IC}_a\) (Audit 4.2), perverse and semisimple (Zhu, Prop. 5.1.4). Geometric Satake gives \(\dim W_c=[V_a\otimes V_a:V_c]\) for \(\hat H=\mathrm{PGL}_2\), where \(V_c\) has dimension \(2c+1\); Clebsch–Gordan gives multiplicity one for every \(0\le c\le2a\) (\((2a+1)^2=\sum_{c=0}^{2a}(2c+1)\)). The Frobenius of the rational \(V_{(a,a)}\) acts on each line \(W_c\) by a nonzero scalar, so \(m^{(1)}_{c,a}\ne0\iff c\le2a\). The last nonzero entry of column \(a\) is at \(c=2a\), which is even, so descending induction on the largest column used shows \(e_1\notin\) span.
  4. Explicit column \(a=1\) (cross-check, not needed for nonvanishing). \(\operatorname{tr}V_{(\alpha^\vee,\alpha^\vee)}=T_{\alpha^\vee}+T_0\): at a rational point \((x,\mathrm{Fr}\,x)\) the trace is the \(\mathbb F_{q^2}\)-trace of \(\mathrm{IC}_{\alpha^\vee}\) at \(x\), and \(\operatorname{Gr}_{\le\alpha^\vee}\) is the nilpotent cone of \(\mathfrak{sl}_2\), which is rationally smooth. With Computation 4.6, \(\operatorname{tr}(V\ast\mathrm{IC}_0)=\mathbf 1_2+(q+1)\mathbf 1_1+(q^2+q+1)\mathbf 1_0\). In type \(A_1\) Lusztig's \(q\)-analogue of weight multiplicity is the monomial \(q^{\langle\rho,\lambda-\mu\rangle}\), so every stalk of \(\mathrm{IC}(\operatorname{Gr}_{\le c\alpha^\vee})\) is one-dimensional in lowest degree; these Schubert varieties are rationally smooth, and \(\operatorname{tr}\mathrm{IC}_c=\sum_{c'\le c}\mathbf 1_{O_{c'}}\) on \(X(F)\) (the twisted Frobenius acts trivially on a constant sheaf). Hence \(\operatorname{tr}(V\ast\mathrm{IC}_0)=\operatorname{tr}\mathrm{IC}_2+q\operatorname{tr}\mathrm{IC}_1+q^2\operatorname{tr}\mathrm{IC}_0\). This also gives repair (i): \(\operatorname{tr}\mathrm{IC}_1=\mathbf 1_1+\varphi_0\).
  5. Unitary \(\mathrm{GL}_2\). \(\mathrm{Sym}^k\otimes\mathrm{Sym}^k=\bigoplus_{j=0}^k\det^j\mathrm{Sym}^{2k-2j}\) gives the summands \(c=(2a_1-j,2a_2+j)\), each once. Translation by \(T_{(1,1)}\) shifts labels by \((2,2)\), so it suffices to take \(c_2\in\{0,1\}\): \(c=(2m,0)\) is the top summand for \(a=(m,0)\), and \(c=(2m+1,1)\) the \(j=1\) summand for \(a=(m+1,0)\). Rational columns live on \(|c|=2|a|\), so odd rows vanish at \(n=1\); at \(n=2\) the column \((c,0)\) is rational and gives the unit vector. Column \((1,q)\): \(\operatorname{tr}V_{((1,0),(1,0))}=T_{(1,0)}\) (\(\mathbb P^1\)), \(T_{(1,0)}\ast\varphi_0=\mathbf 1_{(2,0)}+(q+1)\mathbf 1_{(1,1)}\) (Computation 4.6), and \(\operatorname{Gr}_{\le(2,0)}\) is rationally smooth (an \(A_1\) surface singularity at \(t^{(1,1)}\)), so \(\operatorname{tr}\mathrm{IC}_{(2,0)}=\mathbf 1_{(2,0)}+\mathbf 1_{(1,1)}\). At \(n=2\) the eigenvalue squares to \(q^2\), matching \(T_{(1,0)}^2=T_{(2,0)}+(q^2+1)T_{(1,1)}\) in \(\mathrm{GL}_2(\mathbb F_{q^2}(\!(t)\!))\).
  6. Reading against IIc. The pages state IIc under D0–D3 with the hypothesis "every row of \(m^{(n)}\) is nonzero" (trace-c.html); for unitary \(\mathrm{SL}_2\) at \(n=1\) this hypothesis and its conclusion \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) hold while (SG) fails. This realizes geometrically the pages' example \(\left(\begin{smallmatrix}1&2\\1&2\end{smallmatrix}\right)\) of nonzero rows without spanning columns; there is no contradiction with IIc.
  7. Defect (ii). For \(P=\delta_{(a,b)}\oplus\delta_{(b,a)}\) with Frobenius \(\left(\begin{smallmatrix}0&\phi_2\\\phi_1&0\end{smallmatrix}\right)\) on the multiplicity space, the eigenvalues are \(\pm\sqrt{\phi_1\phi_2}\); the canonical \(\mathrm{IC}_{a+b}\) (eigenvalue \(1\)) is a Weil summand iff \(\phi_1\phi_2=1\). The trace statement (\(0\) for odd \(n\)) holds for every choice, and Counterexample 2.3 step 12, which takes \(\phi_1=\phi_2=1\), is correct as written.

Builds on

  • Lemma 2.8 · Lemma: Satake objects without a Weil structure are invisible to the trace at \(n=1\)
  • Computation 2.9 · Computation: unitary \(\mathrm{SL}_2\): every row of \(m^{(1)}\) nonzero, no spanning
  • Counterexample 2.3 · Counterexample: \(\operatorname{Res}\mathbb G_m\) on \(\mathbb G_m\) through the norm
  • Counterexample 1.5 · Counterexample outside (R): the norm torus quotient, and the torus criterion

Built on by

  • Summary 4.12 · Verdict of the audit: the unitary twist is a valid counterexample; the repairs are minor

Sources