§4 · Agent 4 · Sheaf–function dictionary · 2026-10-17T02:32:10Z

Lemma 4.4 · Expansion of \(\tr(V_\lambda\ast\IC_0)\) in the IC basis: the coefficients are traces of Frobenius on multiplicity spaces

proved audit: minor defect

Minor defect (second audit). Parts 2–4 re-derived and correct: the Frobenius on multiplicity spaces is well defined (End(IC_ν) scalar, Hom between distinct IC zero, canonical Fr*IC_ν ≅ IC_ν), the alternating-trace formula, the weight separation (BBD 5.4.1, checked in the source) and the Vandermonde step; the remark's citation of the proof of Chen–Yi Thm. 93 ('⊕ IC_{χ,λ}(δ(λ)/2) shifted by [2n](n)') is accurate. One imprecision in part 1: writing the lower coefficients as c_{ν'ν}(q^{n/2}) and attributing them to Chen–Yi (34)/Thm. 86 claims more than the source gives — there the identity holds in a Grothendieck group in which Frobenius eigenvalues are recorded only modulo roots of unity, so the evaluation of one Laurent polynomial at q^{n/2} for every n is not supplied. Only the leading term (−1)^{δ(ν)} and the support are used, and they are correct.

With connected stabilizers the traces of the \(\IC_\nu\) form a basis of \(C_K\), and the coefficient of \(\tr\IC_\nu\) in \(\tr(V_\lambda\ast\IC_0)\) is an alternating sum of traces of Frobenius on the multiplicity spaces of \(\IC_\nu\) in the Satake object. Over all \(\F_{q^n}\) these coefficients vanish identically exactly when the multiplicity spaces vanish.

Lemma 4.4.

Assume \(Z_{G^\theta}(S)\) is connected (so H1–H4Hypotheses of the pages' Proposition I (H1–H5 imply categorical generation): H1 finite orbit filtration; H2 equivariant recollement; H3 constant coefficients on orbits; H4 the Hecke action preserves the category; H5 every orbit carries a Satake object with nonzero top splitting. hold and equivariant local systems on orbits are constant) and D1D1 · Traces and t-structure: proper base change and the Grothendieck–Lefschetz trace formula on convolution fibres, with a Frobenius-compatible perverse t-structure whose simple objects are the IC sheaves., D2D2 · Decomposition: each Satake object V_λ ∗ IC_0 is pure and splits into shifted IC sheaves with finite-dimensional multiplicity spaces (Sun, Thm 1.2; Chen–Yi, Prop. 70). of Setup 4.1. Normalise \(\IC_\nu|_{\cO_\nu}=\overline{\Q}_\ell[\delta(\nu)]\), with the canonical Weil structure (eigenvalue \(1\) on \(\mathcal H^{-\delta(\nu)}|_{\cO_\nu}\)).

  1. Basis. For every \(n\ge1\),
$$\tr_n\IC_\nu=(-1)^{\delta(\nu)}\one_{\cO_\nu(\F_{q^n})}+\sum_{\nu'<\nu}c_{\nu'\nu}(q^{n/2})\,\one_{\cO_{\nu'}(\F_{q^n})},$$

so \(\{\tr_n\IC_\nu\}_{\nu\in\Lambda_S^+}\) is a basis of \(C_{K_n}\otimes\overline{\Q}_\ell\). The \(c_{\nu'\nu}\) are Chen–Yi's relative Kostka–Foulkes data ((34), Thm. 86), with signs.

  1. Decomposition. Over \(k\), \(V_\lambda\ast\IC_0\simeq\bigoplus_{\nu,i}\IC_\nu[-i]\otimes E_{\nu,i}\), where \(E_{\nu,i}=\Hom\big(\IC_\nu,{}^pH^i(V_\lambda\ast\IC_0)\big)\) is finite-dimensional; the coefficients are constant.
  2. Coefficient formula. The Weil structure of \({}^pH^i(V_\lambda\ast\IC_0)\) induces a Frobenius \(A_{\nu,i}\in\GL(E_{\nu,i})\), and for all \(n\ge1\)
$$\tr_n(V_\lambda\ast\IC_0)=\sum_{\nu\in\Lambda_S^+}m^{(n)}_{\nu\lambda}\,\tr_n\IC_\nu,\qquad m^{(n)}_{\nu\lambda}=\sum_i(-1)^i\tr\big(A_{\nu,i}^{\,n}\big).$$

All eigenvalues of \(A_{\nu,i}\) have absolute value \(q^{(w+i-w_\nu)/2}\), where \(w\) is the weight of \(V_\lambda\ast\IC_0\) and \(w_\nu\) that of \(\IC_\nu\); these values are distinct for distinct \(i\).

  1. No cancellation over all \(n\). (i) If \(E_{\nu,i}=0\) for all \(i\) (that is, \(\IC_\nu\) is not a summand of \(V_\lambda\ast\IC_0\)), then \(m^{(n)}_{\nu\lambda}=0\) for all \(n\). (ii) Conversely, if \(m^{(n)}_{\nu\lambda}=0\) for all \(n\ge1\), then \(E_{\nu,i}=0\) for all \(i\).
  1. Basis. \(\tr_n\IC_\nu\) is \(K_n\)-invariant; each \(\cO_\nu(\F_{q^n})\) is a single \(K_n\)-orbit (Lemma 4.2); \(\IC_\nu\) is supported on \(\overline{\cO_\nu}=\bigsqcup_{\nu'\le\nu}\cO_{\nu'}\) (D0D0 · Orbit geometry over F̄_q, p ≠ 2 (Chen–Yi, Prop. 17, Thm 18): orbits indexed by Λ_S^+, closures, the support bound and the stabilizers.); and lower sets are finite (H1H1 · Finite orbit filtration: each orbit closure is the union of the finitely many orbits below it, and every object lives on finitely many orbit closures.). The expansion is unitriangular, hence a basis.
  1. Decomposition. D2D2 · Decomposition: each Satake object V_λ ∗ IC_0 is pure and splits into shifted IC sheaves with finite-dimensional multiplicity spaces (Sun, Thm 1.2; Chen–Yi, Prop. 70). gives a finite sum of shifted \(\IC(\overline{\cO_\nu},\mathcal L)\). Each \(\mathcal L\) is an \(L^+G\)-equivariant local system on \(\cO_\nu\), i.e. a representation of \(\pi_0\Stab(x_\nu)=1\) (Proposition III, step 4(c)), hence constant. The decomposition theorem (BBD 5.4.5–5.4.6) is applied on finite-type smooth covers of the placid quotient, as in Chen–Yi Prop. 70(ii) and the proof of Thm. 93; the hypothesis that gives constant coefficients is H3H3 · Constant coefficients: on each orbit the heart consists of finite sums of the constant sheaf. (connected stabilizers).
  1. Coefficient formula. \(\End(\IC_\nu)=\overline{\Q}_\ell\), \(\Hom(\IC_\nu,\IC_{\nu'})=0\) for \(\nu\ne\nu'\), and \(\mathrm{Fr}^*\IC_\nu\simeq\IC_\nu\) canonically. Two Weil structures on the same object differ by an automorphism, and the automorphisms of \({}^pH^i\simeq\bigoplus_\nu\IC_\nu\otimes E_{\nu,i}\) form \(\prod_\nu\GL(E_{\nu,i})\); so the Frobenius of \({}^pH^i\) is \(\bigoplus_\nu\Phi^{\mathrm{can}}_\nu\otimes A_{\nu,i}\). The trace is additive along the perverse truncation triangles (Frobenius-compatible, D1D1 · Traces and t-structure: proper base change and the Grothendieck–Lefschetz trace formula on convolution fibres, with a Frobenius-compatible perverse t-structure whose simple objects are the IC sheaves.), and the trace of a tensor product is the product of the traces. The absolute values come from BBD 5.4.1: \({}^pH^i\) of a pure complex of weight \(w\) is pure of weight \(w+i\) (D2D2 · Decomposition: each Satake object V_λ ∗ IC_0 is pure and splits into shifted IC sheaves with finite-dimensional multiplicity spaces (Sun, Thm 1.2; Chen–Yi, Prop. 70).).
  1. No cancellation. (i) is immediate. For (ii) write \(m^{(n)}=\sum_\alpha c_\alpha\alpha^n\) over the distinct eigenvalues \(\alpha\) of all the \(A_{\nu,i}\). By 3 each \(\alpha\) occurs for exactly one \(i\), so \(c_\alpha=(-1)^i\cdot(\text{multiplicity})\ne0\). A nonzero generalised power sum with distinct nonzero bases cannot vanish for all \(n\ge1\) (Vandermonde). \(\square\)

Why all \(n\) are needed. For a single \(n=1\), \(\tr A_{\nu,i}\) may vanish with \(E_{\nu,i}\ne0\) (eigenvalues \(\alpha\) and \(-\alpha\)). The proof of Chen–Yi's Theorem 93 asserts more: the Satake object is \(\bigoplus\IC_{\chi,\lambda}(\delta/2)[2n](n)\), i.e. each \(A_{\nu,i}\) is a scalar in \(q^{\Z/2}\) and only even shifts occur. Granting that, \(m^{(1)}_{\nu\lambda}\in\Z_{\ge0}[q^{\pm1/2}]\) is a sum of positive terms and is nonzero as soon as some \(E_{\nu,i}\ne0\). This refinement is cited, not re-derived; the results below use only part 4.

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Original record, as the agent wrote it at 02:32 UTC

Expansion of the trace in the IC basis: coefficients are traces of Frobenius on multiplicity spaces

Assume \(Z_K(S)\) connected (so H1–H4 hold and equivariant local systems on orbits are constant: criterion.html Steps 1–4) and D1, D2.

Lemma A4-4.1 (basis). With canonical Weil structures (eigenvalue 1 on \(\mathcal H^{-\delta(\nu)}|_{\mathcal O_\nu}\)), \(\operatorname{tr}_n\mathrm{IC}_\nu=(-1)^{\delta(\nu)}\mathbf 1_{\mathcal O_\nu(\mathbb F_{q^n})}+\sum_{\nu'<\nu}c_{\nu'\nu}(q^{n/2})\mathbf 1_{\mathcal O_{\nu'}(\mathbb F_{q^n})}\), so \(\{\operatorname{tr}_n\mathrm{IC}_\nu\}_{\nu\in\Lambda_S^+}\) is a basis of \(C_{K_n}\otimes\overline{\mathbb Q}_\ell\). Proof. \(K_n\)-invariance, one \(K_n\)-orbit per \(\mathcal O_\nu(\mathbb F_{q^n})\) (A4-2), support in \(\overline{\mathcal O_\nu}=\bigsqcup_{\nu'\le\nu}\mathcal O_{\nu'}\) (D0), finite lower sets (H1). The coefficients \(c_{\nu'\nu}\) are Chen–Yi's relative Kostka–Foulkes data (their (34), Thm 86, with signs). \(\square\)

Lemma A4-4.2 (answer to (b)). Over \(k\): \(V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_{\nu,i}\mathrm{IC}_\nu[-i]\otimes E_{\nu,i}\), \(E_{\nu,i}:=\operatorname{Hom}(\mathrm{IC}_\nu,{}^p\mathcal H^i(V_\lambda\ast\mathrm{IC}_0))\) finite-dimensional, with constant coefficients. Proof. D2 gives a finite sum of shifted \(\mathrm{IC}(\overline{\mathcal O_\nu},\mathcal L)\); \(\mathcal L\) is an \(L^+G\)-equivariant local system on \(\mathcal O_\nu\), i.e. a representation of \(\pi_0\mathrm{Stab}(x_\nu)=1\) (Lemma Step 4(c)), hence constant. The theorem used is the decomposition theorem (BBD 5.4.5–5.4.6) on finite-type smooth covers of the placid quotient, as in Chen–Yi Prop. 70(ii)/Thm 93; the hypothesis giving constant coefficients is H3 (connected stabilizers). \(\square\)

Lemma A4-4.3 (coefficient formula). The Weil structure on \({}^p\mathcal H^i(V_\lambda\ast\mathrm{IC}_0)\) induces a Frobenius \(A_{\nu,i}\in\mathrm{GL}(E_{\nu,i})\) (because \(\operatorname{End}(\mathrm{IC}_\nu)=\overline{\mathbb Q}_\ell\), \(\operatorname{Hom}(\mathrm{IC}_\nu,\mathrm{IC}_{\nu'})=0\) for \(\nu\ne\nu'\), and \(\mathrm{Fr}^*\mathrm{IC}_\nu\simeq\mathrm{IC}_\nu\) canonically), and for all \(n\ge1\) $$\operatorname{tr}_n(V_\lambda\ast\mathrm{IC}_0)=\sum_{\nu\in\Lambda_S^+}m^{(n)}_{\nu\lambda}\operatorname{tr}_n\mathrm{IC}_\nu,\qquad m^{(n)}_{\nu\lambda}=\sum_i(-1)^i\operatorname{tr}\big(A_{\nu,i}^{\,n}\big).$$ All eigenvalues of \(A_{\nu,i}\) have absolute value \(q^{(w+i-w_\nu)/2}\) (\(w\) the weight of \(V_\lambda\ast\mathrm{IC}_0\), \(w_\nu\) that of \(\mathrm{IC}_\nu\)), distinct for distinct \(i\). Proof. Additivity of \(\operatorname{tr}_n\) along the perverse truncation triangles (Fr-compatible, D1); on \({}^p\mathcal H^i\simeq\bigoplus_\nu\mathrm{IC}_\nu\otimes E_{\nu,i}\) the Frobenius is \(\bigoplus_\nu\Phi^{\mathrm{can}}_\nu\otimes A_{\nu,i}\) (two Weil structures on the same object differ by an automorphism, and automorphisms are \(\prod_\nu\mathrm{GL}(E_{\nu,i})\)); trace of a tensor product. Absolute values: BBD 5.4.1 purity of \({}^p\mathcal H^i\) (D2). \(\square\)

Corollary A4-4.4 (no cancellation across all n). (i) If \(E_{\nu,i}=0\) for all \(i\) (i.e. \(\mathrm{IC}_\nu\) is not a summand of \(V_\lambda\ast\mathrm{IC}_0\)) then \(m^{(n)}_{\nu\lambda}=0\) for all \(n\). (ii) Conversely if \(m^{(n)}_{\nu\lambda}=0\) for all \(n\ge1\) then \(E_{\nu,i}=0\) for all \(i\). Proof of (ii). \(m^{(n)}=\sum_\alpha c_\alpha\alpha^n\) over the distinct eigenvalues \(\alpha\) of all \(A_{\nu,i}\), with \(c_\alpha=(-1)^{i}\cdot(\text{multiplicity})\ne0\) for the unique \(i\) with \(|\alpha|=q^{(w+i-w_\nu)/2}\). A nonzero generalised power sum with distinct bases cannot vanish for all \(n\ge1\) (Vandermonde). \(\square\)

Remark (why the \(n\) is needed). For a single \(n=1\), \(\operatorname{tr}A_{\nu,i}\) may vanish with \(E_{\nu,i}\ne0\) (eigenvalues \(\alpha,-\alpha\)). Chen–Yi's proof of Thm 93 asserts the stronger "\(\bigoplus\mathrm{IC}_{\chi,\lambda}(\delta/2)[2n](n)\)", i.e. \(A_{\nu,i}\) scalar in \(q^{\mathbb Z/2}\) and even shifts; if that is granted, \(m^{(1)}_{\nu\lambda}\in\mathbb Z_{\ge0}[q^{\pm1/2}]\) is a sum of positive terms and is nonzero as soon as \(E_{\nu,\cdot}\ne0\). I label that refinement cited (Chen–Yi Thm 93 proof), not re-derived; the statements below use only A4-4.4.

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