II · The trace dictionary

The trace dictionary

The definition of $\operatorname{tr}_n$ (Proposition II uses $\operatorname{tr}=\operatorname{tr}_1$), what it does to each operation on sheaves, how the sources state it, and what does not transfer to the loop space $LX$. Sources checked 17 October 2026.

§1 definition · §2 dictionary · §3 examples · §4 why $n$ varies · §5 conventions in the sources · §6 what does not transfer · §7 where Proposition II uses it · §8 references.

1. Definition

Definition 1.1 (trace function over $\mathbb F_{q^n}$)

Let $Y_0$ be a scheme over $\mathbb F_q$, $Y=Y_0\otimes k$, $k=\overline{\mathbb F}_q$, and $\mathrm{Fr}:Y\to Y$ the $\mathbb F_q$-linear Frobenius endomorphism (coordinates to the $q$-th power), so that $Y_0(\mathbb F_{q^n})=\{y\in Y(k):\mathrm{Fr}^n(y)=y\}$. A Weil object is a pair

$$(M,\Phi),\qquad M\in D^b_c(Y,\overline{\mathbb Q}_\ell),\qquad\Phi:\mathrm{Fr}^*M\xrightarrow{\sim}M .$$

At a geometric point $\bar y$, $\Phi_{\bar y}:\mathcal H^i(M)_{\mathrm{Fr}(\bar y)}\to\mathcal H^i(M)_{\bar y}$. For $x\in Y_0(\mathbb F_{q^n})$ and $\bar x$ over it,

$$\Phi^{(n)}_{\bar x}=\Phi_{\bar x}\circ\Phi_{\mathrm{Fr}\bar x}\circ\cdots\circ\Phi_{\mathrm{Fr}^{n-1}\bar x}\in\operatorname{End}\mathcal H^i(M)_{\bar x},$$ $$\operatorname{tr}_n(M):Y_0(\mathbb F_{q^n})\to\overline{\mathbb Q}_\ell,\qquad\boxed{\operatorname{tr}_n(M)(x)=\sum_i(-1)^i\operatorname{tr}\big(\Phi^{(n)}_{\bar x},\ \mathcal H^i(M)_{\bar x}\big).}$$

Geometric Frobenius on stalks; no Tate twist; $\operatorname{tr}=\operatorname{tr}_1$; for $x\in Y_0(\mathbb F_q)$, $\Phi^{(n)}_{\bar x}=\Phi_{\bar x}^n$. On the Proposition II pages $Y=LX$, $M\in\mathcal C_{K,k}$ and $n=1$: $Y_0(\mathbb F_q)=X(F)$, $F=\mathbb F_q(\!(t)\!)$. General $n$ gives the same statements over $F_n=\mathbb F_{q^n}(\!(t)\!)$, $Y_0(\mathbb F_{q^n})=X(F_n)$; it is used only for the converse.

Why the composite $\Phi^{(n)}$

The $n$-fold iterate $\Phi\circ\mathrm{Fr}^*\Phi\circ\cdots\circ(\mathrm{Fr}^{n-1})^*\Phi:(\mathrm{Fr}^n)^*M\to M$ is the Weil structure relative to $\mathbb F_{q^n}$ (BBD 5.1.1, "par itération").
Its stalk at $\bar x$ is $\Phi^{(n)}_{\bar x}$, since the stalk of $(\mathrm{Fr}^j)^*\Phi$ at $\bar x$ is $\Phi_{\mathrm{Fr}^j\bar x}$; it maps $\mathcal H^i(M)_{\mathrm{Fr}^n\bar x}=\mathcal H^i(M)_{\bar x}$ to itself.
When $M$ comes from $M_0$ on $Y_0$, $\Phi^{(n)}_{\bar x}$ is the geometric Frobenius of $x$ acting on the stalk (Weil II (1.1.14), (1.1.8)), and $\operatorname{tr}_n(M)$ is Laumon's $t_{M_0|Y_0\otimes\mathbb F_{q^n}}$.

2. The dictionary

Table 2. 1. Operations on Weil objects and on their trace functions (for schemes of finite type over $\mathbb F_q$).
Sheaves	Functions	Source
triangle $M'\to M\to M''\to$	$\operatorname{tr}_nM=\operatorname{tr}_nM'+\operatorname{tr}_nM''$	Laumon (1.1.1.1)
shift $M[1]$; perverse cohomology	$-\operatorname{tr}_nM$; $\ \operatorname{tr}_nM=\sum_j(-1)^j\operatorname{tr}_n{}^p\mathcal H^jM$	Laumon (1.1.1.1)
Tate twist $\overline{\mathbb Q}_\ell(m)$	constant $q^{-nm}$	Laumon (1.1.1.0)
$M_1\otimes M_2$	$\operatorname{tr}_nM_1\cdot\operatorname{tr}_nM_2$	Laumon (1.1.1.2)
pullback $f^*M$	$\operatorname{tr}_nM\circ f$	Laumon (1.1.1.4)
$Rf_!M$	$y\mapsto\sum_{f(x)=y}\operatorname{tr}_nM(x)$	Laumon (1.1.1.3) (Grothendieck); for $Rf_*$ only when $f$ is proper, (1.1.1.5)
convolution $A_1\star A_2$ on $\operatorname{Gr}$	$f_{A_1}\ast f_{A_2}$ in $\mathcal H_K$, $\operatorname{vol}K=1$	Zhu, Lemma 5.6.1 ($n=1$)
$V_\lambda\ast\mathrm{IC}_0$ on $LX$	$\operatorname{tr}_n(V_\lambda)\ast\varphi_0$	not in the sources; proved in IIa
class $[M]\in \mathrm{K}(Y_0)$	$(\operatorname{tr}_nM)_{n\ge1}$ determines it	Laumon, Thm 1.1.2
semisimple perverse $M$	$(\operatorname{tr}_nM)_{n\ge1}$ determines $M$ up to isomorphism	Laumon, Prop. 1.1.2.1

Laumon proves 1.1.2.1 and says he knows no reference for it; Theorem 1.1.2 uses all $n$ at once.

3. Examples

Example 3.1 (constant sheaf and shifts)

For $M=\overline{\mathbb Q}_\ell[d](m)$ on any $Y_0$ with its standard Weil structure:

$$\operatorname{tr}_n(M)\equiv(-1)^dq^{-nm}.$$

So $M$ and $M[2]$ have the same trace, and $M\oplus M[1]$ has trace $0$: trace functions see classes in $\mathrm{K}(Y_0)$, not objects.

Example 3.2 (Grothendieck–Lefschetz on $\mathbb P^1$)

Let $\pi:\mathbb P^1\to\operatorname{Spec}\mathbb F_q$ and $M=\overline{\mathbb Q}_\ell[1]$. Both sides of $\operatorname{tr}_n(R\pi_!M)=\pi_!\operatorname{tr}_n(M)$ equal $-(q^n+1)$:

$$\operatorname{tr}_n(R\pi_!M)=-\big(1+q^n\big),\qquad\sum_{x\in\mathbb P^1(\mathbb F_{q^n})}(-1)=-(q^n+1).$$

How this is computed

$R\pi_!M$ has $\mathcal H^{-1}=H^0(\mathbb P^1)=\overline{\mathbb Q}_\ell$ and $\mathcal H^{1}=H^2(\mathbb P^1)=\overline{\mathbb Q}_\ell(-1)$, with geometric Frobenius $1$ and $q$.
Alternating sum: $(-1)^{-1}\cdot1+(-1)^{1}\cdot q^n=-(1+q^n)$.
$\#\mathbb P^1(\mathbb F_{q^n})=q^n+1$, each point contributing $\operatorname{tr}_n(\overline{\mathbb Q}_\ell[1])=-1$.

Example 3.3 (Satake sheaves, as used in IIa)

With Zhu's $f_\lambda=\operatorname{tr}(\mathrm{IC}_\lambda[-\langle2\rho,\lambda\rangle])$ and $c_\mu$ the indicator of $Kt^\mu K$, over $\mathbb F_{q^n}$:

$$\operatorname{tr}_nV_\lambda=(-1)^{\langle2\rho,\lambda\rangle}\Big(c_\lambda+\sum_{\mu\lt\lambda}a_{\lambda\mu}(q^n)\,c_\mu\Big),$$

with $a_{\lambda\mu}$ Kazhdan–Lusztig polynomials (Zhu (5.6.2)). Proposition II uses only the leading term $(-1)^{\langle2\rho,\lambda\rangle}c_\lambda$. This untwisted function is not the Satake transform $S^{-1}(\chi_\lambda)$, which needs the half twist (Zhu, Lemma 5.6.3).

4. Why $n$ varies

A multiplicity space $E$ with Frobenius $A$ contributes $\operatorname{tr}(A^n)$. One $n$ can cancel a nonzero space, and all $n$ cannot:

$$A=\begin{pmatrix}\alpha&0\\0&-\alpha\end{pmatrix}:\qquad\operatorname{tr}(A^n)=\begin{cases}0&n\text{ odd},\\2\alpha^n&n\text{ even}.\end{cases}$$

Proposition II uses only $n=1$, and in (c) only "zero space ⇒ zero coefficient", which holds for every $n$. Its converse (limits) would need all $n$: that is the only place $n$ varies.

5. Conventions in the sources

Table 3. 1. How each source sets up the trace; translate to Definition 1.1 as stated.
Source	Frobenius and structure	Twist	To read it here
Laumon (0.9), (1.1.1)	$t_K(x)=\operatorname{tr}(\mathrm{Frob}_x,K)$, geometric; $K$ a complex on $X$ over $\mathbb F_q$	none; $t_{\overline{\mathbb Q}_\ell(n)}=q^{-n}$	$\operatorname{tr}_n=t$ on $X\otimes\mathbb F_{q^n}$, for Weil objects that come from $X$
Deligne, Weil II (1.1.7)–(1.1.14)	$F=\varphi^{-1}$ geometric; Weil sheaf $=(\mathcal F,F^*\mathcal F\xrightarrow{\sim}\mathcal F)$, $F$ the $\mathbb F_q$-linear Frobenius	none	same as here
BBD 5.1.1–5.1.2	$\mathrm{Fr}_q^*\mathcal F\xrightarrow{\sim}\mathcal F$; $\mathbb F_{q^n}$-structure by iteration	none	same; Galois → Weil is fully faithful, not essentially surjective
Zhu §5.6	geometric $\sigma_x$, only $n=1$	$f_\mu$ uses $\mathrm{IC}_\mu[-(2\rho,\mu)]$; half twist for $S^{-1}$	replace $q$ by $q^n$; multiply $f_\lambda$ by $(-1)^{\langle2\rho,\lambda\rangle}$
Chen–Yi §8.2, A.8	$\Phi_n:(F^n)^*\mathcal F\xrightarrow{\sim}\mathcal F$ for $n$ sufficiently divisible; classes in Grothendieck groups	$q^{-n}[L]=[L(n)]$, fixed $q^{1/2}$	restrict $\Phi$ to their equivalence class; their triangularity is for classes
Frenkel §3.3, (5.2)	geometric $\mathrm{Fr}_x$, informal	implicit half twist	use for orientation only; exact formulas from Zhu

All checked sources use the geometric Frobenius and $\Phi:\mathrm{Fr}^*M\to M$; none uses $M\to\mathrm{Fr}^*M$.

6. What does not transfer to $LX$

Ind-schemes. Every checked statement is for schemes of finite type, or for $\operatorname{Gr}$ through its finite-type closed pieces (Zhu). On $LX$ the pages use it on finite-type models: objects of $\mathcal C_{K,k}$ are supported on finitely many orbit closures, and $L^+G$ acts through finite-type quotients (Zhu, §A.1.4).
Weil versus Galois. Laumon's injectivity is for complexes defined over $\mathbb F_q$. The Weil objects of Proposition II come from $\mathbb F_q$-structures (R), so this applies; a Weil structure such as multiplication by a non-unit $u\in\overline{\mathbb Q}_\ell^\times$ on $\overline{\mathbb Q}_\ell$ would not (Weil II (1.1.14)).
Convolution on $LX$. $\operatorname{tr}_n(V_\lambda\ast\mathrm{IC}_0)=\operatorname{tr}_n(V_\lambda)\ast\varphi_0$ is in no checked source; IIa proves it from Laumon (1.1.1.2)–(1.1.1.4) and Lang's theorem, as Zhu does on $\operatorname{Gr}$.
Objects versus classes. Trace functions determine classes in $\mathrm{K}(Y_0)$ (all $n$), and semisimple perverse sheaves; not complexes (Example 3.1).

7. Where Proposition II uses each entry

Table 4. 1. Dictionary entries and the steps that use them.
Entry	Step
$Rf_!$, $\otimes$, pullback	IIa, step 4: trace of a convolution
open-cell value, untwisted	IIa, step 5 and IIb, step 3: leading signs $(-1)^{\langle2\rho,\lambda\rangle}$, $(-1)^{\delta(\nu)}$
triangles, shifts, $\otimes$	IIb, step 5: $m_{\nu\lambda}=\sum_i(-1)^i\operatorname{tr}A_{\nu,i}(\lambda)$
injectivity over all $n$	Limits: why the converse is not reached

8. References

G. Laumon, Transformation de Fourier, constantes d'équations fonctionnelles et conjecture de Weil, Publ. Math. IHÉS 65 (1987) 131–210, numdam: (0.9) p. 136; (1.1.1.0)–(1.1.1.5) p. 137; Theorem 1.1.2 and Proposition 1.1.2.1 p. 138. Read in full for these sections.
P. Deligne, La conjecture de Weil II, Publ. Math. IHÉS 52 (1980) 137–252, numdam: (1.1.7)–(1.1.8) pp. 150–151, (1.1.14) p. 153. Read for §1.1.7–1.1.15.
A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100 (1982): 5.1.1 p. 122, Proposition 5.1.2 and Remarque p. 123. Read from page images.
X. Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence, arXiv:1603.05593v2: Remark 5.5.15 p. 80; Lemma 5.6.1 and (5.6.2) p. 83; Lemma 5.6.3 p. 84; §A.1.4. Read in full for §5.6.
T.-H. Chen, L. Yi, Singularities of orbit closures in loop spaces of symmetric varieties, arXiv:2310.20006v4: §8.2–8.2.3 pp. 40–42, (27) p. 41, App. A.8 p. 63.
E. Frenkel, Lectures on the Langlands program and conformal field theory, arXiv:hep-th/0512172v1: §3.3 pp. 37–38, (5.2) p. 61.
Not checked: SGA 4½ (Rapport; Sommes trigonométriques), which Laumon cites for §1.1; Kiehl–Weissauer and Achar (no open copy).

Sheaves	Functions	Source
triangle \(M'\to M\to M''\to\)	\(\operatorname{tr}_nM=\operatorname{tr}_nM'+\operatorname{tr}_nM''\)	Laumon (1.1.1.1)
shift \(M[1]\); perverse cohomology	\(-\operatorname{tr}_nM\); \(\ \operatorname{tr}_nM=\sum_j(-1)^j\operatorname{tr}_n{}^p\mathcal H^jM\)	Laumon (1.1.1.1)
Tate twist \(\overline{\mathbb Q}_\ell(m)\)	constant \(q^{-nm}\)	Laumon (1.1.1.0)
\(M_1\otimes M_2\)	\(\operatorname{tr}_nM_1\cdot\operatorname{tr}_nM_2\)	Laumon (1.1.1.2)
pullback \(f^*M\)	\(\operatorname{tr}_nM\circ f\)	Laumon (1.1.1.4)
\(Rf_!M\)	\(y\mapsto\sum_{f(x)=y}\operatorname{tr}_nM(x)\)	Laumon (1.1.1.3) (Grothendieck); for \(Rf_*\) only when \(f\) is proper, (1.1.1.5)
convolution \(A_1\star A_2\) on \(\operatorname{Gr}\)	\(f_{A_1}\ast f_{A_2}\) in \(\mathcal H_K\), \(\operatorname{vol}K=1\)	Zhu, Lemma 5.6.1 (\(n=1\))
\(V_\lambda\ast\mathrm{IC}_0\) on \(LX\)	\(\operatorname{tr}_n(V_\lambda)\ast\varphi_0\)	not in the sources; proved in IIa
class \([M]\in \mathrm{K}(Y_0)\)	\((\operatorname{tr}_nM)_{n\ge1}\) determines it	Laumon, Thm 1.1.2
semisimple perverse \(M\)	\((\operatorname{tr}_nM)_{n\ge1}\) determines \(M\) up to isomorphism	Laumon, Prop. 1.1.2.1