§2 · Agent 2 · Non-split forms · 2026-10-17T21:23:39Z

Lemma 2.8 · Lemma: Satake objects without a Weil structure are invisible to the trace at \(n=1\)

proved audit: correct

\(V_\lambda\) has a Weil structure iff \(\sigma\lambda=\lambda\). In any Weil object of the Satake category, the isotypic parts with \(\sigma\lambda\ne\lambda\) contribute zero trace. So \(\mathcal H_K\ast\varphi_0\) is spanned by traces of rational Satake objects, while in the unitary twist every \(V_\lambda\ast\mathrm{IC}_0\), rational or not, is Frobenius-stable in \(\mathcal C_k\) and carries Weil structures that come from no Satake object.

Lemma 2.8.

Let \(G\) be unramified over \(\mathbb F_q\), with Frobenius \(*\)-action \(\sigma\) on \(\Lambda_T\) preserving \(\Lambda_T^+\). Let \(X=G/H\) satisfy the trace setting of Proposition II, except that \(T\) need not be split.

  1. \(\mathrm{Fr}^*V_\lambda\cong V_{\sigma^{-1}\lambda}\), so \(V_\lambda\) admits a Weil structure iff \(\sigma\lambda=\lambda\).
  2. For \(\sigma^n\lambda=\lambda\), IIa holds: \(\operatorname{tr}_n(V_\lambda\ast\mathrm{IC}_0)=\operatorname{tr}_n(V_\lambda)\ast\varphi_0\) with \(\operatorname{tr}_nV_\lambda=\pm T_\lambda+\) lower terms. Hence
$$\mathcal H_{K_n}\ast\varphi_0\otimes\overline{\mathbb Q}_\ell=\operatorname{span}\{\operatorname{tr}_n(V_\lambda\ast\mathrm{IC}_0):\ \sigma^n\lambda=\lambda\}.$$
  1. For every Weil object \((V,\beta)\) of \(\mathrm{Sat}_{G_k}\),
$$\operatorname{tr}_1(V\ast\mathrm{IC}_0)=\sum_{\sigma\lambda=\lambda}\operatorname{tr}(B_\lambda)\,\operatorname{tr}_1(V_\lambda\ast\mathrm{IC}_0),$$

where \(B_\lambda\) is the endomorphism of \(\operatorname{Hom}(V_\lambda,V)\) induced by \(\beta\). A \(\sigma\)-orbit of length \(r\ge2\) contributes \(0\) to \(\operatorname{tr}_n\) whenever \(r\nmid n\).

  1. In the unitary twist of Setup 2.4, \(V_\lambda\ast\mathrm{IC}_0\cong V_{\sigma\lambda}\ast\mathrm{IC}_0\) in \(\mathcal C_k\), so every \(V_\lambda\ast\mathrm{IC}_0\) carries Weil structures as an object of \(\mathcal C_k\). For \(\lambda=(c,0)\) with \(c\notin2\Lambda_H\), no such Weil structure is induced from a Weil structure on a Satake object, because \(\operatorname{tr}_1\) of it is a nonzero multiple of \(\operatorname{tr}_1(\mathrm{IC}_c)\notin\mathcal H_K\ast\varphi_0\).
  1. Weil structures. \(\mathrm{Fr}\) maps \(\operatorname{Gr}^\lambda\) onto \(\operatorname{Gr}^{\sigma\lambda}\), since \(\mathrm{Fr}(t^\lambda)=t^{\sigma\lambda}\). Pullback along the universal homeomorphism \(\mathrm{Fr}\) sends the IC sheaf of \(\operatorname{Gr}_{\le\sigma\lambda}\) to the IC sheaf of its preimage \(\operatorname{Gr}_{\le\lambda}\). So \(\mathrm{Fr}^*V_{\sigma\lambda}\cong V_\lambda\). An isomorphism \(\mathrm{Fr}^*V_\lambda\cong V_\lambda\) forces \(\operatorname{Gr}_{\le\sigma^{-1}\lambda}=\operatorname{Gr}_{\le\lambda}\), i.e. \(\sigma\lambda=\lambda\).
  2. Rational \(\lambda\). For \(\sigma\lambda=\lambda\), \(\operatorname{Gr}_{\le\lambda}\) is defined over \(\mathbb F_q\). The proof of IIa (Lang for the connected \(L^+G\); Grothendieck–Lefschetz on the projective convolution fibres) uses nothing else. The unramified Cartan decomposition gives \(K\backslash G(F)/K=(\Lambda_T^+)^\sigma\), and the open cell of \(\operatorname{Gr}_{\le\lambda}(\mathbb F_q)\) is \(Kt^\lambda K/K\), so \(\operatorname{tr}V_\lambda\) is unitriangular in the basis \(T_\lambda\). Over \(F_n\), replace \(\sigma\) by \(\sigma^n\).
  3. Induced objects. Write \(V\cong\bigoplus_\lambda V_\lambda\otimes M_\lambda\) (Zhu, Prop. 5.1.1). Fix isomorphisms \(\mathrm{Fr}^*V_\lambda\cong V_{\sigma^{-1}\lambda}\), canonical for \(\sigma\)-fixed \(\lambda\). Schur's lemma (\(\operatorname{Hom}(V_\lambda,V_\mu)=\delta_{\lambda\mu}\overline{\mathbb Q}_\ell\)) writes \(\beta=\bigoplus_\lambda(\mathrm{iso}\otimes B_\lambda)\) with \(B_\lambda:M_\lambda\to M_{\sigma^{-1}\lambda}\). Convolution with \(\mathrm{IC}_0\) is additive, so \(V\ast\mathrm{IC}_0=\bigoplus_\lambda(V_\lambda\ast\mathrm{IC}_0)\otimes M_\lambda\) with the induced Weil structure. At \(x\in X(F_n)\), \(\Phi^{(n)}_x\) maps the \(\lambda\)-summand of the stalk to the \(\sigma^{-n}\lambda\)-summand. When \(\sigma^n\lambda\ne\lambda\) these summands are distinct and contribute \(0\) to the trace. For \(n=1\) each \(\sigma\)-fixed \(\lambda\) contributes \(\operatorname{tr}(B_\lambda)\operatorname{tr}_1(V_\lambda\ast\mathrm{IC}_0)\).
  4. Unitary twist. By Lemma 2.5(2), \(V_{(a,b)}\ast\mathrm{IC}_0\cong\mathrm{IC}_a\star\mathrm{IC}_b\) and \(V_{(b,a)}\ast\mathrm{IC}_0\cong\mathrm{IC}_b\star\mathrm{IC}_a\); the commutativity constraint of \(\mathrm{Sat}_H\) (Zhu, Thm 5.2.9; Cor. 5.4.11) identifies them. Hence \(\mathrm{Fr}^*(V_\lambda\ast\mathrm{IC}_0)\cong V_{\sigma^{-1}\lambda}\ast\mathrm{IC}_0\cong V_\lambda\ast\mathrm{IC}_0\). Decomposing into the \(\mathrm{IC}_c\), each with its canonical Weil structure, and choosing any automorphisms of the multiplicity spaces gives Weil structures. For \(\lambda=(c,0)\), \(V_\lambda\ast\mathrm{IC}_0=\mathrm{IC}_c\) and \(\operatorname{End}(\mathrm{IC}_c)=\overline{\mathbb Q}_\ell\), so every Weil structure is \(u\cdot\mathrm{can}\) with \(u\ne0\). Its trace is \(u\operatorname{tr}_1(\mathrm{IC}_c)\), which is supported on \(\{c'\le c\}\) and is \(\pm u\) on \(O_c\). If it lay in \(\mathcal H_K\ast\varphi_0\), then by Lemma 2.2(b) it would lie in \(\operatorname{span}\{T_a\ast\varphi_0:2a\le c\}\). Since \(c\) is not even, every such \(T_a\ast\varphi_0\) has \(2a<c\) and vanishes on \(O_c\), a contradiction. By 3, the trace of any Weil object induced from \(\mathrm{Sat}_{G_k}\) lies in \(\mathcal H_K\ast\varphi_0\). So this Weil structure is not induced.
  5. Consequence for the matrix of the pages. At level \(n\), the columns of \(m^{(n)}\) are indexed by \((\Lambda_T^+)^{\sigma^n}\), not by \(\Lambda_T^+\). A summand \(\mathrm{IC}_\mu\) of a non-rational \(V_\lambda\ast\mathrm{IC}_0\) enters \(m^{(1)}\) only if it also occurs in a rational column. In the unitary twist the summand \(\mathrm{IC}_c=V_{(c,0)}\ast\mathrm{IC}_0\), which thick generation uses, has no rational column of its own.

Builds on

  • Lemma 2.5 · Lemma: \(V_{(c,0)}\ast\mathrm{IC}_0\cong\mathrm{IC}_c\), so sheaves generate for all forms
  • Theorem 2.6 · Theorem: the unitary twist of \((H\times H)/H\) fails (SG) over \(\mathbb F_q(\!(t)\!)\)

Built on by

  • Computation 2.9 · Computation: unitary \(\mathrm{SL}_2\): every row of \(m^{(1)}\) nonzero, no spanning
  • Remark 2.12 · Remark: why the trace misses thick generation, and what (R) removes
  • Audit 4.8 · Audit: Weil structures, rational columns and \(m^{(1)}\) for unitary twists

Sources