§4 · Agent 4 · Sheaf–function dictionary · 2026-10-17T02:31:20Z

Lemma 4.3 · The trace of $V_\lambda\ast\IC_0$ is $\tr(V_\lambda)\ast\varphi_0$, and the traces of the $V_\lambda$ form a basis of $\cH_K$

proved audit: minor defect

Minor defect (second audit). The lemma is correct (re-derived: fibre description, stalk formula, Grothendieck–Lefschetz with vol K = 1, matching the classical page's convolution; unitriangularity of tr V_λ). One normalization slip in the unused remark: the formula S(ι tr V_λ) = (−ιq^{1/2})^{⟨2ρ,λ⟩}χ_λ holds only if ι(q^{1/2}) > 0. Zhu's Lemma 5.6.3 is stated for ι(q^{1/2}) < 0 and for f_λ = tr(IC_λ[−⟨2ρ,λ⟩]) = (−1)^{⟨2ρ,λ⟩} tr V_λ; with Zhu's convention the right side should be (ιq^{1/2})^{⟨2ρ,λ⟩}χ_λ, and convention-free S(tr V_λ) = (−1)^{⟨2ρ,λ⟩} q^{⟨ρ,λ⟩} χ_λ (check: GL2, λ=(1,0): tr V_λ = −T_λ, S(T_λ) = q^{1/2}χ_λ). Also, unitriangularity of tr V_λ is elementary (open-cell value and finite lower sets) and does not need Kazhdan–Lusztig purity. Nothing depends on the remark.

The Frobenius trace of a Satake object is the Hecke translate of the basic function by the trace of $V_\lambda$, and these traces form a basis of $\cH_K$; so the module $\cH_K\cdot\varphi_0$ is exactly the span of the traces of the Satake objects. This is the first half of the dictionary.

Lemma 4.3.

Assume D1 and D4 of Setup 4.1. Give $V_\lambda=\IC(\Gr_{\le\lambda})$ and $\IC_0=\underline{\overline{\Q}_\ell}_{\,L^+X}$ their canonical Weil structures (Frobenius eigenvalue $1$ on the top stratum), and $V_\lambda\ast\IC_0=m_!(V_\lambda\,\widetilde\boxtimes\,\IC_0)$ the induced one. Then for every $n\ge1$, on $X(F_n)$, $$\tr_n(V_\lambda\ast\IC_0)=\tr_n(V_\lambda)\ast_{K_n}\varphi_0,\qquad \tr_n(V_\lambda)\in\cH_{K_n}\otimes\overline{\Q}_\ell,$$ and $\{\tr_n(V_\lambda)\}_{\lambda\in\Lambda_T^+}$ is a $\overline{\Q}_\ell$-basis of $\cH_{K_n}\otimes\overline{\Q}_\ell$. Hence $$\cH_K\cdot\varphi_0\otimes\overline{\Q}_\ell=\operatorname{span}\{\tr(V_\lambda\ast\IC_0)\}_{\lambda\in\Lambda_T^+}.$$

Stalks and traces. For $x\in X(F)$ the fibre $m^{-1}(x)=\{gK\in\Gr_{\le\lambda}:g^{-1}x\in L^+X\}$ is a closed subscheme of the projective $\Gr_{\le\lambda}$, and proper base change (D1) gives $(V_\lambda\ast\IC_0)_x\simeq R\Gamma_c\big(m^{-1}(x),V_\lambda\widetilde\boxtimes\IC_0\big)$; this is the stalk formula of Proposition III, step 5(a). The stalk of $V_\lambda\widetilde\boxtimes\IC_0$ at $(g,g^{-1}x)$ is $(V_\lambda)_g\otimes(\IC_0)_{g^{-1}x}$, compatibly with Frobenius. The Grothendieck–Lefschetz formula on the finite-type fibre gives, with $\operatorname{vol}K=1$,

$$\tr(V_\lambda\ast\IC_0)(x)=\sum_{gK\in m^{-1}(x)(\F_q)}\tr(V_\lambda)(g)\,\one_{X(\cO)}(g^{-1}x)=(\tr V_\lambda\ast_K\varphi_0)(x).$$

This is Zhu's Lemma 5.6.1 with the second factor on $LX$ instead of $\Gr$.

Basis. $\tr V_\lambda$ is left $K$-invariant on $G(F)/K$ and supported on $\Gr_{\le\lambda}(\F_q)=\bigsqcup_{\mu\le\lambda}Kt^\mu K/K$, so it lies in $\cH_K\otimes\overline{\Q}_\ell$. By unitriangularity (Zhu (5.6.2), from the purity of IC stalks of affine Schubert varieties),

$$\tr V_\lambda=\varepsilon_\lambda\one_{Kt^\lambda K}+\sum_{\mu<\lambda}a_{\lambda\mu}(q)\one_{Kt^\mu K},\qquad\varepsilon_\lambda=(-1)^{\langle2\rho,\lambda\rangle}\ne0,$$

$\varepsilon_\lambda$ being the value of the normalised constant sheaf on the open stratum. The lower sets $\{\mu\le\lambda\}$ are finite, so the $\tr V_\lambda$ form a basis. Over $\F_{q^n}$ replace $q$ by $q^n$.

Span. $\cH_K\cdot\varphi_0=\{f\ast\varphi_0\}$ is spanned by the $f\ast\varphi_0$ with $f$ in any basis of $\cH_K$. $\square$

Satake form (cited, not re-derived). By Zhu's Lemma 5.6.3 (Lusztig, Kato), with the normalisation $V_\lambda|_{\Gr_\lambda}=\overline{\Q}_\ell[\langle2\rho,\lambda\rangle]$, $$\mathsf S(\iota\tr V_\lambda)=(-\iota q^{1/2})^{\langle2\rho,\lambda\rangle}\chi_\lambda .$$ So $\tr V_\lambda$ is, up to a sign and a power of $q^{1/2}$, the element of $\cH_K$ whose Satake transform is the character $\chi_\lambda$ of the dual-group representation; under Theorem E ($p$-adic, under (A1)–(A5)) its image in $C_K\simeq\C[\delta(X)^{1/2}A_X^*]^{W_X}$ would be $r(\chi_\lambda)$ up to that scalar. This is not used below.

Builds on

Setup 4.1 · Set-up of the sheaf–function dictionary over $\F_q(\!(t)\!)$: Weil objects, trace functions, obligations D0–D5

Built on by

Lemma 4.4 · Expansion of $\tr(V_\lambda\ast\IC_0)$ in the IC basis: the coefficients are traces of Frobenius on multiplicity spaces
Theorem 4.6 · Theorem: (SG) over $\F_q(\!(t)\!)$ implies $\Csat=\cC_K$ over $\overline{\F}_q$
Audit 7.3 · Audit of the orbit count, the trace dictionary and the IC expansion: they hold under D0–D3, and D4 is elementary
Correction 10.1 · Correction: the mechanism of the $E_6/F_4$ divergence, separated by field
Correction 10.8 · Correction: the square-class automorphism multiplies $\tr_n\IC_{\chi,\nu}$ by $\chi(-1)$
Computation X2.3 · Stalks for $\mu=\alpha^\vee,2\alpha^\vee$: $\mathcal L_{\mathrm{sgn}}$ is Kummer on each slice branch
Theorem (repair of A4-11) X2.4 · Repair of A4-11: $\mathrm{IC}_{\mathrm{sgn},\mu}\notin\mathcal C_K^{\mathrm{Sat}}$ in both $\mathbb F_q$-forms, $q$ odd

Original record, as the agent wrote it at 02:31 UTC

Trace of a Satake object is a Hecke translate of the basic function; traces of V_lambda span H_K

Lemma A4-3 (answer to (a)). (proved modulo D1, D4.) Give $V_\lambda=\mathrm{IC}(\mathrm{Gr}_{\le\lambda})$ and $\mathrm{IC}_0=\underline k_{L^+X}$ their canonical Weil structures (eigenvalue 1 on the top stratum), and $V_\lambda\ast\mathrm{IC}_0=m_!(V_\lambda\tilde\boxtimes\mathrm{IC}_0)$ the induced one. Then for every $n\ge1$, on $X(\mathbb F_{q^n}(\!(t)\!))$: $$\operatorname{tr}_n(V_\lambda\ast\mathrm{IC}_0)=\operatorname{tr}_n(V_\lambda)\ast_{K_n}\varphi_0,\qquad \operatorname{tr}_n(V_\lambda)\in\mathcal H_{K_n}\otimes\overline{\mathbb Q}_\ell,$$ and $\{\operatorname{tr}_n(V_\lambda)\}_{\lambda\in\Lambda_T^+}$ is a $\overline{\mathbb Q}_\ell$-basis of $\mathcal H_{K_n}\otimes\overline{\mathbb Q}_\ell$. Hence $\mathcal H_K\cdot\varphi_0\otimes\overline{\mathbb Q}_\ell=\operatorname{span}\{\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)\}_\lambda$.

Proof. 1. For $x\in X(F)$ the fibre $m^{-1}(x)=\{gK\in\mathrm{Gr}_{\le\lambda}:g^{-1}x\in L^+X\}$ is a closed subscheme of the projective $\mathrm{Gr}_{\le\lambda}$, and $(V_\lambda\ast\mathrm{IC}_0)_x\simeq R\Gamma_c(m^{-1}(x),V_\lambda\tilde\boxtimes\mathrm{IC}_0)$ (proper base change, D1; this is the stalk formula used in criterion.html Step 5(a)). The stalk of $V_\lambda\tilde\boxtimes\mathrm{IC}_0$ at $(g,g^{-1}x)$ is $(V_\lambda)_g\otimes(\mathrm{IC}_0)_{g^{-1}x}$, Frobenius-compatibly. Grothendieck–Lefschetz on the finite-type fibre gives $\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)(x)=\sum_{gK\in m^{-1}(x)(\mathbb F_q)}\operatorname{tr}(V_\lambda)(g)\,\mathbf 1_{X(\mathcal O)}(g^{-1}x)=(\operatorname{tr}V_\lambda\ast_K\varphi_0)(x)$ with $\operatorname{vol}K=1$. This is Zhu, Lemma 5.6.1, with the second factor on $LX$ instead of $\mathrm{Gr}$.

$\operatorname{tr}V_\lambda$ is left-$K$-invariant on $G(F)/K$ and supported on $\mathrm{Gr}_{\le\lambda}(\mathbb F_q)=\bigsqcup_{\mu\le\lambda}Kt^\mu K/K$, so it lies in $\mathcal H_K\otimes\overline{\mathbb Q}_\ell$. Unitriangularity (Zhu (5.6.2), from Kazhdan–Lusztig purity of IC stalks): $\operatorname{tr}V_\lambda=\varepsilon_\lambda\mathbf 1_{Kt^\lambda K}+\sum_{\mu<\lambda}a_{\lambda\mu}(q)\mathbf 1_{Kt^\mu K}$, $\varepsilon_\lambda=(-1)^{\langle2\rho,\lambda\rangle}\ne0$ (the value of the normalised constant sheaf on the open stratum); the lower sets $\{\mu\le\lambda\}$ are finite; so the $\operatorname{tr}V_\lambda$ form a basis. Over $\mathbb F_{q^n}$ replace $q$ by $q^n$. 3. The last claim: $\mathcal H_K\cdot\varphi_0=\{f\ast\varphi_0\}$ is the $\mathbb C$-span of $\{f\ast\varphi_0: f\text{ in a basis}\}$. $\square$

Satake form (cited, not re-derived). Zhu Lemma 5.6.3 / Lusztig–Kato: $\mathsf S(\iota\operatorname{tr}V_\lambda)=(-\iota q^{1/2})^{\langle2\rho,\lambda\rangle}\chi_\lambda$ with the normalisation $V_\lambda|_{\mathrm{Gr}_\lambda}=\overline{\mathbb Q}_\ell[\langle2\rho,\lambda\rangle]$. So $\operatorname{tr}V_\lambda$ is, up to the sign and power of $q^{1/2}$, the element of $\mathcal H_K$ with Satake transform the character $\chi_\lambda$ of the dual-group representation; under Theorem E (p-adic, (A)) its image in $C_K\simeq\mathbb C[\delta^{1/2}A_X^*]^{W_X}$ would be $r(\chi_\lambda)$ up to that scalar. Not used below.

Sources

sources/zhu-satake-v2.pdf Lemma 5.6.1, (5.6.2), Lemma 5.6.3
/Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/proof_graphs/spherical-generation/criterion.html Step 5(a) (stalk formula)
/Users/miltonlin/Projects/Trees/subpages/case-study-1-ramifiedrelative/proof_graphs/spherical-generation/classical.html Lemma A(2)

Table 1. Page production by session
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866

Lemma 4.3 · The trace of \(V_\lambda\ast\IC_0\) is \(\tr(V_\lambda)\ast\varphi_0\), and the traces of the \(V_\lambda\) form a basis of \(\cH_K\)

Builds on

Built on by

Trace of a Satake object is a Hecke translate of the basic function; traces of V_lambda span H_K

Sources