Theorem 1.3 · Theorem T: trace of Frobenius on \(\mathcal C_k\) is \(C_K\); the missing retract classes

Proof.

1. LT is well defined. At a Frobenius-fixed \(\bar x\), \((g\circ\mathrm{Fr}^*f)_{\bar x}=g_{\bar x}f_{\bar x}\) and \((f\circ g)_{\bar x}=f_{\bar x}g_{\bar x}\) have the same graded trace. For a Weil object \(\mathrm{LT}[M,\Phi]=\mathrm{tr}(M,\Phi)\).

2. One orbit. The equivariant category of \(\mathcal O_\mu\) is \(\mathcal C_\mu\simeq\mathrm{Shv}_c(BL_\mu)\), \(L_\mu=Z_{G^\theta}(\mu)\), the pro-unipotent part of the stabilizer being irrelevant (D0). \(L_\mu\) is connected by (C), so every object has constant cohomology sheaves and \(\mathcal C_\mu=\mathrm{Thick}(\underline{\overline{\mathbb Q}_\ell})\simeq\mathrm{Perf}\,C^*(BL_\mu)\) (Schwede–Shipley, Lurie HA Thm 7.1.2.1). \(H^*(BL_\mu)=\operatorname{Sym}V\) with \(V\) spanned by Chevalley generators of degrees \(2d_j\), on which Frobenius has eigenvalues of absolute value \(q^{d_j}\) (Deligne's purity); the cochain algebra is Frobenius-equivariantly formal by the same weight argument (as used by Chen–Yi, Thm 93). The Koszul resolution computes \(\mathrm{HH}(\operatorname{Sym}V,(\operatorname{Sym}V)_{\mathrm{Fr}})\) as the Koszul complex of the sequence \((1-\mathrm{Fr})v_j\); since \(1-\mathrm{Fr}\) is invertible on \(V\), this sequence is regular and \(\mathrm{HH}=\operatorname{Sym}V/(V)=\overline{\mathbb Q}_\ell\) in degree 0, spanned by \([\underline{\overline{\mathbb Q}_\ell},\mathrm{can}]\). (Finite check: in the one-generator complex \(m\mapsto(1-q)um\), every degree truncation has a one-dimensional cokernel.)

3. Gluing. For a closed finite union \(Z\) of orbits with open orbit \(\mathcal O_\mu\), \(j_!\) and \(i_*\) give a Frobenius-stable semi-orthogonal decomposition \(\mathcal C_Z=\langle i_*\mathcal C_{Z\smallsetminus\mathcal O_\mu},j_!\mathcal C_\mu\rangle\) (recollement, H2). Hochschild homology with coefficients in an endofunctor preserving both pieces is additive for such decompositions (Kuznetsov's additivity, Keller's localization; the twisted case has the same proof), and commutes with the filtered union over \(Z\). Hence \(\mathrm{HH}(\mathcal C_k,\mathrm{Fr})=\bigoplus_\mu\overline{\mathbb Q}_\ell[\Delta_\mu,\mathrm{can}]\) in degree 0, \(\Delta_\mu=j_{\mu!}\underline{\overline{\mathbb Q}_\ell}\). Since \(\mathcal O_\mu(\mathbb F_q)\) is one \(K\)-orbit (IIb step 1), \(\mathrm{LT}[\Delta_\mu,\mathrm{can}]=\mathbf 1_{\mathcal O_\mu(\mathbb F_q)}\); these form a basis of \(C_K\), so LT is an isomorphism. For \(\mathcal H\) run the same argument on Schubert cells (stabilizers: Levi \(Z_G(\lambda)\) times pro-unipotent, connected).

4. (b). Both sides are linear; \(\mathrm{HH}_0(\mathcal H,\mathrm{Fr})\) is spanned by the Weil classes \([V_\lambda,\mathrm{can}]\) (step 3 and unitriangularity), on which \(\mathrm{LT}\circ\mathrm{HH}_0(\mathrm{act})=(-\ast\varphi_0)\circ\mathrm{LT}\) is IIa. The equivalence with (SG) follows since both LT are isomorphisms.

5. (c). Since LT is injective, compute in \(C_K\): \(\mathrm{LT}[\mathrm{IC}_\nu,\mathrm{can}]=\mathrm{tr}\,\mathrm{IC}_\nu\), a basis (IIb (i)). Cyclicity converts the class \([M,e\Phi]\) of a Frobenius-equivariant idempotent \(e=\iota\pi\) (\(\pi\iota=\mathrm{id}_N\)) into \([N,\pi\Phi\,\mathrm{Fr}^*\iota]\), the summand with its induced structure: take \(f=\pi\), \(g=e\Phi\,\mathrm{Fr}^*\iota\), and use \(e\Phi=\Phi\,\mathrm{Fr}^*e\). The isotypic projectors of \({}^p\mathcal H^i(V_\lambda\ast\mathrm{IC}_0)\) and the generalized eigenprojectors of \(A_{\nu,i}(\lambda)\) are Frobenius-equivariant, and \({}^p\mathcal H^i(V_\lambda\ast\mathrm{IC}_0)[-i]\) is a retract of \(V_\lambda\ast\mathrm{IC}_0\) over \(k\) (D2). By Proposition I every \(\mathrm{IC}_\nu\) occurs this way when \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\). The formula for the full class is additivity along the perverse filtration. By Morita invariance of HH (Keller), \(\mathrm{HH}(\mathcal C_k,\mathrm{Fr})\) can be computed on the full subcategory of finite sums of Satake objects, where it is spanned by classes \([V,\beta]\) with \(\beta\) arbitrary: the idempotent-twisted classes are exactly the extra generators.

6. (d). \(\mathcal H\) is rigid (duals by inversion composed with Verdier duality), so \(\mathrm{act}^R\) is \(\mathcal H\)-linear: \(\operatorname{Hom}(G,\mathrm{act}^R(F\ast M))=\operatorname{Hom}(F^\vee\ast G\ast\mathrm{IC}_0,M)=\operatorname{Hom}(G,F\ast\mathrm{act}^RM)\). Hence \(\operatorname{Hom}(F\ast\mathrm{IC}_0,F'\ast\mathrm{IC}_0)=\operatorname{Hom}(F,F'\ast A)\), and \(a\mapsto a\ast\mathrm{IC}_0\) is \(a\mapsto a\otimes(1\to A)\). Barr–Beck–Lurie (HA Thm 4.7.3.5): \(\mathrm{act}^R\) is continuous, so monadicity is equivalent to conservativity, i.e. to \(\{F\ast\mathrm{IC}_0\}\) generating \(\operatorname{Ind}\mathcal C_k\), i.e. to \(\mathrm{Thick}(\mathrm{act}\,\mathcal H)=\mathcal C_k\); and \(\mathrm{Thick}(\mathrm{act}\,\mathcal H)=\mathcal C^{\mathrm{Sat}}_k\) because \(\mathcal H=\mathrm{Thick}\{V_\lambda\}\) (Schubert cells have connected stabilizers, so each cell category is generated by its constant sheaf, and the standard objects \(\Delta_\lambda\) and the \(V_\lambda\) generate the same thick subcategory by triangularity along the finite lower sets). Up to \(\lambda\mapsto\lambda^\vee\), \(A\) is Chen–Yi's de-equivariantized Ext algebra (41), formal for symmetric \(X\) (their Theorem 93).

7. What is cited, and what is not. Laumon (1.1.1.1)–(1.1.1.4) are statements about trace functions of complexes on schemes of finite type over \(\mathbb F_q\); Zhu, Lemma 5.6.1, is the convolution formula on \(\mathrm{Gr}\) for \(n=1\); IIa extends it to \(LX\). The categorical upgrade (a)–(b) is proved here, for these categories only, from Morita invariance, additivity along semi-orthogonal decompositions and the Koszul computation. Ponto–Shulman (local copy, arXiv:1107.6032v2), Prop. 2.4 (cyclicity) and Prop. 6.2 with Ex. 6.3 (traces commute with strong monoidal functors, \(\operatorname{tr}(f\otimes_RS)=\phi(\operatorname{tr}f)\)), are the formal principle behind Theorem 1.2(ii); they concern single endomorphisms and say nothing about generation. AGKRRV ("Automorphic functions as the trace of Frobenius") identify the trace of Frobenius on \(\mathrm{Shv}_{\mathrm{Nilp}}(\mathrm{Bun}_G)\) with compactly supported automorphic functions (a global, singular-support-restricted category); Gaitsgory–Varshavsky compare true and naive local terms; Zhu computes traces of Frobenius on Hecke categories, and Ben-Zvi–Francis–Nadler and Ben-Zvi–Nadler identify traces on \(\mathrm{QCoh}\) of perfect stacks with functions on (twisted) loop spaces. None of these is in the local sources, none is used, and none asserts surjectivity of a trace map induced by a generating functor. \(\square\)