§4 · Agent 4 · The group: sign and weights · 2026-10-18T18:43:35Z

Lemma 4.4 · Trace expansion of $M_{x,y}$ with pure multiplicity Frobenii

proved

Expands the trace of each generator IC_x ∗ IC_{w_0} ∗ IC_y in the IC-trace basis; the coefficient of tr IC_z is an alternating sum of traces of the Frobenius on the multiplicity spaces E_{z,n}, and these Frobenii are pure of a weight that determines n. This weight control is what rules out cancellation.

Lemma 4.4.

Over \(\overline{\mathbb F}_q\), fix \(x,y\in\widetilde W\), put \(w=\ell(x)+\ell(w_0)+\ell(y)\), and let \(A_{z,n}:E_{z,n}\to E_{z,n}\) be the Frobenius of the multiplicity space (trace.tex, Definition A: \(f\mapsto\Phi_n\circ\mathrm{Fr}^*f\circ\Phi_z^{-1}\)). For every \(m\ge1\): $$\text{(i)}\ \ \operatorname{tr}_mM_{x,y}=\operatorname{tr}_m\mathrm{IC}_x\ast\operatorname{tr}_m\mathrm{IC}_{w_0}\ast\operatorname{tr}_m\mathrm{IC}_y,\qquad \text{(ii)}\ \ \operatorname{tr}_mM_{x,y}=\sum_{z,n}(-1)^n\operatorname{tr}\big(A_{z,n}^m\big)\,\operatorname{tr}_m\mathrm{IC}_z,$$ (iii) \(A_{z,n}\) is invertible and every eigenvalue \(\alpha\) of \(A_{z,n}\) satisfies \(|\iota(\alpha)|=q^{(w+n-\ell(z))/2}\) for every isomorphism \(\iota:\overline{\mathbb Q}_\ell\cong\mathbb C\).

  1. (i) is Lemma 4.3(c) applied twice; \(M_{x,y}\) is an iterated convolution in \(\mathcal H_{c,I_H}\) (Setup 4.1(d)).
  2. (ii). The perverse t-structure on \(\mathrm{Fl}_{\le u}\) is \(\mathrm{Fr}\)-stable, so \(P_n={}^pH^n(M_{x,y})\) carries a Weil structure \(\Phi_n\). By Setup 4.1(c),(e) (\(P_n\) geometrically semisimple, \(\operatorname{End}\mathrm{IC}_z=\overline{\mathbb Q}_\ell\), \(\operatorname{Hom}(\mathrm{IC}_z,\mathrm{IC}_{z'})=0\) for \(z\ne z'\)), the evaluation map \(\bigoplus_z\mathrm{IC}_z\otimes E_{z,n}\to P_n\) is an isomorphism carrying \(\bigoplus_z\Phi_z\otimes A_{z,n}\) to \(\Phi_n\) (trace.tex, Lemma frob). Over \(\mathbb F_{q^m}\) the \(m\)-th iterate gives \(A_{z,n}^m\) (trace.tex, footnote 1). The trace is additive on the \(\mathrm{Fr}\)-compatible perverse truncation triangles and \(\operatorname{tr}(P[-n])=(-1)^n\operatorname{tr}P\) (laumon-1987.pdf, (1.1.1.1)); on stalks \(\operatorname{tr}(\Phi_z\otimes A)=\operatorname{tr}\Phi_z\cdot\operatorname{tr}A\).
  3. Invertibility. \(A_{z,n}\) is a composite of the bijections \(f\mapsto\mathrm{Fr}^*f\), \(\Phi_n\circ-\), \(-\circ\Phi_z^{-1}\).
  4. Upper weight bound. \(M_{x,y}\) is pure of weight \(w\) (Setup 4.1(e)), so \(P_n\) is pure of weight \(w+n\) (bbd-perverse.pdf, 5.4.1–5.4.4: perverse cohomology of a pure complex). Take the rational point \(\dot zI\in\mathrm{Fl}_z\). In degree \(-\ell(z)\) its stalk is \(\bigoplus_{z'}E_{z',n}\otimes\mathcal H^{-\ell(z)}(\mathrm{IC}_{z'})_{\dot z}\); the term \(z'=z\) is \(E_{z,n}\otimes\overline{\mathbb Q}_\ell\) with Frobenius \(A_{z,n}\); for \(z'\ne z\) with \(\mathrm{Fl}_z\subseteq\overline{\mathrm{Fl}_{z'}}\) the strict support condition gives \(\mathcal H^{j}(\mathrm{IC}_{z'})|_{\mathrm{Fl}_z}=0\) for \(j\ge-\ell(z)\); all other terms vanish. Hence \(\mathcal H^{-\ell(z)}(P_n)_{\dot z}\cong(E_{z,n},A_{z,n})\), and mixedness of weight \(\le w+n\) bounds its weights by \(w+n-\ell(z)\).
  5. Lower bound. \(\mathbb DP_n\) is pure of weight \(-(w+n)\) and \(\mathbb DP_n\cong\bigoplus_z\mathbb D\mathrm{IC}_z\otimes E_{z,n}^\vee\) with \(\mathbb D\mathrm{IC}_z\cong\mathrm{IC}_z(\ell(z))\) (\(\mathbb D\) commutes with \(j_{!*}\) and \(\mathbb D(\overline{\mathbb Q}_\ell[d])=\overline{\mathbb Q}_\ell[d](d)\) on a smooth \(d\)-dimensional cell). Step 4 for \(\mathbb DP_n\): the Frobenius eigenvalues \(q^{-\ell(z)}\alpha^{-1}\) have weight \(\le-(w+n)-\ell(z)\), i.e. \(\operatorname{wt}(\alpha)\ge w+n-\ell(z)\). With step 4, \(\operatorname{wt}(\alpha)=w+n-\ell(z)\).

Remark. Steps 4–5 only use that \(M_{x,y}\) is pure; they do not use that the multiplicity spaces are Tate (that stronger fact, with parity, is Bezrukavnikov–Yun Prop. 3.2.5 and gives a second route, see Theorem 4.5).

Builds on

  • Setup 4.1 · Group case: cell pieces, IC heart, two-sided generators $M_{x,y}$
  • Lemma 4.3 · Sign character on IC traces: $\varepsilon(\operatorname{tr}\mathrm{IC}_z)=[z\in\Omega]$

Built on by

  • Theorem 4.5 · No $\Delta_\omega$ in any generator $M_{x,y}$; so $\Delta_e\notin\mathcal C_I^{\rm Iwa}$
  • Corollary 4.6 · (IG) fails by the same character; (II$_I$) holds for the group
  • Proposition 4.7 · $\mathcal M_I$ for the group: top cells in, $\Omega$ out, exact in rank one
  • Audit 8.4 · Audit of the trace expansion (A4-4): $E_{z,n}$ has weight $w+n-\ell(z)$
Original record, as the agent wrote it at 18:43 UTC

Trace expansion of $M_{x,y}$ with pure multiplicity Frobenii

  1. (i) is Lemma 4.3(c) applied twice; \(M_{x,y}\) is an iterated convolution in \(\mathcal H_{c,I_H}\) (Setup 4.1(d)).
  2. (ii). The perverse t-structure on \(\mathrm{Fl}_{\le u}\) is \(\mathrm{Fr}\)-stable, so \(P_n={}^pH^n(M_{x,y})\) carries a Weil structure \(\Phi_n\). By Setup 4.1(c),(e) (\(P_n\) geometrically semisimple, \(\operatorname{End}\mathrm{IC}_z=\overline{\mathbb Q}_\ell\), \(\operatorname{Hom}(\mathrm{IC}_z,\mathrm{IC}_{z'})=0\) for \(z\ne z'\)), the evaluation map \(\bigoplus_z\mathrm{IC}_z\otimes E_{z,n}\to P_n\) is an isomorphism carrying \(\bigoplus_z\Phi_z\otimes A_{z,n}\) to \(\Phi_n\) (trace.tex, Lemma frob). Over \(\mathbb F_{q^m}\) the \(m\)-th iterate gives \(A_{z,n}^m\) (trace.tex, footnote 1). The trace is additive on the \(\mathrm{Fr}\)-compatible perverse truncation triangles and \(\operatorname{tr}(P[-n])=(-1)^n\operatorname{tr}P\) (laumon-1987.pdf, (1.1.1.1)); on stalks \(\operatorname{tr}(\Phi_z\otimes A)=\operatorname{tr}\Phi_z\cdot\operatorname{tr}A\).
  3. Invertibility. \(A_{z,n}\) is a composite of the bijections \(f\mapsto\mathrm{Fr}^*f\), \(\Phi_n\circ-\), \(-\circ\Phi_z^{-1}\).
  4. Upper weight bound. \(M_{x,y}\) is pure of weight \(w\) (Setup 4.1(e)), so \(P_n\) is pure of weight \(w+n\) (bbd-perverse.pdf, 5.4.1–5.4.4: perverse cohomology of a pure complex). Take the rational point \(\dot zI\in\mathrm{Fl}_z\). In degree \(-\ell(z)\) its stalk is \(\bigoplus_{z'}E_{z',n}\otimes\mathcal H^{-\ell(z)}(\mathrm{IC}_{z'})_{\dot z}\); the term \(z'=z\) is \(E_{z,n}\otimes\overline{\mathbb Q}_\ell\) with Frobenius \(A_{z,n}\); for \(z'\ne z\) with \(\mathrm{Fl}_z\subseteq\overline{\mathrm{Fl}_{z'}}\) the strict support condition gives \(\mathcal H^{j}(\mathrm{IC}_{z'})|_{\mathrm{Fl}_z}=0\) for \(j\ge-\ell(z)\); all other terms vanish. Hence \(\mathcal H^{-\ell(z)}(P_n)_{\dot z}\cong(E_{z,n},A_{z,n})\), and mixedness of weight \(\le w+n\) bounds its weights by \(w+n-\ell(z)\).
  5. Lower bound. \(\mathbb DP_n\) is pure of weight \(-(w+n)\) and \(\mathbb DP_n\cong\bigoplus_z\mathbb D\mathrm{IC}_z\otimes E_{z,n}^\vee\) with \(\mathbb D\mathrm{IC}_z\cong\mathrm{IC}_z(\ell(z))\) (\(\mathbb D\) commutes with \(j_{!*}\) and \(\mathbb D(\overline{\mathbb Q}_\ell[d])=\overline{\mathbb Q}_\ell[d](d)\) on a smooth \(d\)-dimensional cell). Step 4 for \(\mathbb DP_n\): the Frobenius eigenvalues \(q^{-\ell(z)}\alpha^{-1}\) have weight \(\le-(w+n)-\ell(z)\), i.e. \(\operatorname{wt}(\alpha)\ge w+n-\ell(z)\). With step 4, \(\operatorname{wt}(\alpha)=w+n-\ell(z)\).

Remark. Steps 4–5 only use that \(M_{x,y}\) is pure; they do not use that the multiplicity spaces are Tate (that stronger fact, with parity, is Bezrukavnikov–Yun Prop. 3.2.5 and gives a second route, see Theorem 4.5).

Sources