§4 · Agent 4 · The group: sign and weights · 2026-10-18T18:42:29Z

Lemma 4.3 · Sign character on IC traces: $\varepsilon(\operatorname{tr}\mathrm{IC}_z)=[z\in\Omega]$

proved

Over every F_{q^m}((t)) the sign character of the Iwahori–Hecke algebra vanishes on the trace function of every IC_z with z of positive length, equals 1 on the length-zero ones, and kills the trace of IC_0. This is the linear functional that the no-cancellation theorem evaluates.

Lemma 4.3.

Let \(m\ge1\), \(F_m=\mathbb F_{q^m}(\!(t)\!)\), \(\mathcal H_{I,m}=C_c(I_H\backslash H(F_m)/I_H)\) with \(\operatorname{vol}(I_H)=1\), \(T_w=\mathbf 1_{I_HwI_H}\), and \(\operatorname{tr}_m\) the trace of Frobenius of a Weil object of \(\mathcal H_{c,I_H}\) at the points \(\mathrm{Fl}(\mathbb F_{q^m})\). (a) \(\varepsilon:\mathcal H_{I,m}\to\mathbb Z\), \(\varepsilon(T_w)=(-1)^{\ell(w)}\), is a ring homomorphism (for any non-archimedean local field). Its value on \(T_\omega\), \(\omega\in\Omega\), is \(1\). (b) \(\mathbf 1_{K_H}=\sum_{w\in W}T_w\) and \(\varepsilon(\mathbf 1_{K_H})=\sum_{w\in W}(-1)^{\ell(w)}=0\) because \(W\ne1\). (c) \(\operatorname{tr}_m(\mathcal A\ast\mathcal B)=\operatorname{tr}_m\mathcal A\ast\operatorname{tr}_m\mathcal B\) (convolution with \(\operatorname{vol}(I_H)=1\)). (d) With canonical Weil structures: \(\operatorname{tr}_m\mathrm{IC}_\omega=T_\omega\), \(\operatorname{tr}_m\mathrm{IC}_{w_0}=(-1)^{\ell(w_0)}\mathbf 1_{K_H}\), \(\operatorname{tr}_m\mathrm{IC}_s=-(T_e+T_s)\) for a simple affine reflection \(s\). (e) For every \(z\in\widetilde W\setminus\Omega\): \(\varepsilon(\operatorname{tr}_m\mathrm{IC}_z)=0\). (a), (b) hold over any non-archimedean \(F\); (c)–(e) are statements about \(\ell\)-adic sheaves over \(\overline{\mathbb F}_q\).

Quadratic relation used: \(T_s^2=(q-1)T_s+qT_e\). It is the specialisation \(v=q^{-1/2}\), \(H_s=vT_s\), of Soergel's \((H_s+v)(H_s-v^{-1})=0\); under it \(\varepsilon\) becomes \(H_s\mapsto-v\), so \(\varepsilon(\underline H_s)=\varepsilon(H_s+v)=0\).

  1. \(\varepsilon\) is a character. Iwahori–Matsumoto relations (haines-kottwitz-prasad-v3.pdf, §7.2, display after Lemma 7.2.1): for \(x\in\widetilde W\), \(s\) simple affine, \(\sigma\in\Omega\): \(T_xT_s=T_{xs}\) if \(x<xs\), \(T_xT_s=qT_{xs}+(q-1)T_x\) if \(xs<x\), \(T_xT_\sigma=T_{x\sigma}\). Check: in the first case \((-1)^{\ell(x)+1}=\varepsilon(T_x)\varepsilon(T_s)\); in the second \(q(-1)^{\ell(x)-1}+(q-1)(-1)^{\ell(x)}=(-1)^{\ell(x)+1}\); in the third \((-1)^{\ell(x)}\). By linearity \(\varepsilon(hT_s)=\varepsilon(h)\varepsilon(T_s)\) and \(\varepsilon(hT_\sigma)=\varepsilon(h)\) for all \(h\). For a reduced expression \(y=s_1\cdots s_k\sigma\), \(T_y=T_{s_1}\cdots T_{s_k}T_\sigma\) (first and third relations), so induction on \(k\) gives \(\varepsilon(hT_y)=\varepsilon(h)\varepsilon(T_y)\); \(\varepsilon(T_e)=1\).
  2. \(\mathbf 1_K\). \(K_H=\bigsqcup_{w\in W}I_HwI_H\), and for a finite simple reflection \(s\) the involution \(w\mapsto ws\) of \(W\) changes the parity of \(\ell\); so the alternating sum is \(0\). (This is classical.html, Proposition E.)
  3. Trace of convolution. For \(\mathcal A,\mathcal B\) supported on \(\mathrm{Fl}_{\le u},\mathrm{Fl}_{\le u'}\), the fibre of the proper map \(m:\mathrm{Fl}_{\le u}\widetilde\times\mathrm{Fl}_{\le u'}\to\mathrm{Fl}\) over \(x\) is \(\{gI\in\mathrm{Fl}_{\le u}:g^{-1}x\in\mathrm{Fl}_{\le u'}\}\) with stalks \(\mathcal A_g\otimes\mathcal B_{g^{-1}x}\). Proper base change and the Grothendieck–Lefschetz formula (laumon-1987.pdf, (1.1.1.3)), with \(\mathrm{Fl}(\mathbb F_{q^m})=H(F_m)/I_H\) (Lang's theorem for the connected pro-algebraic \(I_H\), milne-iag-v2.pdf Cor. 27.56, as in trace.tex input D3), give \(\operatorname{tr}(\mathcal A\ast\mathcal B)(x)=\sum_{gI\in H(F_m)/I}\operatorname{tr}\mathcal A(g)\operatorname{tr}\mathcal B(g^{-1}x)\): the \(\operatorname{vol}(I)=1\) convolution (trace.tex, Lemma conv; zhu-satake-v2.pdf, Lemma 5.6.1). Checklist 5.
  4. Values. \(\mathrm{IC}_\omega\) is the skyscraper \(\overline{\mathbb Q}_\ell\) at the rational point \(\dot\omega I\); \(\overline{\mathrm{Fl}_{w_0}}=K_H/I_H\) is smooth, so \(\mathrm{IC}_{w_0}=\overline{\mathbb Q}_\ell[\ell(w_0)]\); \(\overline{\mathrm{Fl}_s}=P_s/I\cong\mathbb P^1\), \(\mathrm{IC}_s=\overline{\mathbb Q}_\ell[1]\); and \(\operatorname{tr}(\mathcal K[1])=-\operatorname{tr}\mathcal K\) (laumon-1987.pdf (1.1.1.1)). Hence \(\varepsilon(\operatorname{tr}\mathrm{IC}_s)=-(1-1)=0\), \(\varepsilon(\operatorname{tr}\mathrm{IC}_\omega)=1\).
  5. Descent. If \(z\notin\Omega\), write \(z=\omega z'\) with \(z'\in W_{\rm aff}\), \(\ell(z')=\ell(z)>0\); a right descent \(s\) of \(z'\) satisfies \(zs<z\). Then \(\overline{IzI}\) is right \(P_s\)-stable, so \(\overline{\mathrm{Fl}_z}=\pi_s^{-1}\pi_s(\overline{\mathrm{Fl}_z})\) for the \(\mathbb P^1\)-bundle \(\pi_s:\mathrm{Fl}\to LH/P_s\), and \(\mathrm{IC}_z\simeq\pi_s^*\mathcal G[1]\) with \(\mathcal G\) the IC complex of the image (smooth pullback). Right convolution with \(\underline k_{P_s/I}\) is \(\pi_s^*\pi_{s*}\) (base change in \(\mathrm{Fl}\times_{LH/P_s}\mathrm{Fl}\)), and \(\pi_{s*}\pi_s^*\mathcal G=\mathcal G\otimes\pi_{s*}\overline{\mathbb Q}_\ell\) with a Weil triangle \(\overline{\mathbb Q}_\ell\to\pi_{s*}\overline{\mathbb Q}_\ell\to\overline{\mathbb Q}_\ell(-1)[-2]\). So \(\mathrm{IC}_z\ast\mathrm{IC}_s\simeq\pi_s^*(\mathcal G\otimes\pi_{s*}\overline{\mathbb Q}_\ell)[2]\) and
$$\operatorname{tr}_m(\mathrm{IC}_z)\ast\big(-(T_e+T_s)\big)=\operatorname{tr}_m(\mathrm{IC}_z\ast\mathrm{IC}_s)=-(1+q^m)\operatorname{tr}_m(\mathrm{IC}_z).$$

Applying \(\varepsilon\) and step 4: \(0=-(1+q^m)\,\varepsilon(\operatorname{tr}_m\mathrm{IC}_z)\), hence \(\varepsilon(\operatorname{tr}_m\mathrm{IC}_z)=0\).

Hecke-algebra dictionary (not used in the proof): \(\operatorname{tr}_m\mathrm{IC}_z=(-1)^{\ell(z)}\sum_{y\le z}P_{y,z}(q^m)T_y=(-1)^{\ell(z)}q^{m\ell(z)/2}C'_z|_{q^m}\) for the Kazhdan–Lusztig basis \(C'_z=q^{-\ell(z)/2}\sum_yP_{y,z}(q)T_y\) (pointwise purity of Schubert varieties, cited in Proposition 4.7), and step 5 is the identity \(C'_zC'_s=(q^{1/2}+q^{-1/2})C'_z\) for \(zs<z\); with Soergel's \(\underline H_s=H_s+v\), the sign character \(H_s\mapsto-v\) kills \(\underline H_s\).

Builds on

  • Setup 4.1 · Group case: cell pieces, IC heart, two-sided generators $M_{x,y}$

Built on by

  • Lemma 4.4 · Trace expansion of $M_{x,y}$ with pure multiplicity Frobenii
  • 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 8.3 · Audit of the sign character (A4-3): $\varepsilon$ kills IC traces off $\Omega$
Original record, as the agent wrote it at 18:42 UTC

Sign character on IC traces: $\varepsilon(\operatorname{tr}\mathrm{IC}_z)=[z\in\Omega]$

Quadratic relation used: \(T_s^2=(q-1)T_s+qT_e\). It is the specialisation \(v=q^{-1/2}\), \(H_s=vT_s\), of Soergel's \((H_s+v)(H_s-v^{-1})=0\); under it \(\varepsilon\) becomes \(H_s\mapsto-v\), so \(\varepsilon(\underline H_s)=\varepsilon(H_s+v)=0\).

  1. \(\varepsilon\) is a character. Iwahori–Matsumoto relations (haines-kottwitz-prasad-v3.pdf, §7.2, display after Lemma 7.2.1): for \(x\in\widetilde W\), \(s\) simple affine, \(\sigma\in\Omega\): \(T_xT_s=T_{xs}\) if \(x<xs\), \(T_xT_s=qT_{xs}+(q-1)T_x\) if \(xs<x\), \(T_xT_\sigma=T_{x\sigma}\). Check: in the first case \((-1)^{\ell(x)+1}=\varepsilon(T_x)\varepsilon(T_s)\); in the second \(q(-1)^{\ell(x)-1}+(q-1)(-1)^{\ell(x)}=(-1)^{\ell(x)+1}\); in the third \((-1)^{\ell(x)}\). By linearity \(\varepsilon(hT_s)=\varepsilon(h)\varepsilon(T_s)\) and \(\varepsilon(hT_\sigma)=\varepsilon(h)\) for all \(h\). For a reduced expression \(y=s_1\cdots s_k\sigma\), \(T_y=T_{s_1}\cdots T_{s_k}T_\sigma\) (first and third relations), so induction on \(k\) gives \(\varepsilon(hT_y)=\varepsilon(h)\varepsilon(T_y)\); \(\varepsilon(T_e)=1\).
  2. \(\mathbf 1_K\). \(K_H=\bigsqcup_{w\in W}I_HwI_H\), and for a finite simple reflection \(s\) the involution \(w\mapsto ws\) of \(W\) changes the parity of \(\ell\); so the alternating sum is \(0\). (This is classical.html, Proposition E.)
  3. Trace of convolution. For \(\mathcal A,\mathcal B\) supported on \(\mathrm{Fl}_{\le u},\mathrm{Fl}_{\le u'}\), the fibre of the proper map \(m:\mathrm{Fl}_{\le u}\widetilde\times\mathrm{Fl}_{\le u'}\to\mathrm{Fl}\) over \(x\) is \(\{gI\in\mathrm{Fl}_{\le u}:g^{-1}x\in\mathrm{Fl}_{\le u'}\}\) with stalks \(\mathcal A_g\otimes\mathcal B_{g^{-1}x}\). Proper base change and the Grothendieck–Lefschetz formula (laumon-1987.pdf, (1.1.1.3)), with \(\mathrm{Fl}(\mathbb F_{q^m})=H(F_m)/I_H\) (Lang's theorem for the connected pro-algebraic \(I_H\), milne-iag-v2.pdf Cor. 27.56, as in trace.tex input D3), give \(\operatorname{tr}(\mathcal A\ast\mathcal B)(x)=\sum_{gI\in H(F_m)/I}\operatorname{tr}\mathcal A(g)\operatorname{tr}\mathcal B(g^{-1}x)\): the \(\operatorname{vol}(I)=1\) convolution (trace.tex, Lemma conv; zhu-satake-v2.pdf, Lemma 5.6.1). Checklist 5.
  4. Values. \(\mathrm{IC}_\omega\) is the skyscraper \(\overline{\mathbb Q}_\ell\) at the rational point \(\dot\omega I\); \(\overline{\mathrm{Fl}_{w_0}}=K_H/I_H\) is smooth, so \(\mathrm{IC}_{w_0}=\overline{\mathbb Q}_\ell[\ell(w_0)]\); \(\overline{\mathrm{Fl}_s}=P_s/I\cong\mathbb P^1\), \(\mathrm{IC}_s=\overline{\mathbb Q}_\ell[1]\); and \(\operatorname{tr}(\mathcal K[1])=-\operatorname{tr}\mathcal K\) (laumon-1987.pdf (1.1.1.1)). Hence \(\varepsilon(\operatorname{tr}\mathrm{IC}_s)=-(1-1)=0\), \(\varepsilon(\operatorname{tr}\mathrm{IC}_\omega)=1\).
  5. Descent. If \(z\notin\Omega\), write \(z=\omega z'\) with \(z'\in W_{\rm aff}\), \(\ell(z')=\ell(z)>0\); a right descent \(s\) of \(z'\) satisfies \(zs<z\). Then \(\overline{IzI}\) is right \(P_s\)-stable, so \(\overline{\mathrm{Fl}_z}=\pi_s^{-1}\pi_s(\overline{\mathrm{Fl}_z})\) for the \(\mathbb P^1\)-bundle \(\pi_s:\mathrm{Fl}\to LH/P_s\), and \(\mathrm{IC}_z\simeq\pi_s^*\mathcal G[1]\) with \(\mathcal G\) the IC complex of the image (smooth pullback). Right convolution with \(\underline k_{P_s/I}\) is \(\pi_s^*\pi_{s*}\) (base change in \(\mathrm{Fl}\times_{LH/P_s}\mathrm{Fl}\)), and \(\pi_{s*}\pi_s^*\mathcal G=\mathcal G\otimes\pi_{s*}\overline{\mathbb Q}_\ell\) with a Weil triangle \(\overline{\mathbb Q}_\ell\to\pi_{s*}\overline{\mathbb Q}_\ell\to\overline{\mathbb Q}_\ell(-1)[-2]\). So \(\mathrm{IC}_z\ast\mathrm{IC}_s\simeq\pi_s^*(\mathcal G\otimes\pi_{s*}\overline{\mathbb Q}_\ell)[2]\) and
$$\operatorname{tr}_m(\mathrm{IC}_z)\ast\big(-(T_e+T_s)\big)=\operatorname{tr}_m(\mathrm{IC}_z\ast\mathrm{IC}_s)=-(1+q^m)\operatorname{tr}_m(\mathrm{IC}_z).$$

Applying \(\varepsilon\) and step 4: \(0=-(1+q^m)\,\varepsilon(\operatorname{tr}_m\mathrm{IC}_z)\), hence \(\varepsilon(\operatorname{tr}_m\mathrm{IC}_z)=0\).

Hecke-algebra dictionary (not used in the proof): \(\operatorname{tr}_m\mathrm{IC}_z=(-1)^{\ell(z)}\sum_{y\le z}P_{y,z}(q^m)T_y=(-1)^{\ell(z)}q^{m\ell(z)/2}C'_z|_{q^m}\) for the Kazhdan–Lusztig basis \(C'_z=q^{-\ell(z)/2}\sum_yP_{y,z}(q)T_y\) (pointwise purity of Schubert varieties, cited in Proposition 4.7), and step 5 is the identity \(C'_zC'_s=(q^{1/2}+q^{-1/2})C'_z\) for \(zs<z\); with Soergel's \(\underline H_s=H_s+v\), the sign character \(H_s\mapsto-v\) kills \(\underline H_s\).

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