§3 · Agent 3 · Functions ⇒ sheaves by traces · 2026-10-18T18:40:04Z

Lemma 3.3 · Trace of $*_I$ is convolution (vol $I=1$); IC traces span $\mathcal H_I$

proved

The trace of Frobenius intertwines the sheaf action $*_I$ with function convolution normalized by $\mathrm{vol}(I)=1$, and the traces of the $\mathrm{IC}_w$ are unitriangular in the $T_w$, so the Hecke translates of $\varphi_0$ are spanned by traces of the generators $\mathrm{IC}_w*_I\mathrm{IC}_0$.

Lemma 3.3.

Under Setup 3.1 (with (D1), (D2)), for Weil objects $\mathcal G\in\mathcal H_{c,I}$ supported on $\mathrm{Fl}{\le w}$ and $M\in\mathcal C_I$, and $x\in X(F)=LX(\mathbb F_q)$, with $\mathrm{tr}(M)(x)=\sum_i(-1)^i\mathrm{tr}(\Phi{\bar x},\mathcal H^i(M)_{\bar x})$:

(a) $\displaystyle \mathrm{tr}(\mathcal G*IM)(x)=\sum{gI\in G(F)/I}\mathrm{tr}(\mathcal G)(g)\,\mathrm{tr}(M)(g^{-1}x)=(\mathrm{tr}\,\mathcal G*\mathrm{tr}\,M)(x)$, the convolution of $\mathcal H_I$ on $C_I$ with $\mathrm{vol}(I)=1$; in particular $\mathrm{tr}(\mathrm{IC}_w*_I\mathrm{IC}_0)=\mathrm{tr}(\mathrm{IC}_w)*\varphi_0$.

(b) With canonical Weil structures, $\mathrm{tr}\,\mathrm{IC}w=(-1)^{\ell(w)}T_w+\sum{v<w}c_{vw}T_v$ (Bruhat order), so $\{\mathrm{tr}\,\mathrm{IC}w\}{w\in\widetilde W}$ is a basis of $\mathcal H_I\otimes\overline{\mathbb Q}_\ell$.

(c) $\mathcal H_I*\varphi_0\otimes\overline{\mathbb Q}\ell=\mathrm{span}\{\mathrm{tr}(\mathrm{IC}_w*_I\mathrm{IC}_0):w\in\widetilde W\}$, where the left side is the $\overline{\mathbb Q}\ell$-span of the $\mathbb Z$-valued functions $T_w*\varphi_0$.

  1. Shearing. $LG\times^ILX\xrightarrow{\sim}\mathrm{Fl}\times LX$, $[g,y]\mapsto(gI,gy)$, with inverse $(gI,x)\mapsto[g,g^{-1}x]$, turns $m$ into the second projection. So $\mathcal G*IM=\mathrm{pr}{2!}K$ with $K$ supported on $\mathrm{Fl}{\le w}\times Y{N'}$ (D1) and $K_{(gI,x)}\simeq\mathcal G_g\otimes M_{g^{-1}x}$, compatibly with Frobenius at rational points (the stalk of $M$ at $g^{-1}x$ depends only on $gI$ by $I$-equivariance).
  2. Finite-type model. By (D1), $(gI,x)\mapsto\pi_{N,M}(g^{-1}x)$ factors through $\mathrm{Fl}{\le w}\times Y{N',M''}$, so $K$ is pulled back from a constructible complex $\bar K$ on the finite-type scheme $\mathrm{Fl}{\le w}\times Y{N',M''}$, and $\mathcal G*IM$ is the pullback of $\mathrm{pr}{2!}\bar K$, where $\mathrm{pr}2$ is proper ($\mathrm{Fl}{\le w}$ is projective).
  3. Lefschetz. Proper base change and the trace formula [Laumon (1.1.1.3)] on the fibre $\mathrm{Fl}{\le w}\times\{\pi{N',M''}(x)\}$ give $\mathrm{tr}(\mathcal G*IM)(x)=\sum{gI\in\mathrm{Fl}_{\le w}(\mathbb F_q)}\mathrm{tr}(\mathcal G)(g)\,\mathrm{tr}(M)(g^{-1}x)$, using $\mathrm{tr}(A\otimes B)=\mathrm{tr}A\cdot\mathrm{tr}B$ [Laumon (1.1.1.2)].
  4. Points. $\mathrm{Fl}(\mathbb F_q)=G(F)/I(\mathcal O)$ since $H^1(\mathbb F_q,I)=1$ (D2), and $\mathrm{Fl}{\le w}(\mathbb F_q)\supseteq\mathrm{supp}\,\mathrm{tr}\,\mathcal G$. The sum over $G(F)/I$ of $f(g)\varphi(g^{-1}x)$ is $\int{G(F)}f(g)\varphi(g^{-1}x)\,dg$ for the Haar measure with $\mathrm{vol}(I)=1$. With $\mathrm{tr}\,\mathrm{IC}0=\mathbf 1{L^+X(\mathbb F_q)}=\mathbf 1_{X(\mathcal O)}=\varphi_0$ this is (a). (Group case: $\mathrm{Fl}_G=\mathrm{Fl}_H\times\mathrm{Fl}_H$ and the same computation on both sides.)
  5. (b). $\mathrm{IC}w$ is $I$-equivariant and supported on $\mathrm{Fl}{\le w}$, whose rational points are $\bigsqcup_{v\le w}I(\mathcal O)vI(\mathcal O)/I(\mathcal O)$ (Lang for the connected $I\cap vIv^{-1}$); on the open cell it is $\overline{\mathbb Q}\ell[\ell(w)]$ with Frobenius $1$, value $(-1)^{\ell(w)}$. Bruhat lower sets $\{v\le w\}$ in $\widetilde W=W{\mathrm{aff}}\rtimes\Omega$ are finite, so the transition matrix to $\{T_v\}$ is invertible.
  6. (c). By (b), $\mathcal H_I*\varphi_0\otimes\overline{\mathbb Q}_\ell$ is spanned by $\mathrm{tr}(\mathrm{IC}_w)*\varphi_0=\mathrm{tr}(\mathrm{IC}_w*_I\mathrm{IC}_0)$ by (a). $\square$

Check (vol $I=1$). The normalization matters only in (a): with $\mathrm{vol}(K)=1$ instead, the right side of (a) acquires the factor $[K:I]^{-1}$; spans are unaffected.

Builds on

  • Setup 3.1 · Set-up over $\mathbb F_q$: levels, the perverse heart of $\mathcal C_I$, Weil structures
  • Lemma 3.2 · Functions at a finite level: bases $\{\mathbf 1_{P(\mathbb F_q)}\}$ by Lang

Built on by

  • Lemma 3.4 · Unitriangular IC-trace basis of $C_I$ and the decomposition matrix $m_{S,w}$
  • Theorem 3.6 · (II$_I$): (IG) over $\mathbb F_q(\!(t)\!)$ implies $\mathcal M_I=\mathrm{Irr}$
  • Audit 7.6 · Audit of A3-1, A3-2, A3-3: glued Weil heart, piece-indicator bases by Lang, $\mathrm{tr}(\mathcal G\ast_IM)=\mathrm{tr}\,\mathcal G\ast\mathrm{tr}\,M$ with $\mathrm{vol}(I)=1$
Original record, as the agent wrote it at 18:40 UTC

Trace of $*_I$ is convolution (vol $I=1$); IC traces span $\mathcal H_I$

  1. Shearing. $LG\times^ILX\xrightarrow{\sim}\mathrm{Fl}\times LX$, $[g,y]\mapsto(gI,gy)$, with inverse $(gI,x)\mapsto[g,g^{-1}x]$, turns $m$ into the second projection. So $\mathcal G*IM=\mathrm{pr}{2!}K$ with $K$ supported on $\mathrm{Fl}{\le w}\times Y{N'}$ (D1) and $K_{(gI,x)}\simeq\mathcal G_g\otimes M_{g^{-1}x}$, compatibly with Frobenius at rational points (the stalk of $M$ at $g^{-1}x$ depends only on $gI$ by $I$-equivariance).
  2. Finite-type model. By (D1), $(gI,x)\mapsto\pi_{N,M}(g^{-1}x)$ factors through $\mathrm{Fl}{\le w}\times Y{N',M''}$, so $K$ is pulled back from a constructible complex $\bar K$ on the finite-type scheme $\mathrm{Fl}{\le w}\times Y{N',M''}$, and $\mathcal G*IM$ is the pullback of $\mathrm{pr}{2!}\bar K$, where $\mathrm{pr}2$ is proper ($\mathrm{Fl}{\le w}$ is projective).
  3. Lefschetz. Proper base change and the trace formula [Laumon (1.1.1.3)] on the fibre $\mathrm{Fl}{\le w}\times\{\pi{N',M''}(x)\}$ give $\mathrm{tr}(\mathcal G*IM)(x)=\sum{gI\in\mathrm{Fl}_{\le w}(\mathbb F_q)}\mathrm{tr}(\mathcal G)(g)\,\mathrm{tr}(M)(g^{-1}x)$, using $\mathrm{tr}(A\otimes B)=\mathrm{tr}A\cdot\mathrm{tr}B$ [Laumon (1.1.1.2)].
  4. Points. $\mathrm{Fl}(\mathbb F_q)=G(F)/I(\mathcal O)$ since $H^1(\mathbb F_q,I)=1$ (D2), and $\mathrm{Fl}{\le w}(\mathbb F_q)\supseteq\mathrm{supp}\,\mathrm{tr}\,\mathcal G$. The sum over $G(F)/I$ of $f(g)\varphi(g^{-1}x)$ is $\int{G(F)}f(g)\varphi(g^{-1}x)\,dg$ for the Haar measure with $\mathrm{vol}(I)=1$. With $\mathrm{tr}\,\mathrm{IC}0=\mathbf 1{L^+X(\mathbb F_q)}=\mathbf 1_{X(\mathcal O)}=\varphi_0$ this is (a). (Group case: $\mathrm{Fl}_G=\mathrm{Fl}_H\times\mathrm{Fl}_H$ and the same computation on both sides.)
  5. (b). $\mathrm{IC}w$ is $I$-equivariant and supported on $\mathrm{Fl}{\le w}$, whose rational points are $\bigsqcup_{v\le w}I(\mathcal O)vI(\mathcal O)/I(\mathcal O)$ (Lang for the connected $I\cap vIv^{-1}$); on the open cell it is $\overline{\mathbb Q}\ell[\ell(w)]$ with Frobenius $1$, value $(-1)^{\ell(w)}$. Bruhat lower sets $\{v\le w\}$ in $\widetilde W=W{\mathrm{aff}}\rtimes\Omega$ are finite, so the transition matrix to $\{T_v\}$ is invertible.
  6. (c). By (b), $\mathcal H_I*\varphi_0\otimes\overline{\mathbb Q}_\ell$ is spanned by $\mathrm{tr}(\mathrm{IC}_w)*\varphi_0=\mathrm{tr}(\mathrm{IC}_w*_I\mathrm{IC}_0)$ by (a). $\square$

Check (vol $I=1$). The normalization matters only in (a): with $\mathrm{vol}(K)=1$ instead, the right side of (a) acquires the factor $[K:I]^{-1}$; spans are unaffected.

Sources