§1 · Agent 1 · Literature and Table 1 audit · 2026-10-18T18:47:23Z

Lemma 1.6 · Dictionary inputs at Iwahori level: rational orbits, Lang, trace bases, convolution

proved ★ key step

Collects, with exact citations, the function–sheaf inputs an Iwahori-level bridge needs, proves them for rows 1–3, and isolates the one input not proved for the symmetric rows: that every \(I\)-orbit is Frobenius-stable for the given \(q\).

Lemma 1.6.

Over \(k=\overline{\mathbb F}_q\), \(p\) odd, \(F=\mathbb F_q((t))\), \(\operatorname{vol}(I)=1\).

(D1) Convolution. For Weil objects \(\mathcal G\in\mathcal H_{c,I}\), \(M\in\mathcal C_I\): \(\operatorname{tr}(\mathcal G\ast_IM)=\operatorname{tr}\mathcal G\ast\operatorname{tr}M\), with \((f\ast\varphi)(x)=\sum_{g\in G(F)/I}f(g)\varphi(g^{-1}x)\). (D2) Hecke basis. \(\operatorname{tr}\mathrm{IC}_w=(-1)^{\ell(w)}T_w+\sum_{y<w}c_{yw}T_y\); \(\{\operatorname{tr}\mathrm{IC}_w\}\) is a basis of \(\mathcal H_I\otimes\overline{\mathbb Q}_\ell\), and \(\operatorname{tr}\mathrm{IC}_0=\pm\varphi_0\). Hence \(\mathcal H_I\ast\varphi_0\otimes\overline{\mathbb Q}_\ell=\operatorname{span}\{\operatorname{tr}(\mathrm{IC}_w\ast_I\mathrm{IC}_0)\}\). (D3) Orbits and Lang. If an \(I\)-orbit \(O\) of a finite level is Frobenius-stable and its stabilizer \(S\) has \(\pi_0S\) with trivial Frobenius action, then \(O(\mathbb F_q)\) is non-empty and splits into \(|\pi_0S|\) orbits of \(I(\mathbb F_q[[t]])\); for connected \(S\) it is one orbit. (D4) IC basis at a common level. If all pieces of level \((N,M)\) are Frobenius-stable with Frobenius trivial on the \(\pi_0\)'s, the traces \(\operatorname{tr}\mathrm{IC}_{\xi,P}\) of the simple objects of that level form a basis of the level-\((N,M)\) functions, unitriangular for the closure order (diagonal: a character \(\xi\) on the orbits in \(P(\mathbb F_q)\) times \(\pm1\)). (D5) Rows 1–3: every piece of every level contains an \(\mathbb F_q\)-point and has connected stabilizers; (D3)–(D4) hold with \(n=1\). (D6) Symmetric rows under (R): the \(K\)-orbits are Frobenius-stable; for \(I\)-orbits, Frobenius acts trivially on every \(\pi_0T_v\), but Frobenius-stability of each \(I\)-orbit for the given \(q\) is not proved (gap).

Proof.

  1. (D1). As zhu-satake-v2.pdf, Lemma 5.6.1 (spherical case), with \(\mathrm{Fl}\) in place of \(\mathrm{Gr}\): on a finite-type model the fibre of \(m:LG\times^ILX\to LX\) over \(x\in X(F)\) has \(\mathbb F_q\)-points \(\{gI\in G(F)/I:g^{-1}x\in\operatorname{supp}M\}\) (using \(\mathrm{Fl}(\mathbb F_q)=G(F)/I\), i.e. \(H^1(\mathbb F_q,I)=1\) by Lang applied to the connected finite-dimensional quotients \(I/I_m\), milne-iag-v2.pdf, Cor. 27.56, and passage to the limit), and \((\mathcal G\mathbin{\widetilde\boxtimes}M)_{(g,g^{-1}x)}=\mathcal G_g\otimes M_{g^{-1}x}\) with Frobenius; then laumon-1987.pdf, (1.1.1.3) (Grothendieck trace formula for \(f_!\)) and (1.1.1.2) (tensor products). The counting measure on \(G(F)/I\) is \(\operatorname{vol}(I)=1\).
  2. (D2). \(\operatorname{tr}\mathrm{IC}_w\) is supported on \(\overline{IwI}(\mathbb F_q)=\bigsqcup_{y\le w}IyI\), with value \((-1)^{\ell(w)}\) on \(IwI\) since \(\mathrm{IC}_w|_{IwI}=\overline{\mathbb Q}_\ell[\ell(w)]\) with Frobenius \(1\); Bruhat lower sets are finite. \(L^+X(\mathbb F_q)=X(\mathbb F_q[[t]])\), so \(\operatorname{tr}\underline{\overline{\mathbb Q}_\ell}_{L^+X}=\varphi_0\) (sign \((-1)^{s}\) for the shift \(s\) chosen). Then (D1).
  3. (D3). At a finite level \(O=H/S\) with \(H=I/I_{N,M}\) connected. Frobenius-stable homogeneous spaces under connected groups have rational points (milne-iag-v2.pdf, Cor. 27.55, surjectivity of the Lang map); the \(H(\mathbb F_q)\)-orbits on \(O(\mathbb F_q)\) are classified by \(\ker(H^1(\mathbb F_q,S)\to H^1(\mathbb F_q,H))=H^1(\mathbb F_q,S)\) (Cor. 27.56 for \(H\)), and \(H^1(\mathbb F_q,S)=H^1(\mathbb F_q,\pi_0S)\) (Cor. 27.56 for \(S^\circ\)), which is \(\pi_0S\) when Frobenius acts trivially. \(I(\mathbb F_q[[t]])\to H(\mathbb F_q)\) is onto (Lang for the connected kernel), so these are the \(I(\mathcal O)\)-orbits of the piece.
  4. (D4). On \(P(\mathbb F_q)\) the trace of \(\mathrm{IC}_{\xi,P}\) is \(\pm\) the function \(c\mapsto\xi(c)\) on the classes \(c\in\pi_0S\) of step 3 (the standard description of the trace of the local system attached to a character of \(\pi_0S\)); characters of the abelian group \(\pi_0S\) (Chen–Yi, Lemma 71: \(\pi_0S\) commutative) form a basis of functions on \(\pi_0S\). Off \(P\) the trace is supported on \(\partial P\), a union of lower pieces of the same level; finitely many pieces (I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).) give unitriangularity.
  5. (D5). Rows 1–3: Lemma 1.2, Lemma 1.3, Lemma 1.4 list rational points (\(\dot w\), \(t^me_j\) and \(0\), \(t^m\)) and connected stabilizers.
  6. (D6). \(K\)-orbits: satake-two-levels props/pdf/trace.tex, Lemma "Orbits" (lem:lang), under (R) and (C). Frobenius on \(\pi_0T_v\): if \(O_v\) is Frobenius-stable choose a rational \(x_v\) (step 3); then \(\psi_v\) is defined over \(\mathbb F_q\), Frobenius acts on the split \(T_0\) by \(t\mapsto t^q\), and on \(\pi_0(T_v)\simeq H^1(\langle\psi_v\rangle,X_*(T_0))\) (Proposition 1.5), a group killed by \(2\), as multiplication by the odd number \(q\), i.e. trivially. Frobenius-stability itself: the \(I\)-orbits inside \(O_\mu\) are \(I\backslash K/\operatorname{Stab}_K(x_\mu)=B\backslash G/\bar H_\mu\), \(\bar H_\mu\) the image of the stabilizer in \(G\), a finite set permuted by Frobenius. Chen–Yi (§8.2) and Mars–Springer (mars-springer-1998.pdf, 3.1.2: "take \(q\) big enough") replace \(q\) by a power instead of proving stability. Heuristic: if \(\bar H_\mu\) depends only on the facet of \(\mu\), one power \(q^n\) works for all \(\mu\) simultaneously, and the bridge holds with \((IG)\) over \(\mathbb F_{q^n}((t))\).

Remark (Laumon 1.1.1.5). The trace does not determine a complex, even virtually; the bridge must therefore go from functions to summands (a nonzero coefficient forces a summand), never from summands to functions without positivity.

Builds on

Built on by

  • Audit 1.7 · Every row of Table 1 at Iwahori level: what is proved, over which field, by which argument
  • Summary 1.10 · Verdict of the audit: rows 1–3 agree, the rest reduce to one module computation
Original record, as the agent wrote it at 18:47 UTC

Dictionary inputs at Iwahori level: rational orbits, Lang, trace bases, convolution

Proof.

  1. (D1). As zhu-satake-v2.pdf, Lemma 5.6.1 (spherical case), with \(\mathrm{Fl}\) in place of \(\mathrm{Gr}\): on a finite-type model the fibre of \(m:LG\times^ILX\to LX\) over \(x\in X(F)\) has \(\mathbb F_q\)-points \(\{gI\in G(F)/I:g^{-1}x\in\operatorname{supp}M\}\) (using \(\mathrm{Fl}(\mathbb F_q)=G(F)/I\), i.e. \(H^1(\mathbb F_q,I)=1\) by Lang applied to the connected finite-dimensional quotients \(I/I_m\), milne-iag-v2.pdf, Cor. 27.56, and passage to the limit), and \((\mathcal G\mathbin{\widetilde\boxtimes}M)_{(g,g^{-1}x)}=\mathcal G_g\otimes M_{g^{-1}x}\) with Frobenius; then laumon-1987.pdf, (1.1.1.3) (Grothendieck trace formula for \(f_!\)) and (1.1.1.2) (tensor products). The counting measure on \(G(F)/I\) is \(\operatorname{vol}(I)=1\).
  2. (D2). \(\operatorname{tr}\mathrm{IC}_w\) is supported on \(\overline{IwI}(\mathbb F_q)=\bigsqcup_{y\le w}IyI\), with value \((-1)^{\ell(w)}\) on \(IwI\) since \(\mathrm{IC}_w|_{IwI}=\overline{\mathbb Q}_\ell[\ell(w)]\) with Frobenius \(1\); Bruhat lower sets are finite. \(L^+X(\mathbb F_q)=X(\mathbb F_q[[t]])\), so \(\operatorname{tr}\underline{\overline{\mathbb Q}_\ell}_{L^+X}=\varphi_0\) (sign \((-1)^{s}\) for the shift \(s\) chosen). Then (D1).
  3. (D3). At a finite level \(O=H/S\) with \(H=I/I_{N,M}\) connected. Frobenius-stable homogeneous spaces under connected groups have rational points (milne-iag-v2.pdf, Cor. 27.55, surjectivity of the Lang map); the \(H(\mathbb F_q)\)-orbits on \(O(\mathbb F_q)\) are classified by \(\ker(H^1(\mathbb F_q,S)\to H^1(\mathbb F_q,H))=H^1(\mathbb F_q,S)\) (Cor. 27.56 for \(H\)), and \(H^1(\mathbb F_q,S)=H^1(\mathbb F_q,\pi_0S)\) (Cor. 27.56 for \(S^\circ\)), which is \(\pi_0S\) when Frobenius acts trivially. \(I(\mathbb F_q[[t]])\to H(\mathbb F_q)\) is onto (Lang for the connected kernel), so these are the \(I(\mathcal O)\)-orbits of the piece.
  4. (D4). On \(P(\mathbb F_q)\) the trace of \(\mathrm{IC}_{\xi,P}\) is \(\pm\) the function \(c\mapsto\xi(c)\) on the classes \(c\in\pi_0S\) of step 3 (the standard description of the trace of the local system attached to a character of \(\pi_0S\)); characters of the abelian group \(\pi_0S\) (Chen–Yi, Lemma 71: \(\pi_0S\) commutative) form a basis of functions on \(\pi_0S\). Off \(P\) the trace is supported on \(\partial P\), a union of lower pieces of the same level; finitely many pieces (I1) give unitriangularity.
  5. (D5). Rows 1–3: Lemma 1.2, Lemma 1.3, Lemma 1.4 list rational points (\(\dot w\), \(t^me_j\) and \(0\), \(t^m\)) and connected stabilizers.
  6. (D6). \(K\)-orbits: satake-two-levels props/pdf/trace.tex, Lemma "Orbits" (lem:lang), under (R) and (C). Frobenius on \(\pi_0T_v\): if \(O_v\) is Frobenius-stable choose a rational \(x_v\) (step 3); then \(\psi_v\) is defined over \(\mathbb F_q\), Frobenius acts on the split \(T_0\) by \(t\mapsto t^q\), and on \(\pi_0(T_v)\simeq H^1(\langle\psi_v\rangle,X_*(T_0))\) (Proposition 1.5), a group killed by \(2\), as multiplication by the odd number \(q\), i.e. trivially. Frobenius-stability itself: the \(I\)-orbits inside \(O_\mu\) are \(I\backslash K/\operatorname{Stab}_K(x_\mu)=B\backslash G/\bar H_\mu\), \(\bar H_\mu\) the image of the stabilizer in \(G\), a finite set permuted by Frobenius. Chen–Yi (§8.2) and Mars–Springer (mars-springer-1998.pdf, 3.1.2: "take \(q\) big enough") replace \(q\) by a power instead of proving stability. Heuristic: if \(\bar H_\mu\) depends only on the facet of \(\mu\), one power \(q^n\) works for all \(\mu\) simultaneously, and the bridge holds with \((IG)\) over \(\mathbb F_{q^n}((t))\).

Remark (Laumon 1.1.1.5). The trace does not determine a complex, even virtually; the bridge must therefore go from functions to summands (a nonzero coefficient forces a summand), never from summands to functions without positivity.

Sources