The argument

1. The action map \(\mathrm{act}:\mathrm{Sph}_k\to\mathcal C_{K,k}\), \(F\mapsto F\ast\mathrm{IC}_0\), commutes with Frobenius.
2. \(\mathcal C^{\mathrm{Sat}}_{K,k}=\mathcal C_{K,k}\): the objects \(\mathrm{act}(F)\) generate \(\mathcal C_{K,k}\) under shifts, cones and retracts.
3. The categorical trace of Frobenius is functorial, so it produces a linear map \(\operatorname{Tr}(\mathrm{act}):\operatorname{Tr}(\mathrm{Fr}_*,\mathrm{Sph}_k)\to\operatorname{Tr}(\mathrm{Fr}_*,\mathcal C_{K,k})\).
4. The two traces are \(\mathcal H_K\) and \(C_K\), and \(\operatorname{Tr}(\mathrm{act})\) is \(f\mapsto f\ast\varphi_0\).
5. Hence \(\mathcal H_K\ast\varphi_0=C_K\).

Definition 2.1 (dualizable object, trace) [GKRV, §3.1.1]

Let \((\mathcal O,\otimes,\mathbf 1)\) be a symmetric monoidal category. An object \(o\) is dualizable if there are an object \(o^\vee\) and maps

satisfying the two zigzag identities. The trace of an endomorphism \(F\) of \(o\) is the composite

Definition 3.1 (the categorical trace) [GKRV, §3.4.1; AGKRRV3, §0.2.2; GR1, Ch. 1]

\(\mathrm{DGCat}\) is the symmetric monoidal 2-category whose objects are the presentable stable \(\overline{\mathbb Q}_\ell\)-linear categories, whose 1-morphisms are the colimit-preserving (continuous) functors, and whose 2-morphisms are the natural transformations. Its tensor product is Lurie's, and its unit is \(\mathrm{Vect}\). Since \(\operatorname{End}(\mathrm{Vect})=\mathrm{Vect}\), the trace of a continuous endofunctor \(F\) of a dualizable \(\mathcal C\) is a chain complex:

Proposition 3.2 (the trace is Hochschild homology) [GKRV, §3.5.1]

A compactly generated \(\mathcal C\) is dualizable, with \(\mathcal C^\vee=\operatorname{Ind}((\mathcal C^c)^{\mathrm{op}})\). Let \(g\) be a compact generator and \(E=\operatorname{End}(g)\), so that \(\operatorname{Hom}(g,-)\) identifies \(\mathcal C\) with right \(E\)-modules. A continuous endofunctor \(F\) corresponds to the \(E\)-bimodule \(Q=\operatorname{Hom}(g,Fg)\), with \(a\cdot q\cdot b=F(a)\circ q\circ b\). Then

If \(E\) and \(Q\) are concentrated in degree \(0\), then \(\mathrm{HH}_\bullet\) lives in degrees \(\ge0\), and

The left side is defined by \((\mathcal C,F)\) alone, so the right side does not depend on the choice of generator (Morita invariance). With several generators \(g_i\), replace \(E\) by the category on the \(g_i\) and \(Q\) by \(Q(i,j)=\operatorname{Hom}(g_i,Fg_j)\).

Definition 4.1 (the class of an object) [GKRV, §3.4.3; AGKRRV3, Remark 0.6.5]

Let \(c\in\mathcal C\) and \(\alpha:c\to F(c)\). The functor \(T_c:\mathrm{Vect}\to\mathcal C\), \(V\mapsto V\otimes c\), has a continuous right adjoint exactly when \(c\) is compact, and \(\alpha\) is a 2-morphism \(T_c\circ\mathrm{Id}\Rightarrow F\circ T_c\). Its trace (Definition 4.2) is a map \(\overline{\mathbb Q}_\ell=\operatorname{Tr}(\mathrm{Id},\mathrm{Vect})\to\operatorname{Tr}(F,\mathcal C)\), that is, an element

In the model of Proposition 3.2, \(\operatorname{cl}(g,q)\) is the class of \(q\in Q\). For \(F=\mathrm{Fr}_*\), adjunction turns \(\alpha:M\to\mathrm{Fr}_*M\) into \(\alpha^\flat:\mathrm{Fr}^*M\to M\). A Weil object \((M,\Phi)\) is the case where \(\alpha^\flat=\Phi\) is invertible, and we write \(\operatorname{cl}(M,\Phi)\). AGKRRV allow any \(\alpha\) ("possibly lax Weil sheaves").

Definition 4.2 (the trace of a functor) [GKRV, §3.2.1, §3.4.1]

A 1-morphism \((\Psi,\gamma):(\mathcal C,F_{\mathcal C})\to(\mathcal D,F_{\mathcal D})\) consists of two things.

• A continuous functor \(\Psi:\mathcal C\to\mathcal D\) with a continuous right adjoint. For compactly generated categories this means that \(\Psi\) preserves compact objects.
• A natural transformation \(\gamma:\Psi\circ F_{\mathcal C}\Rightarrow F_{\mathcal D}\circ\Psi\), not necessarily invertible.

Pasting \(\gamma\) and the adjunction \((\Psi,\Psi^R)\) into the unit–trace–counit composite gives a linear map

GKRV follow Ben-Zvi–Nadler and Kondyrev–Prikhodko for this construction.

Theorem 4.3 (functoriality) [GKRV, §3.2.3, §3.3.1, §3.4.2–3.4.3]

1. Composition. \(\operatorname{Tr}(\Psi'\circ\Psi,\gamma'\circ\gamma)=\operatorname{Tr}(\Psi',\gamma')\circ\operatorname{Tr}(\Psi,\gamma)\). More precisely, \(\operatorname{Tr}\) is a symmetric monoidal functor from pairs (dualizable category, endofunctor), with these 1-morphisms, to \(\mathrm{Vect}\).
2. Classes go to classes. \(\operatorname{Tr}(\Psi,\gamma)\big(\operatorname{cl}(c,\alpha)\big)=\operatorname{cl}\big(\Psi c,\ \gamma_c\circ\Psi(\alpha)\big)\).
3. Additivity and cyclicity. \(\operatorname{cl}(c\oplus c',\alpha\oplus\alpha')=\operatorname{cl}(c,\alpha)+\operatorname{cl}(c',\alpha')\). For \(u:c\to c'\) and \(\beta:c'\to F(c)\), \(\operatorname{cl}(c,\beta\circ u)=\operatorname{cl}(c',F(u)\circ\beta)\).
4. Vector spaces. For \(\mathcal C=\mathrm{Vect}\), \(F=\mathrm{Id}\) and \(V\) perfect, \(\operatorname{cl}(V,\alpha)=\sum_i(-1)^i\operatorname{tr}H^i(\alpha)\).

Proposition 4.4 (traces of module categories) [GKRV, §3.3.2–3.3.3, §3.4.4, Proposition 3.4.6]

Let a monoidal \(\mathcal R\) act on \(\mathcal M\), with multiplication and action preserving compact objects. Let \(F_{\mathcal R}\) be right-lax monoidal and \(F_{\mathcal M}\) right-lax compatible with the action. Then \(\operatorname{Tr}(F_{\mathcal R},\mathcal R)\) is an associative algebra and \(\operatorname{Tr}(F_{\mathcal M},\mathcal M)\) a module over it. For compact \(r\) with \(\alpha:r\to F_{\mathcal R}(r)\), the class \(\operatorname{cl}(r,\alpha)\) acts by \(\operatorname{Tr}(H_r,\alpha_{r,\mathcal M})\), with \(H_r=r\ast-\).

Corollary 4.5 (the trace of the action map)

Take \(\mathcal R=\operatorname{Ind}\mathrm{Sph}_k\), \(\mathcal M=\operatorname{Ind}\mathcal C_{K,k}\) and \(F=\mathrm{Fr}_*\) on both, and let \(\Phi_0\) be the canonical Weil structure of \(\mathrm{IC}_0\). Then

Definition 5.1 (local terms) [AGKRRV1, §22.2; AGKRRV3, (0.9) and Remark 0.6.5]

For a quasi-compact algebraic stack \(Y\) over \(\mathbb F_q\), there is a canonical local-term map \(\mathrm{LT}_Y:\operatorname{Tr}((\mathrm{Frob}_Y)_*,\mathrm{Shv}(Y))\to\mathrm{Funct}(Y(\mathbb F_q))\). On classes it is the Grothendieck–Deligne function:

On the loop-space category, the finite stabilizer presentations and rational stalk functors used in Proposition 5.2(c) define the local-term map on the full trace complex. Its value on a Weil class is the stalk trace; checking degree-zero cyclicity alone would not construct this map for an arbitrary differential graded category.

Proposition 5.2 (the square)

1. The square commutes on classes of Weil objects: \(\operatorname{tr}(F\ast\mathrm{IC}_0)=\operatorname{tr}(F)\ast\varphi_0\).
2. Suppose every Frobenius-stable orbit has connected stabilizer. This holds under (C) and in the examples of Proposition IX. Then \(\mathrm{LT}_{\mathrm{Sph}}\) and \(\mathrm{LT}_{\mathcal C_K}\) are onto, and hence, with \(W=\operatorname{span}\{\operatorname{cl}(V_\lambda\ast\mathrm{IC}_0,\Phi):\sigma\lambda=\lambda\}\subseteq\operatorname{Tr}(\mathrm{Fr}_*,\mathcal C_{K,k})\), the span of the classes that Weil Satake objects produce, $$\mathrm{LT}_{\mathcal C_K}(W)=\mathcal H_K\ast\varphi_0,\qquad(\mathrm{SG})\iff\mathrm{LT}_{\mathcal C_K}(W)=C_K .$$
3. Assume (R), (C) and D0–D3. Then both traces are concentrated in degree \(0\) and spanned by classes, both local-term maps are isomorphisms, and $$(\mathrm{SG})\iff\operatorname{Tr}(\mathrm{act})\ \text{is onto}.$$

Theorem 6.1 (the image of a trace map, in degree 0)

Let \((\Psi,\gamma):(\mathcal C,F_{\mathcal C})\to(\mathcal D,F_{\mathcal D})\) be as in Definition 4.2. Let \(\{c_i\}\) be compact objects with

and suppose all the spaces \(\operatorname{Hom}(c_i,c_j)\), \(\operatorname{Hom}(c_i,F_{\mathcal C}c_j)\), \(\operatorname{Hom}(\Psi c_i,\Psi c_j)\) and \(\operatorname{Hom}(\Psi c_i,F_{\mathcal D}\Psi c_j)\) are concentrated in degree \(0\). Then:

1. \(H_0\operatorname{Tr}(F_{\mathcal D},\mathcal D)=\bigoplus_i\operatorname{Hom}(\Psi c_i,F_{\mathcal D}\Psi c_i)\big/\big\langle q\circ f-F_{\mathcal D}(f)\circ q\big\rangle\), where \(f:\Psi c_i\to\Psi c_j\) and \(q:\Psi c_j\to F_{\mathcal D}\Psi c_i\). Every element is a sum of classes \(\operatorname{cl}(\Psi c_i,q)\).
2. \(\operatorname{im}H_0\operatorname{Tr}(\Psi,\gamma)=\operatorname{span}\big\{\operatorname{cl}\big(\Psi c_i,\gamma\circ\Psi(\alpha)\big):\alpha\in\operatorname{Hom}(c_i,F_{\mathcal C}c_i)\big\}\).
3. Hence \(\operatorname{Tr}(\Psi,\gamma)\) is onto in degree \(0\) iff $$\bigoplus_i\operatorname{Hom}(\Psi c_i,F_{\mathcal D}\Psi c_i)=\bigoplus_i\gamma\circ\Psi\big(\operatorname{Hom}(c_i,F_{\mathcal C}c_i)\big)+\big\langle q\circ f-F_{\mathcal D}(f)\circ q\big\rangle .$$

Toy A (a retract): \(\Psi:\mathrm{Vect}\to\mathrm{Vect}\times\mathrm{Vect}\), \(V\mapsto(V,V)\), \(F=\mathrm{Id}\)

• Generation. \((\overline{\mathbb Q}_\ell,0)\) is a retract of \(\Psi(\overline{\mathbb Q}_\ell)\).
• Image. The diagonal. In 6.1(iii), \(\operatorname{End}(\Psi\overline{\mathbb Q}_\ell)=\overline{\mathbb Q}_\ell^2\supsetneq\Psi(\operatorname{End}\overline{\mathbb Q}_\ell)=\overline{\mathbb Q}_\ell(1,1)\), with no commutators.
• Missing class. The idempotent \((1,0)\), which is mechanism (M2).
• The same toy as modules. \(\mathrm{Vect}^2=(\overline{\mathbb Q}_\ell\times\overline{\mathbb Q}_\ell)\text{-mod}\) and \(\Psi\) is the free-module functor. In the same way, \(\mathcal C^{\mathrm{Sat}}_{K,k}=\mathcal C_{K,k}\) makes \(\operatorname{Ind}\mathcal C_{K,k}\) the category of \(A\)-modules in \(\operatorname{Ind}\mathrm{Sph}_k\) and \(\mathrm{act}\) the free-module functor (Theorem 1.3(d)).
• Variant A′. \(\Psi:\mathrm{Vect}\to\operatorname{Perf}(\overline{\mathbb Q}_\ell[x]/x^2)\) misses the nilpotent class \(\operatorname{cl}(\overline{\mathbb Q}_\ell[x]/x^2,x)\). If Frobenius acts on \(x\) by \(q\ne1\), the relation \(x\sim qx\) kills that class and the map is onto in degree \(0\). Frobenius weights remove nilpotent classes, not idempotent ones.

Toy B (no Weil structure): \(\Psi:(\mathrm{Vect}^2,\mathrm{swap})\to(\mathrm{Vect},\mathrm{Id})\), \((V_1,V_2)\mapsto V_1\oplus V_2\)

Here \(\gamma:V_2\oplus V_1\xrightarrow{\sim}V_1\oplus V_2\) is the canonical isomorphism.

• Generation. \(\Psi(\overline{\mathbb Q}_\ell,0)=\overline{\mathbb Q}_\ell\), even without retracts.
• Source trace. For \(c=(\overline{\mathbb Q}_\ell,\overline{\mathbb Q}_\ell)\), \(\operatorname{Hom}(c,\mathrm{swap}\,c)\ni(q_1,q_2)\). The relation with \(f=(1,0)\) is \(q\circ f-\mathrm{swap}(f)\circ q=(q_1,-q_2)\), which spans everything, so \(H_0=0\).
• Image of a Weil object. \(\alpha=(\phi_1,\phi_2)\) goes to \(\gamma\circ\Psi(\alpha)=\begin{pmatrix}0&\phi_2\\\phi_1&0\end{pmatrix}\), of trace \(0\). In 6.1(iii) the image consists of the off-diagonal matrices, which lie in \([M_2,M_2]=\{\operatorname{tr}=0\}\ne M_2\).
• Missing class. \((\overline{\mathbb Q}_\ell^2,\mathrm{swap})=(\overline{\mathbb Q}_\ell,+1)\oplus(\overline{\mathbb Q}_\ell,-1)\). The Weil object \((\overline{\mathbb Q}_\ell,+1)\) is a summand of an image, but its class \(1\) is not reached; the image class is \(1+(-1)=0\). This is mechanism (M1).
• Squares. For \(F_{\mathcal C}=\mathrm{swap}^2=\mathrm{Id}\) the map is \((a,b)\mapsto a+b\), which is onto.

Toy C (fusion): \(\operatorname{Res}:\operatorname{Rep}\mathrm{SL}_2\to\operatorname{Rep}T\), \(T\) the diagonal torus, \(F=\mathrm{Id}\)

• Generation. The weight \(z^m\) is a summand of \(\operatorname{Res}V_{|m|}\).
• Traces. For a semisimple \(\operatorname{Rep}H\), Theorem 6.1(i) gives one class per irreducible, \(\operatorname{Tr}=R(H)\otimes\overline{\mathbb Q}_\ell\), the class functions. \(\operatorname{Tr}(\operatorname{Res})\) is restriction.
• Image. The \(W\)-invariant Laurent polynomials.
• Missing class. \(\operatorname{cl}(z)\). It is the weight projector of \(\operatorname{Res}V_1=z\oplus z^{-1}\), which is not in \(\operatorname{End}_{\mathrm{SL}_2}V_1=\overline{\mathbb Q}_\ell\): mechanism (M2).
• Why. \(t\) and \(t^{-1}\) in \(T\) are conjugate in \(\mathrm{SL}_2\) but not in \(T\): two classes fuse. For finite groups \(H\subseteq G\), restriction of class functions is onto iff no two \(H\)-classes fuse in \(G\). The smallest instance is \(\mathbb Z/3\subset S_3\).

8.1 The unitary twist: Toy B with \(\mathrm{Vect}\) replaced by \(\mathrm{Sph}_H\)

Let \(G=\operatorname{Res}_{\mathbb F_{q^2}/\mathbb F_q}H\) with \(\theta=\vartheta\circ\sigma\) (Proposition IX(a)). Over \(k\), \(G_k=H\times H\) and Frobenius exchanges the factors, so \(\mathrm{Sph}_{G}=\mathrm{Sph}_H\otimes\mathrm{Sph}_H\) with \(\mathrm{Fr}_*=\mathrm{swap}\circ(\mathrm{Fr}_*\otimes\mathrm{Fr}_*)\).

• Hecke side. The trace of a swap is the trace of the square, by the same string diagram as \(\operatorname{tr}(\mathrm{swap}\circ(f_1\otimes f_2))=\operatorname{tr}(f_2f_1)\). So \(\operatorname{Tr}(\mathrm{Fr}_*,\mathrm{Sph}_G)\) is the Hecke algebra of \(H(\mathbb F_{q^2}(\!(t)\!))=G(F)\), as it must be.
• Generation. \(X_k\) is the group case, and \(V_{(c,0)}\ast\mathrm{IC}_0\simeq\mathrm{IC}_c\) for every \(c\in\Lambda_H^+\). The label \((c,0)\) is not Frobenius-fixed: \(\mathrm{Fr}^*V_\lambda\simeq V_{\sigma^{-1}\lambda}\), so \(V_\lambda\) is Weil iff \(\sigma\lambda=\lambda\) (Lemma 2.8).
• Traces. Weil Satake objects are built from fixed labels and from orbits \(\{\lambda,\sigma\lambda\}\) with the swap structure. Like \((\overline{\mathbb Q}_\ell^2,\mathrm{swap})\) in Toy B, such an orbit contributes \(0\) to \(\operatorname{tr}_n\) for odd \(n\). The objects \(V_{(c,0)}\ast\mathrm{IC}_0\) do carry Weil structures in \(\mathcal C_{K,k}\), but for \(c\notin2\Lambda_H\) none of them comes from \(\mathrm{Sph}_G\).
• Outcome. \(\mathbf 1_{O_c}\notin\mathcal H_K\ast\varphi_0\) for \(c\notin2\Lambda_H\) (Theorem 2.6, audited twice). For even \(n\), \(\mathrm{Fr}^n\) fixes every label and \((\mathrm{SG})_n\) holds, as in Toy B with \(\mathrm{swap}^2\).

8.2 Sakellaridis's \(X=\mathrm{SL}_2^3\backslash(\mathrm{Sp}_4\times\mathrm{Sp}_4)\): Toy C with \(\mathrm{SL}_2\supset T\) replaced by \(\mathrm{SO}_5^2\supset\iota\check G_X\)

Here \(G\) is split and every label is Frobenius-fixed, so every \(V_\lambda\) is Weil and (M1) cannot occur.

• Multiplicities. Given the decomposition theorem for \(X\) (the hypothesis of Corollary 3.9), \(V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_\nu\mathrm{IC}_\nu^{\,c_{\nu\lambda}}\), where \(c_{\nu\lambda}=[\operatorname{Res}^{\mathrm{SO}_5\times\mathrm{SO}_5}_{\iota\check G_X}V_\lambda:V_\nu]\) are branching numbers (Proposition 3.8), and \(m^{(n)}_{\nu\lambda}=c_{\nu\lambda}\,q^{n(\langle2\rho,\lambda\rangle-\delta(\nu))/2}\). Up to these diagonal powers of \(q\), \(\operatorname{Tr}(\mathrm{act})\) is restriction \(R(\check G)\otimes\overline{\mathbb Q}_\ell\to R(\iota\check G_X)\otimes\overline{\mathbb Q}_\ell\), exactly as in Toy C.
• Fusion. For \(a\ne\pm b^{\pm1}\), the points \((a,b,a)\) and \((b,a,b)\) of \(\check G_X\) are \(\check G\)-conjugate but not \(\check G_X\)-conjugate (Lemma 3.3(d)). Restriction is therefore not onto, and, given the same decomposition, \(\operatorname{tr}\mathrm{IC}_{(0,0,2)}\notin\mathcal H_{K_n}\ast\varphi_0\) for every \(n\) (Corollary 3.9(d)): mechanism (M2).
• Status. The function failure is proved unconditionally, by two independent periods (Theorem 3.5, audited). The sheaf half \(\mathcal C^{\mathrm{Sat}}_{K,k}=\mathcal C_{K,k}\) needs Chen–Yi's framework for this \(X\): Sakellaridis–Wang's degeneration in odd characteristic, the dimension theory \(\delta=3d_1+3d_2+e\), and Chen–Yi's Appendix A with Prop. 70 (obligations 5.10).

Proposition 9.1 (split surjections)

Suppose \((\Psi',\gamma'):(\mathcal D,F_{\mathcal D})\to(\mathcal C,F_{\mathcal C})\) is a 1-morphism with \((\Psi,\gamma)\circ(\Psi',\gamma')\simeq(\mathrm{Id},\mathrm{id})\). Then \(\operatorname{Tr}(\Psi,\gamma)\) is split onto: \(\operatorname{Tr}(\Psi,\gamma)\circ\operatorname{Tr}(\Psi',\gamma')=\operatorname{Tr}(\mathrm{Id})=\mathrm{id}\) by Theorem 4.3(a). An instance is a Frobenius-compatible localization whose fully faithful right adjoint preserves compact objects. \(\mathrm{act}\) is not of this kind: it is a free-module functor (Toy A).

Proposition 9.2 (dominant lifts) [run's Proposition 1.8(e)–(f), audited]

Assume (R), (C) and D0–D3. If every orbit label \(\mu\) has a dominant lift, i.e. \(\mu=\varphi(\lambda_\mu)\) for some \(\lambda_\mu\in\Lambda_T^+\), then the columns \(\lambda_\mu\) of \(m^{(n)}\) are triangular with nonzero diagonal entries \(m^{(n)}_{\mu\lambda_\mu}\), and \((\mathrm{SG})_n\) holds for every \(n\).

In terms of §6: \(\mathrm{IC}_\mu\) is the top summand of \(V_{\lambda_\mu}\ast\mathrm{IC}_0\), with nonzero top coefficient, so \(\operatorname{cl}(\mathrm{IC}_\mu)\) is reached modulo lower labels without any projector. A triangular system of summands whose multiplicity spaces are split Tate of one parity suffices. Without dominant lifts the trace argument decides nothing: with \(U\) the labels without a lift, \((\mathrm{SG})\iff\{p_U(T_\lambda\ast\varphi_0)\}\) spans \(\mathbb C^{(U)}\), and for \(E_6/F_4\) it does not (Proposition IX(a)).

The theorem [AGKRRV3, Main Theorem 0.2.6 = Corollary 4.1.4; Theorem 0.6.8 = Theorem 5.2.3; Corollary 0.6.9]

Let \(\Sigma\) be a smooth projective curve over \(\mathbb F_q\) (AGKRRV write \(X\)) and \(G\) a reductive group as in [AGKRRV3, §0.2.1], with \(\mathrm{Bun}_G\) the stack of \(G\)-bundles on \(\Sigma\) and \(\mathrm{Shv}_{\mathrm{Nilp}}\) the ind-constructible sheaves with nilpotent singular support. Then

Assume [AGKRRV1, Conjecture 14.1.8], that compact objects of \(\mathrm{Shv}_{\mathrm{Nilp}}\) are constructible. Then this isomorphism is the composite of the trace of the inclusion into \(\mathrm{Shv}(\mathrm{Bun}_G)\) with the local-term map. The statement is the Trace Conjecture [AGKRRV1, Conjecture 22.3.7].