# Claims to audit (verbatim from the agents' returns)

Each block gives the step id, its title, the claimed statement and the argument as written. Audit each one independently.


## A1-1 · Setup: Weil Grothendieck groups over \(R=\mathbb Z[\overline{\mathbb Q}_\ell^\times]\)

**Kind/status as reported:** lemma / proved

**Builds on:** —

### Statement

Let \(R=\mathbb Z[\overline{\mathbb Q}_\ell^\times]\) be the group ring (\([\alpha][\beta]=[\alpha\beta]\)), with ring maps
$$\mathrm{ev}_n:R\to\overline{\mathbb Q}_\ell,\ [\alpha]\mapsto\alpha^n\ (n\ge1),\qquad \varepsilon:R\to\mathbb Z,\ [\alpha]\mapsto1 .$$
For a finite-dimensional \(E\) with \(A\in\mathrm{GL}(E)\) put \([E,A]=\sum_\alpha d_\alpha[\alpha]\in R\), \(d_\alpha\) the dimension of the generalized \(\alpha\)-eigenspace. Assume (R), (C), D0–D3. Let \(\mathcal C^W\) (resp. \(\mathcal H^W\)) be the abelian category of Weil perverse sheaves \((P,\Phi:\mathrm{Fr}^*P\xrightarrow{\sim}P)\) with \(P\) an \(L^+G\)-equivariant perverse sheaf on \(LX\) of finite support (resp. on \(\mathrm{Gr}\)). Then:

(a) \(K_0(\mathcal C^W)=\bigoplus_{\nu\in\Lambda_S^+}R\,[\mathrm{IC}_\nu]\) and \(K_0(\mathcal H^W)=\bigoplus_{\lambda\in\Lambda_T^+}R\,[V_\lambda]\), free \(R\)-modules, \([\alpha]\) acting by multiplying the Weil structure by \(\alpha\); the \([\mathrm{IC}_\nu]\), \([V_\lambda]\) carry the canonical Weil structures.

(b) Put \([K]=\sum_i(-1)^i[{}^p\mathcal H^iK]\) for a Weil complex \(K\). Then \(F\mapsto[F\ast\mathrm{IC}_0]\) is an \(R\)-linear map \(\mathrm{act}:K_0(\mathcal H^W)\to K_0(\mathcal C^W)\) with
$$\mathrm{act}[V_\lambda]=\sum_\nu M_{\nu\lambda}[\mathrm{IC}_\nu],\qquad M_{\nu\lambda}=\sum_i(-1)^i\big[E_{\nu,i}(\lambda),A_{\nu,i}(\lambda)\big]\in R,$$
each column finitely supported (\(M_{\nu\lambda}=0\) unless \(\nu\le\varphi(\lambda)\)).

(c) \(\mathrm{tr}_n\) induces isomorphisms \(K_0(\mathcal C^W)\otimes_{R,\mathrm{ev}_n}\overline{\mathbb Q}_\ell\cong C_{K_n}\otimes\overline{\mathbb Q}_\ell\) (\([\mathrm{IC}_\nu]\mapsto\mathrm{tr}_n\mathrm{IC}_\nu\)) and \(K_0(\mathcal H^W)\otimes_{R,\mathrm{ev}_n}\overline{\mathbb Q}_\ell\cong\mathcal H_{K_n}\otimes\overline{\mathbb Q}_\ell\), and \(\mathrm{tr}_n\circ\mathrm{act}=(-\ast\varphi_0)\circ\mathrm{tr}_n\). In particular \(m^{(n)}_{\nu\lambda}=\mathrm{ev}_n(M_{\nu\lambda})\).

### Argument

**Proof.**

1. *Simple Weil perverse sheaves.* Let \((P,\Phi)\) be simple in \(\mathcal C^W\). The socle of \(P\) over \(k\) is carried to itself by \(\Phi\circ\mathrm{Fr}^*\), so it is a Weil subobject and equals \(P\): \(P\) is semisimple over \(k\). Under (R) every orbit \(\mathcal O_\mu\) is Frobenius-stable (its representative \(x_\mu\) is rational, IIb step 1), so \(\mathrm{Fr}^*\mathrm{IC}_\nu\simeq\mathrm{IC}_\nu\) with the canonical structure (eigenvalue \(1\) on the lowest stalk of the open stratum); under (C) the simple equivariant perverse sheaves are the \(\mathrm{IC}_\nu\) with constant coefficients (D0, D1). With \(E_\nu=\operatorname{Hom}(\mathrm{IC}_\nu,P)\), Schur's lemma (\(\operatorname{End}\mathrm{IC}_\nu=\overline{\mathbb Q}_\ell\), \(\operatorname{Hom}(\mathrm{IC}_\nu,\mathrm{IC}_{\nu'})=0\)) gives \(P\simeq\bigoplus_\nu\mathrm{IC}_\nu\otimes E_\nu\) with \(\Phi=\bigoplus_\nu\mathrm{can}_\nu\otimes A_\nu\) (as in IIb step 4). Simplicity forces a single \(\nu\) and \(E_\nu\) simple over \(\overline{\mathbb Q}_\ell[A_\nu^{\pm1}]\), hence one-dimensional: \(P=\mathrm{IC}_\nu(\alpha):=(\mathrm{IC}_\nu,\alpha\cdot\mathrm{can})\).

2. *Finite length and \(K_0\).* A chain of Weil subobjects is a chain of subobjects over \(k\), so every object of finite support has finite length. Jordan–Hölder gives \(K_0(\mathcal C^W)=\bigoplus_{\nu,\alpha}\mathbb Z[\mathrm{IC}_\nu(\alpha)]=\bigoplus_\nu R[\mathrm{IC}_\nu]\). A tensor product \(\mathrm{IC}_\nu\otimes(E,A)\) has a filtration by an \(A\)-stable flag with graded pieces \(\mathrm{IC}_\nu(\alpha)\), so its class is \([E,A][\mathrm{IC}_\nu]\). For \(\mathcal H^W\) the same argument runs on Schubert cells: for split \(G\) they are Frobenius-stable with connected stabilizers \(L^+G\cap t^\lambda L^+Gt^{-\lambda}\) (Zhu, §5.1).

3. *The map \(\mathrm{act}\).* Convolution with \(\mathrm{IC}_0\) is triangulated and commutes with \(\mathrm{Fr}^*\) (proper base change, D1). A short exact sequence of Weil perverse sheaves on \(\mathrm{Gr}\) therefore gives a Frobenius-compatible triangle on \(LX\) and a Frobenius-compatible long exact sequence of perverse cohomology, so \([F\ast\mathrm{IC}_0]\) is additive in \(F\). It is \(R\)-linear because \((F\otimes L_\alpha)\ast\mathrm{IC}_0\simeq(F\ast\mathrm{IC}_0)\otimes L_\alpha\) (projection formula; \(L_\alpha\) the rank-one Weil sheaf pulled back from the point). The formula for \(\mathrm{act}[V_\lambda]\) is the Frobenius-equivariant isotypic decomposition \({}^p\mathcal H^i(V_\lambda\ast\mathrm{IC}_0)\simeq\bigoplus_\nu\mathrm{IC}_\nu\otimes E_{\nu,i}(\lambda)\), \(\Phi=\bigoplus\mathrm{can}\otimes A_{\nu,i}(\lambda)\) (IIb step 4), combined with step 2. Column finiteness is the support bound of D0.

4. *Traces.* \(\mathrm{tr}_n\) is additive on short exact sequences and on perverse truncation triangles (Laumon (1.1.1.1)) and multiplicative under \(\otimes L_\alpha\) (Laumon (1.1.1.2)): \(\mathrm{tr}_n\mathrm{IC}_\nu(\alpha)=\alpha^n\,\mathrm{tr}_n\mathrm{IC}_\nu\). So \(\mathrm{tr}_n\) factors through \(-\otimes_{R,\mathrm{ev}_n}\overline{\mathbb Q}_\ell\) and sends the basis \([\mathrm{IC}_\nu]\) to the basis \(\mathrm{tr}_n\mathrm{IC}_\nu\) of \(C_{K_n}\otimes\overline{\mathbb Q}_\ell\) (IIb (i)); likewise \([V_\lambda]\mapsto\mathrm{tr}_nV_\lambda\), a basis of \(\mathcal H_{K_n}\otimes\overline{\mathbb Q}_\ell\) (IIa step 5). The square commutes on the \(R\)-basis \([V_\lambda]\) by IIa, \(\mathrm{tr}_n(V_\lambda\ast\mathrm{IC}_0)=\mathrm{tr}_n(V_\lambda)\ast\varphi_0\), hence everywhere. Finally \(\mathrm{ev}_n(M_{\nu\lambda})=\sum_i(-1)^i\operatorname{tr}A_{\nu,i}(\lambda)^n=m^{(n)}_{\nu\lambda}\) (IIb (ii)). \(\square\)

*Remark.* Chen–Yi, §8.2.1, work in the coarser ring \(\mathbb Z[\overline{\mathbb Q}_\ell^\times/\mu]\) (eigenvalues modulo roots of unity, Weil structures only for \(n\) sufficiently divisible); the difference matters, see [[A1-9]].

**Sources cited:** trace-b.html (IIb steps 1, 3, 4, 5); trace-a.html (IIa steps 4, 5); laumon-1987.pdf, (1.1.1.1)-(1.1.1.2); zhu-satake-v2.pdf, §5.1 and (5.6.2); chen-yi-v4.pdf, §8.2.1


## A1-2 · Theorem A: decategorification is base change of one \(R\)-matrix

**Kind/status as reported:** lemma / proved

**Builds on:** A1-1

### Statement

Assume (R), (C), D0–D3; let \(M\) be the matrix of [[A1-1]] and \(Q=\operatorname{coker}\big(M:R^{(\Lambda_T^+)}\to R^{(\Lambda_S^+)}\big)\).

(i) \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\iff\) no row of \(M\) is zero in \(R\).

(ii) For every \(n\ge1\): \((\mathrm{SG})_n\iff Q\otimes_{R,\mathrm{ev}_n}\overline{\mathbb Q}_\ell=0\iff\) the columns of \(\mathrm{ev}_n(M)\) span \(\bigoplus_{\Lambda_S^+}\overline{\mathbb Q}_\ell\).

(iii) Suppose \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\). The triangulated closure without retracts, \(\mathrm{Tri}\{V_\lambda\ast\mathrm{IC}_0\}\), equals \(\mathcal C_k\iff Q\otimes_{R,\varepsilon}\mathbb Z=0\).

(iv) \(Q=0\Longrightarrow(\mathrm{SG})_n\) for every \(n\).

For an arbitrary \(R\)-matrix with finitely supported columns, (ii) for one \(n\) implies (i) (this is IIc), and the converse fails: two labels with every column equal to \((r,r)^{\mathsf T}\), \(r\neq0\), satisfy (i) and violate (ii) for all \(n\).

### Argument

**Proof.**

1. *(ii).* By IIa, \(\mathcal H_{K_n}\ast\varphi_0\otimes\overline{\mathbb Q}_\ell\) is spanned by the \(\mathrm{tr}_n(V_\lambda\ast\mathrm{IC}_0)\), whose coordinates in the basis \(\mathrm{tr}_n\mathrm{IC}_\nu\) are the columns \(\mathrm{ev}_n(M_{\bullet\lambda})\) ([[A1-1]](c)). Tensor product is right exact: applying \(-\otimes_{R,\mathrm{ev}_n}\overline{\mathbb Q}_\ell\) to the exact sequence \(R^{(\Lambda_T^+)}\xrightarrow{M}R^{(\Lambda_S^+)}\to Q\to0\) gives \(\operatorname{coker}\mathrm{ev}_n(M)=Q\otimes_{R,\mathrm{ev}_n}\overline{\mathbb Q}_\ell\). The column criterion and the independence of the coefficient field are IIb (iii), step 7.

2. *(i).* (\(\Leftarrow\)) If \(M_{\mu\lambda}\neq0\) then some \(E_{\mu,i}(\lambda)\neq0\), so \(\mathrm{IC}_\mu[-i]\) is a direct summand of \(V_\lambda\ast\mathrm{IC}_0\) (D2), and Proposition I gives \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) once this holds for every \(\mu\). (\(\Rightarrow\)) Proposition I gives, for each \(\mu\), some \(\lambda,i\) with \(E_{\mu,i}(\lambda)\neq0\). \(V_\lambda\ast\mathrm{IC}_0\) is pure of some weight \(w\) (D2), so \({}^p\mathcal H^i\) is pure of weight \(w+i\) (BBD, Cor. 5.4.4), and \(\mathrm{IC}_\mu\) with its canonical structure is pure of a weight \(w_\mu\); hence every eigenvalue of \(A_{\mu,i}(\lambda)\) has weight \(w+i-w_\mu\). For different \(i\) the elements \([E_{\mu,i}(\lambda),A_{\mu,i}(\lambda)]\in R\) are supported on disjoint sets of group elements, so the alternating sum \(M_{\mu\lambda}\) is nonzero.

3. *(iii).* Over \(k\), \(K_0(\mathcal C_k)=\bigoplus_\nu\mathbb Z[\mathrm{IC}_\nu]\) (bounded t-structure with finite-length heart whose simples are the \(\mathrm{IC}_\nu\), by (C)), and \([V_\lambda\ast\mathrm{IC}_0]=\sum_\nu\varepsilon(M_{\nu\lambda})[\mathrm{IC}_\nu]\). Thomason's classification (Compositio Math. 105 (1997), Theorem 2.1): in an essentially small triangulated category \(\mathcal T\), dense triangulated subcategories \(\mathcal D\) (every object a direct summand of an object of \(\mathcal D\)) correspond to subgroups of \(K_0(\mathcal T)\) by \(\mathcal D\mapsto\operatorname{im}K_0(\mathcal D)\), with \(\mathcal D=\{x:[x]\in\operatorname{im}K_0(\mathcal D)\}\). The direct summands of objects of \(\mathcal D=\mathrm{Tri}\{V_\lambda\ast\mathrm{IC}_0\}\) form a thick subcategory (a cone of \(f\oplus0:x\oplus x'\to y\oplus y'\) is \(\mathrm{cone}(f)\oplus x'[1]\oplus y'\)), which is \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\); so \(\mathcal D\) is dense, and \(\mathcal D=\mathcal C_k\) iff the columns of \(\varepsilon(M)\) span \(\bigoplus\mathbb Z\), i.e. iff \(Q\otimes_\varepsilon\mathbb Z=0\).

4. *(iv)* is immediate from (ii).

5. *The mechanism.* Thick closure is closure under retracts. The class of a retract \(N\) of \(V_\lambda\ast\mathrm{IC}_0\) is in general not a combination of classes of Satake objects; condition (i) records only which retracts exist (the support of the rows), while (ii) asks that the particular combinations \(\mathrm{ev}_n(M_{\bullet\lambda})\), one per Satake object, span. In the language of [[A1-3]]: the trace sees classes, Morita invariance makes all retract classes available in \(\operatorname{Tr}(\mathrm{Fr},\mathcal C_k)\), but only the full classes lie in the image of \(\mathcal H\).

6. *Independence for \(R\)-matrices.* (ii) for one \(n\) gives (i) because a zero row of \(M\) stays zero under \(\mathrm{ev}_n\). Conversely, with labels \(\nu_1,\nu_2\) and every column equal to \((r,r)^{\mathsf T}\), \(r\neq0\): no row vanishes, and the columns of \(\mathrm{ev}_n(M)\) span at most a line, for every \(n\). With one label and every column in \(Rr\), \(r=[\gamma]+[-\gamma]\), the cokernel is \(R/rR\) and \(\mathrm{ev}_n(r)=\gamma^n(1+(-1)^n)\), so \((\mathrm{SG})_n\) fails exactly for odd \(n\); this pattern is realized geometrically in [[A1-5]].

7. *Relation to Laumon.* For distinct \(\alpha_1,\dots,\alpha_r\in\overline{\mathbb Q}_\ell^\times\) the functions \(n\mapsto\alpha_j^n\) are linearly independent (Artin), so \((\mathrm{ev}_n)_{n\ge1}\) is injective on \(R\otimes\overline{\mathbb Q}_\ell\); this is the content of Laumon, Theorem 1.1.2, that all \(n\) together determine a class. A single \(n\) is a single point of \(\operatorname{Spec}R\), and Laumon states explicitly before 1.1.2 that one \(n\) does not determine a class. \(\square\)

**Sources cited:** trace-a.html (IIa); trace-b.html (IIb step 7); trace-c.html (IIc); _definitions.html (Proposition I); laumon-1987.pdf, Theorem 1.1.2 and the paragraph before it; bbd-perverse.pdf, Cor. 5.4.4 (purity of perverse cohomology); Thomason, Compositio 105 (1997), Theorem 2.1 (not in local sources)


## A1-3 · Theorem T: trace of Frobenius on \(\mathcal C_k\) is \(C_K\); the missing retract classes

**Kind/status as reported:** lemma / proved

**Builds on:** A1-1, A1-2

### Statement

For a small idempotent-complete \(\overline{\mathbb Q}_\ell\)-linear stable category \(\mathcal C\) with autoequivalence \(\mathrm{Fr}^*\), let \(\mathrm{HH}(\mathcal C,\mathrm{Fr})\) be Hochschild homology with coefficients in the bimodule \((M,N)\mapsto\operatorname{Hom}(\mathrm{Fr}^*M,N)\). Its degree-0 part is spanned by classes \([M,\beta]\), \(\beta:\mathrm{Fr}^*M\to M\), subject to \([M,g\circ\mathrm{Fr}^*f]=[N,f\circ g]\) for \(f:M\to N\), \(g:\mathrm{Fr}^*N\to M\). Assume (R), (C), D0–D3.

(a) \(\mathrm{LT}[M,\beta](x)=\sum_i(-1)^i\operatorname{tr}\big(\mathcal H^i(\beta)_{\bar x}\big)\), \(x\in X(F)\), is well defined, and \(\mathrm{LT}:\mathrm{HH}(\mathcal C_k,\mathrm{Fr})\xrightarrow{\sim}C_K\otimes\overline{\mathbb Q}_\ell\), with \(\mathrm{HH}\) concentrated in degree 0; likewise \(\mathrm{HH}(\mathcal H,\mathrm{Fr})\cong\mathcal H_K\otimes\overline{\mathbb Q}_\ell\).

(b) Under these isomorphisms \(\mathrm{HH}_0(\mathrm{act}_{\mathrm{IC}_0}):[F,\beta]\mapsto[F\ast\mathrm{IC}_0,\beta\ast\mathrm{IC}_0]\) is \(f\mapsto f\ast\varphi_0\). Hence \((\mathrm{SG})\iff\mathrm{HH}_0(\mathrm{act}_{\mathrm{IC}_0})\) is surjective.

(c) If \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\), then \(\mathrm{HH}_0(\mathcal C_k,\mathrm{Fr})\) has the basis \([\mathrm{IC}_\nu,\mathrm{can}]\), each the class of a Frobenius-stable retract of a perverse cohomology sheaf of a Satake object (an eigen-isotypic piece \(\mathrm{IC}_\nu\otimes E_{\nu,i}(\lambda)_\alpha\) has class \(d_\alpha\,\alpha\,[\mathrm{IC}_\nu,\mathrm{can}]\)), whereas \(\operatorname{im}\mathrm{HH}_0(\mathrm{act})\) is spanned by the full classes
$$[V_\lambda\ast\mathrm{IC}_0,\Phi]=\sum_{\nu,i}(-1)^i\operatorname{tr}A_{\nu,i}(\lambda)\,[\mathrm{IC}_\nu,\mathrm{can}] .$$

(d) Let \(A=\mathrm{act}^R(\mathrm{IC}_0)\in\operatorname{Ind}\mathcal H\), the right adjoint of the Ind-extended action applied to \(\mathrm{IC}_0\). Then \(\operatorname{End}_{\mathcal C_k}(V_\lambda\ast\mathrm{IC}_0)=\operatorname{Hom}_{\operatorname{Ind}\mathcal H}(V_\lambda,V_\lambda\ast A)\), which contains the image \(\overline{\mathbb Q}_\ell\) of \(\operatorname{End}^0_{\mathcal H}(V_\lambda)\) (via the unit \(1\to A\)); the idempotents of (c) lie outside this image unless \(V_\lambda\ast\mathrm{IC}_0\) is isotypic. Moreover \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\iff\operatorname{Ind}\mathcal C_k\simeq A\text{-}\mathrm{mod}(\operatorname{Ind}\mathcal H)\).

### Argument

**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 [[A1-2]](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. \(\square\)

**Sources cited:** trace-a.html (IIa); trace-b.html (IIb step 1, (i)); _hypotheses.html (D0-D3); chen-yi-v4.pdf, (41) and Theorem 93; zhu-satake-v2.pdf, Lemma 5.6.1; laumon-1987.pdf, (1.1.1.1)-(1.1.1.4); ponto-shulman-v2.pdf (arXiv:1107.6032v2), Prop. 2.4, Prop. 6.2, Ex. 6.3; lurie-ha-2017.pdf, Thm 4.7.3.5 (Barr-Beck-Lurie), Thm 7.1.2.1 (Schwede-Shipley); Keller 1998 (Morita invariance of HH), Kuznetsov 2009 (additivity for semi-orthogonal decompositions): not in local sources


## A1-6 · Forcing criteria: vanishing eigenfunctions, multiplicity two, extra endomorphisms

**Kind/status as reported:** lemma / proved

**Builds on:** A1-2, A1-5

### Statement

Let \(X=G/H\) be homogeneous spherical over a non-archimedean local field \(F\) (for instance \(\mathbb F_q(\!(t)\!)\)), with \(K\), \(\mathcal H_K\), \(C_K\), \(\varphi_0\) as in the setting.

(S) If \(\Lambda:C_K\to\mathbb C\) is nonzero, \(\Lambda(f\ast\varphi)=\chi(f)\Lambda(\varphi)\) for a character \(\chi\) of \(\mathcal H_K\), and \(\Lambda(\varphi_0)=0\), then (SG) fails. Conversely, if \(C_K\) is a finitely generated \(\mathcal H_K\)-module and (SG) fails, such \(\Lambda\) exists. (With \(\Omega(x)=\Lambda(\mathbf 1_{Kx})\): a Hecke eigenfunction on \(K\backslash X(F)\) with \(\sum_{Kx\subseteq X(\mathcal O)}\Omega(x)=0\).)

(d) If (SG) holds, then \(\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\le1\) for every \(\chi\); for every irreducible unramified \(\pi\), \(\dim\operatorname{Hom}_G(\mathcal S(X(F)),\pi)\le1\), and, if \(X(F)\) carries a \(G(F)\)-invariant measure, \(\dim\operatorname{Hom}_G(\pi,C^\infty(X(F)))\le1\) and every nonzero \(K\)-invariant eigenfunction \(\Omega\) has \(\int_{X(\mathcal O)}\Omega\neq0\).

(b) If (SG) holds, \(\operatorname{End}_{\mathcal H_K}(C_K)\) is the image of \(\mathcal H_K\), a commutative ring; so an \(\mathcal H_K\)-linear automorphism of \(C_K\) fixing \(\varphi_0\) and different from the identity forces failure. Sheaf-level source (under (R), (C), D0–D3): a Frobenius-compatible \(\mathcal H\)-linear t-exact autoequivalence \(\sigma\) of \(\mathcal C_k\) with \(\sigma(\mathrm{IC}_0,\mathrm{can})\cong(\mathrm{IC}_0,\mathrm{can})\) and \(\sigma(\mathrm{IC}_\nu)\cong\mathrm{IC}_{s(\nu)}\), \(s(\nu)\neq\nu\) for some \(\nu\), makes the rows \(\nu\) and \(s(\nu)\) of \(M\) proportional by a unit of \(R\), and \((\mathrm{SG})_n\) fails for every \(n\).

(c) \((\mathrm{SG})_n\) fails iff there is \(c\in\prod_{\nu\in\Lambda_S^+}\overline{\mathbb Q}_\ell\), \(c\neq0\), with \(\sum_\nu c_\nu m^{(n)}_{\nu\lambda}=0\) for all \(\lambda\) — a relation that may involve infinitely many rows. In particular, if for a finite set \(N\) of labels all columns restricted to \(N\) lie in one proper subspace of \(\overline{\mathbb Q}_\ell^N\), then \((\mathrm{SG})_n\) fails.

### Argument

**Proofs.**

1. *(S), first half.* \(\Lambda(f\ast\varphi_0)=\chi(f)\Lambda(\varphi_0)=0\) for all \(f\), so \(\Lambda\) vanishes on \(\mathcal H_K\ast\varphi_0\) but not on \(C_K\).

2. *(S), converse.* \(\mathcal H_K\) is a finitely generated commutative \(\mathbb C\)-algebra (Satake), hence Noetherian. If \(Q=C_K/\mathcal H_K\ast\varphi_0\neq0\) is finitely generated, pick a maximal ideal \(\mathfrak m\) with \(Q_{\mathfrak m}\neq0\); by Nakayama \(Q/\mathfrak mQ=Q_{\mathfrak m}/\mathfrak mQ_{\mathfrak m}\neq0\), and \(\mathcal H_K/\mathfrak m=\mathbb C\) (Nullstellensatz). A nonzero linear form on \(Q/\mathfrak mQ\) composed with \(C_K\to Q\) is the required \(\Lambda\), with \(\chi=\chi_{\mathfrak m}\).

3. *(d).* Under (SG), \(\Lambda\mapsto\Lambda(\varphi_0)\) is injective on \(\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\), so the dimension is at most 1. For \(\pi\) irreducible unramified with \(\pi^K=\mathbb Cv_0\) and character \(\chi_\pi\): restriction \(\ell\mapsto\ell|_{C_K}\) embeds \(\operatorname{Hom}_G(\mathcal S(X(F)),\pi)\) into \(\operatorname{Hom}_{\mathcal H_K}(C_K,\pi^K)\), because \(\ell(C_K)=\ell(e_K\mathcal S)=e_K\ell(\mathcal S)=\pi^K\neq0\) when \(\ell\neq0\) (\(e_K\) the projector to \(K\)-invariants; \(\ell(\mathcal S)=\pi\) by irreducibility). For \(A\in\operatorname{Hom}_G(\pi,C^\infty(X(F)))\), \(\Omega=A(v_0)\) determines \(A\), is \(K\)-invariant and satisfies \(f\ast\Omega=\chi_\pi(f)\Omega\); with an invariant measure, \(\Lambda_\Omega(\varphi)=\int_X\varphi\,\Omega\) satisfies \(\Lambda_\Omega(f\ast\varphi)=\Lambda_{f^\vee\ast\Omega}(\varphi)=\chi_\pi(f^\vee)\Lambda_\Omega(\varphi)\) (\(f^\vee(g)=f(g^{-1})\); change of variables), and \(\Omega\mapsto\Lambda_\Omega\) is injective (test on orbit indicators). Hence the bound, and \(\Lambda_\Omega(\varphi_0)=\int_{X(\mathcal O)}\Omega\neq0\) by (S).

4. *(b).* If \(C_K=\mathcal H_K\ast\varphi_0\cong\mathcal H_K/\mathrm{Ann}(\varphi_0)\), an \(\mathcal H_K\)-endomorphism \(D\) is determined by \(D(\varphi_0)=f_D\ast\varphi_0\), and \(D=f_D\ast-\). For the sheaf-level source: \(\sigma(V_\lambda\ast\mathrm{IC}_0)\cong V_\lambda\ast\sigma(\mathrm{IC}_0)\cong V_\lambda\ast\mathrm{IC}_0\) as Weil objects, and \(\sigma\) maps \({}^p\mathcal H^i\) to \({}^p\mathcal H^i\) and the \(\mathrm{IC}_\nu\)-isotypic part to the \(\mathrm{IC}_{s(\nu)}\)-isotypic part. If \(\sigma\) carries the canonical structure of \(\mathrm{IC}_\nu\) to \(c_\nu\) times that of \(\mathrm{IC}_{s(\nu)}\), then \([E_{s(\nu),i}(\lambda)]=[c_\nu][E_{\nu,i}(\lambda)]\) in \(R\), so \(M_{s(\nu),\lambda}=[c_\nu]M_{\nu,\lambda}\) for all \(\lambda\), and the functional \(e^*_{s(\nu)}-c_\nu^ne^*_\nu\) kills every column of \(m^{(n)}\). Its trace on \(C_{K_n}\) is an \(\mathcal H_{K_n}\)-automorphism fixing \(\varphi_0\) and moving \(\mathrm{tr}_n\mathrm{IC}_\nu\).

5. *(c).* The algebraic dual of \(\bigoplus_\nu\overline{\mathbb Q}_\ell\) is \(\prod_\nu\overline{\mathbb Q}_\ell\); the columns fail to span iff some nonzero functional kills them all. Each column has finite support, so each sum is finite. The finite-block version: extend by zero a nonzero \(c\in W^\perp\subset(\overline{\mathbb Q}_\ell^N)^*\).

6. *Examples.* In [[A1-5]] the eigenspaces are 2-dimensional, \(\delta_j\mapsto(-1)^j\delta_j\) is a nontrivial \(\mathcal H_K\)-automorphism fixing \(\varphi_0\), and \(c=\mathbf 1_{\text{odd}}\) is an infinite row relation. For \(\mathrm{SL}_2/T\) outside (C), Proposition J(2) of the classical page is the automorphism \(\sigma\) of (b), induced by \(\mathrm{diag}(u,1)\) with \(u\) a non-square unit.

7. *Relation to Sakellaridis.* His Cor. 8.0.4 (char 0, under (A1)–(A5)) derives principal generation, multiplicity one and good test vectors from surjectivity of \(r\), and his remark after it notes multiplicity \(>1\) at glued points; his Cor. 8.0.3 identifies the geometric endomorphisms with \(\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\), larger than the image of \(\mathcal H_K\) when \(r\) is not onto. Here the implications from (SG) to multiplicity one and good test vectors are proved for every \(X\) and every \(F\), with no hypothesis. \(\square\)

**Sources cited:** sakellaridis-v3.pdf, Thm 8.0.2, Cor. 8.0.3, Cor. 8.0.4 and the remark after it (sak.txt near line 3880); classical.html (Proposition B, Proposition J(2)); trace-b.html (IIb (iii))


## A1-7 · Close-field transfer: witnesses of (SG) persist over highly ramified \(p\)-adic fields

**Kind/status as reported:** lemma / proved

**Builds on:** A1-6

### Statement

Hypotheses. (H-int) \(G\supseteq T\supseteq S\) split, \(\theta\), \(H=G^\theta\), \(X=G/H\) have smooth affine models over \(\Lambda=W(\mathbb F_q)\) (\(p\) odd), with a \(G\)-equivariant closed embedding \(X\subset V\) into a free \(\Lambda\)-module of finite rank on which \(T\) acts diagonally, and \(x_0\in X(\Lambda)\). For a complete discrete valuation ring \(\mathcal O\) which is a \(\Lambda\)-algebra with residue field \(\mathbb F_q\) and uniformizer \(\varpi\) (\(\mathbb F_q[\![t]\!]\) via \(\Lambda\to\mathbb F_q\), or the integers of a finite extension of \(W(\mathbb F_q)[1/p]\) with residue field \(\mathbb F_q\)), put \(x_\mu=\mu(\sqrt\varpi)\,x_0\) (\(\mu\in\Lambda_S^+\)); it lies in \(X(F)\) and \(\tau(x_\mu)=\mu(\varpi)\). (H-Cartan for \(\mathcal O\)) every \(K\)-orbit in \(G(F)X(\mathcal O)\) contains some \(x_\mu\). [For \(\mathcal O=\mathbb F_q[\![t]\!]\) this is IIb step 1 under (R), (C), D0, D3.]

(a) For \(\lambda\in\Lambda_T^+\), \(\mu\in\Lambda_S^+\) there is \(m_0\) such that \(N_{\mathcal O}(\lambda,\mu)=(T_\lambda\ast\varphi_0)(x_\mu)\) depends only on the \(\Lambda\)-algebra \(\mathcal O/\varpi^{m_0}\) together with the class of \(\varpi\).

(b) If \(\mathbf 1_{Kx_\mu}=\sum_{\lambda\in\Lambda_0}c_\lambda\,T_\lambda\ast\varphi_0\) over \(\mathbb F_q(\!(t)\!)\) (\(\Lambda_0\) finite, \(c_\lambda\in\mathbb Q\)), the same identity holds over every \(\mathcal O'\) with residue field \(\mathbb F_q\), ramification index \(e(\mathcal O')\ge m_1(\Lambda_0,\mu)\), and (H-Cartan).

(c) Consequently, if for some \(\mu\) one has \(\mathbf 1_{K'x'_\mu}\notin\mathcal H_{K'}\ast\varphi_0'\) over \(p\)-adic \(\mathcal O'\) with residue field \(\mathbb F_q\), satisfying (H-Cartan), of arbitrarily large ramification index, then (SG) fails at \(\mu\) over \(\mathbb F_q(\!(t)\!)\).

### Argument

**Proof.**

1. *\(x_\mu\) is rational and universal.* For \(\gamma\in\mathrm{Gal}(F^{\mathrm{sep}}/F)\), \(\gamma(\mu(\sqrt\varpi))=\mu(\pm1)\mu(\sqrt\varpi)\) and \(\mu(-1)\in S[2]\subseteq H\) fixes \(x_0\), so \(x_\mu\in X(F)\); \(\tau(x_\mu)=\mu(\sqrt\varpi)\theta(\mu(\sqrt\varpi))^{-1}=\mu(\varpi)\). Writing \(x_0=\sum_\chi v_\chi\) in \(T\)-weight vectors, \(x_\mu=\sum_\chi\sqrt\varpi^{\langle\chi,\mu\rangle}v_\chi\); rationality forces \(\langle\chi,\mu\rangle\) even whenever \(v_\chi\neq0\). So the coordinates of \(x_\mu\) are Laurent monomials in \(\varpi\) with coefficients in \(\Lambda\), the same for every \(\mathcal O\).

2. *Cosets.* \(Kt^\lambda K/K\cong K/K_\lambda\), \(K_\lambda=K\cap t^\lambda Kt^{-\lambda}\). With \(c=\max_\alpha|\langle\alpha,\lambda\rangle|\) over roots, the congruence subgroup \(K(c)=\ker(G(\mathcal O)\to G(\mathcal O/\varpi^c))\) lies in \(K_\lambda\) (Iwahori factorization of \(K(c)\) into \(T(1+\varpi^c\mathcal O)\) and root groups \(U_\alpha(\varpi^c\mathcal O)\), conjugated by \(t^{-\lambda}\) into \(U_\alpha(\varpi^{c-\langle\alpha,\lambda\rangle}\mathcal O)\subseteq K\)). The subgroup \(K_\lambda/K(m)\subseteq G(\mathcal O/\varpi^m)\) (\(m\ge c\)) is cut out by divisibility of matrix coefficients by powers \(\varpi^{\langle\chi-\chi',\lambda\rangle}\) in a faithful representation, a condition on the ring \(\mathcal O/\varpi^m\) and \(\varpi\) only.

3. *The incidence condition.* For \(g=kt^\lambda\), \(g^{-1}x_\mu\in X(\mathcal O)\iff t^{-\lambda}k^{-1}x_\mu\in V(\mathcal O)\) (\(X\) is closed in \(V\)). Its coordinates are \(\varpi^{-\langle\chi,\lambda\rangle}\sum_{\chi'}\rho(k^{-1})_{\chi\chi'}\varpi^{\langle\chi',\mu\rangle/2}v_{\chi'}\); with \(b\) the largest exponent of \(\varpi^{-1}\) occurring, the condition is that finitely many polynomials over \(\Lambda\) in the entries of \(k^{-1}\) and in \(\varpi\) vanish modulo \(\varpi^b\). It depends on \(k\) modulo \(K(b)\) and on \((\mathcal O/\varpi^b,\varpi)\). So for \(m_0\ge\max(c,b)\), \(N_{\mathcal O}(\lambda,\mu)\) is the number of \(\bar k\in G(\mathcal O/\varpi^{m_0})/(K_\lambda/K(m_0))\) satisfying conditions defined over \((\mathcal O/\varpi^{m_0},\bar\varpi)\): (a).

4. *Close rings.* If \(e(\mathcal O')\ge m\), then \(p\in\varpi'^m\mathcal O'\), the Teichmüller section makes \(\mathcal O'/\varpi'^m\) an \(\mathbb F_q\)-algebra, and \(\mathcal O'/\varpi'^m=\mathbb F_q[\varpi']/(\varpi'^m)\cong\mathbb F_q[t]/(t^m)\), compatibly with \(\Lambda\to\mathbb F_q\) and \(\varpi'\mapsto t\).

5. *(b).* Let \(f'=\sum_{\lambda\in\Lambda_0}c_\lambda T_\lambda\ast\varphi_0'-\mathbf 1_{K'x'_\mu}\) over \(\mathcal O'\). Its support lies in \(\bigcup_\lambda K't^\lambda X(\mathcal O')\cup K'x'_\mu\subseteq G(F')X(\mathcal O')\). Since \(\tau(kx)=k\tau(x)\theta(k)^{-1}\) and \(\theta(K')=K'\), \(\tau\) maps \(K'\)-orbits into double cosets, \(\tau(K't^\lambda X(\mathcal O'))\subseteq K't^\lambda K't^{-\theta\lambda}K'\subseteq\bigcup_{\nu\le\lambda+(-\theta\lambda)^+}K't^\nu K'\) (convolution of double cosets, valid over any local field), and \(\tau(K'x'_\nu)\subseteq K't^\nu K'\) with distinct dominant \(\nu\) giving disjoint double cosets. With (H-Cartan), the support of \(f'\) is contained in the finite set of orbits \(K'x'_\nu\), \(\nu\in N_0=\{\nu:\nu\le\lambda+(-\theta\lambda)^+,\ \lambda\in\Lambda_0\}\cup\{\mu\}\). For \(\nu\in N_0\), \(f'(x'_\nu)=\sum c_\lambda N_{\mathcal O'}(\lambda,\nu)-\delta_{\mu\nu}=\sum c_\lambda N_{\mathbb F_q[\![t]\!]}(\lambda,\nu)-\delta_{\mu\nu}=0\) once \(e(\mathcal O')\) exceeds the \(m_0\) of (a) for all pairs in \(\Lambda_0\times N_0\). So \(f'=0\). The coefficients may be taken in \(\mathbb Q\) by IIb step 7.

6. *(c)* is the contrapositive of (b).

7. *Scope.* This is the precise bridge from a \(p\)-adic failure to a failure over \(\mathbb F_q(\!(t)\!)\): failures at bounded ramification do not transfer. Sakellaridis's Theorem 8.0.2 is stated for \(p\)-adic fields, but his axioms (A3) are proved only at almost all places of a global field; to transfer a failure of \(r\) by (c) one needs his theorem, with (A2)–(A4) and (H-Cartan), at places with residue field \(\mathbb F_q\) and unbounded ramification. No such case is established here. \(\square\)

**Sources cited:** trace-b.html (IIb step 1, step 7); criterion.html (Proposition III, step 2: support bound via tau); classical.html (Lemma A; Sakellaridis's hypotheses A1-A5); sakellaridis-v3.pdf, §1.7, §2.4, Thm 8.0.2


## A1-8 · Under (R), (C), D0–D3: three mechanisms vanish; the obligation sits on uncovered labels

**Kind/status as reported:** lemma / proved

**Builds on:** A1-2, A1-5, A1-6

### Statement

Assume (R), (C), D0–D3, \(q\) odd.

(a) Frobenius acts trivially on \(\Lambda_T^+\) and \(\Lambda_S^+\); the label mechanism of [[A1-5]] is void.

(b) \(X(F_n)=G(F_n)\cdot x_0\) for every \(n\); the several-orbit mechanism (\(\mathrm{GL}_n/\mathrm O_n\), unitary instance of [[A1-5]]) is void.

(c) An automorphism \(\epsilon\) of \((G,\theta)\) over \(\mathbb F_q\) with \(\epsilon^*V_\lambda\cong V_\lambda\) for all \(\lambda\) fixes every \(K_n\)-orbit, hence acts trivially on \(C_{K_n}\); the geometric form of criterion (b) of [[A1-6]] is void, and so are the degree-0 automorphisms of \(\mathrm{id}_{\mathcal C_k}\) coming from \(\pi_0\) of stabilizers (the \(\varepsilon\) of \(\mathrm{SL}_2/\mathrm{SO}_2\)).

(d) For every \(\lambda\) and \(n\),
$$m^{(n)}_{\varphi(\lambda),\lambda}=(-1)^{\delta(\varphi(\lambda))+\langle2\rho,\lambda\rangle}\sum_{\mu\le\lambda}a_{\lambda\mu}(q^n)\,(T_\mu\ast\varphi_0)(t^\lambda x_0)\neq0,$$
with \(a_{\lambda\lambda}=1\) and \(a_{\lambda\mu}\in\mathbb N[x]\) the Kazhdan–Lusztig polynomials of Zhu (5.6.2).

(e) Choose \(\lambda_\mu\in\varphi^{-1}(\mu)\) for each covered \(\mu\). Then \(\{T_{\lambda_\mu}\ast\varphi_0\}_{\mu\notin U}\cup\{\mathbf 1_{\mathcal O_\nu}\}_{\nu\in U}\) is a basis of \(C_K\); with \(p_U\) the projection onto \(\mathbb C^{(U)}\) along the first family,
$$(\mathrm{SG})\iff\{p_U(T_\lambda\ast\varphi_0):\lambda\in\Lambda_T^+\}\ \text{spans}\ \mathbb C^{(U)} .$$
In particular \(U=\varnothing\Rightarrow(\mathrm{SG})_n\) for all \(n\) (Corollary C of the classical page).

(f) If for some \(\mu,\lambda\) every eigenvalue of \(A_{\mu,i}(\lambda)\) equals \(q^{(w+i-w_\mu)/2}\) (no root-of-unity factor) and \(E_{\mu,i}(\lambda)\neq0\) only for \(i\) of one parity, then \(m^{(n)}_{\mu\lambda}\neq0\) for every \(n\). Hence a triangular summand system — a map \(\mu\mapsto\lambda_\mu\) and an order \(\prec\) with finite lower sets, with \(M_{\nu\lambda_\mu}\neq0\Rightarrow\nu\preceq\mu\) and \(E_{\mu,\bullet}(\lambda_\mu)\neq0\) of this split Tate type — gives \((\mathrm{SG})_n\) for all \(n\).

(g) The exact remaining obligation for "\(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\Rightarrow(\mathrm{SG})\)" here is the spanning condition (e) at the given \(q\); by [[A1-6]](S) it fails iff (given finite generation of \(C_K\)) some Hecke eigenfunctional vanishes at \(\varphi_0\). The mechanisms not excluded by (a)–(c) are gluing (two spectral points over one character, [[A1-10]]) and cancellation at \(\mathrm{ev}_1\) inside the multiplicity spaces (an alternating sum \(\sum_i(-1)^i\operatorname{tr}A_{\mu,i}(\lambda)\) vanishing at \(n=1\): root-of-unity factors, e.g. Frobenius permuting top-dimensional components of a convolution fibre as in [[A1-4]](E4), non-Tate classes, or coincidences between degrees of opposite parity; all excluded under the hypothesis of (f)).

### Argument

**Proofs.**

1. *(a).* \(G\) and \(S\) are split, so Frobenius is trivial on \(X_*(T)\) and \(X_*(S)\).

2. *(b).* Put \(T_\theta=(T^\theta)^\circ\). Since \(L=Z_G(S)=S\cdot Z_{G^\theta}(S)\) with finite intersection \(S[2]\) (criterion page, step 4), \(\operatorname{rank}Z_{G^\theta}(S)=\dim T-\dim S=\dim T_\theta\); so \(T_\theta\) is a maximal torus of the connected (by (C)) reductive group \(Z_{G^\theta}(S)\), and \(T^\theta\subseteq Z_{Z_{G^\theta}(S)}(T_\theta)=T_\theta\): \(T^\theta\) is connected, and split as a subtorus of the split \(T\). For \(\mu\in\Lambda_S\), \(x_\mu=\mu(\sqrt t)x_0\) and the cocycle \(\gamma\mapsto\mu(\sqrt t)^{-1}\gamma(\mu(\sqrt t))=\mu(\pm1)\) takes values in \(S[2]\subseteq T^\theta=T_\theta\subseteq H\). Its class in \(H^1(F_n,H)\) is the image of a class in \(H^1(F_n,T_\theta)=0\) (Hilbert 90), so \(x_\mu\in G(F_n)x_0\). By IIb step 1, \(X(F_n)=\bigsqcup_\mu K_nx_\mu\); hence \(X(F_n)=G(F_n)x_0\).

3. *(c).* \(\epsilon^*V_\lambda\cong V_\lambda\) forces \(\epsilon\) to fix every dominant coweight (the supports \(\mathrm{Gr}_{\le\lambda}\) are distinct). Then \(\tau(\epsilon x_\mu)=\epsilon(t^\mu)\in L^+Gt^\mu L^+G\), and since \(\tau(\mathcal O_\nu)\subseteq L^+Gt^\nu L^+G\) with disjoint double cosets, \(\epsilon(\mathcal O_\mu)=\mathcal O_\mu\). The \(K_n\)-orbits are the \(\mathcal O_\mu(\mathbb F_{q^n})\) (IIb step 1), so each is preserved. Under (C), \(\pi_0\operatorname{Stab}(x_\mu)=1\), so no degree-0 automorphism of \(\mathrm{id}_{\mathcal C_k}\) arises from stabilizers.

4. *(d).* \(\varphi(\lambda)\) is the top label in the support (D0), and \(\mathrm{tr}_n\mathrm{IC}_\nu(x_{\varphi(\lambda)})=0\) for \(\nu<\varphi(\lambda)\), so evaluating at \(x_{\varphi(\lambda)}=t^\lambda x_0\): \(m^{(n)}_{\varphi(\lambda),\lambda}=(-1)^{\delta(\varphi(\lambda))}\big(\mathrm{tr}_nV_\lambda\ast\varphi_0\big)(t^\lambda x_0)\). By Zhu (5.6.2) over \(\mathbb F_{q^n}\), \(\mathrm{tr}_nV_\lambda=(-1)^{\langle2\rho,\lambda\rangle}\sum_{\mu\le\lambda}a_{\lambda\mu}(q^n)T_\mu\) with \(a_{\lambda\mu}\in\mathbb N[x]\), \(a_{\lambda\lambda}=1\). All \((T_\mu\ast\varphi_0)(t^\lambda x_0)\) are non-negative integers and the \(\mu=\lambda\) term is \(\ge1\) (Lemma A(2) of the classical page): the sum is positive.

5. *(e).* By D0 and Lemma A(2), \(T_{\lambda_\mu}\ast\varphi_0=N_\mu\mathbf 1_{\mathcal O_\mu}+\sum_{\nu<\mu}(\ast)\mathbf 1_{\mathcal O_\nu}\), \(N_\mu\ge1\). Lower sets are finite, so by induction the stated family spans each \(\operatorname{span}\{\mathbf 1_{\mathcal O_\nu}:\nu\le\kappa\}\) and is independent: a basis. \(\mathcal H_K\ast\varphi_0\) contains \(\ker p_U\), so it equals \(C_K\) iff its image under \(p_U\) is \(\mathbb C^{(U)}\), and \(p_U(\mathcal H_K\ast\varphi_0)\) is spanned by the \(p_U(T_\lambda\ast\varphi_0)\) (those with \(\lambda=\lambda_\mu\) vanish). For \(U=\varnothing\) the same holds over every \(F_n\).

6. *(f).* Let \(i_0\) be one index with \(E_{\mu,i_0}(\lambda)\neq0\). On the indices with \(E_{\mu,i}(\lambda)\neq0\), \((-1)^i=(-1)^{i_0}\) and \((i-i_0)/2\in\mathbb Z\), so \(m^{(n)}_{\mu\lambda}=(-1)^{i_0}\big(q^{1/2}\big)^{n(w+i_0-w_\mu)}\sum_i\dim E_{\mu,i}(\lambda)\,q^{n(i-i_0)/2}\), a nonzero number times a sum of positive terms. Given the triangular system, \(\mathrm{tr}_n(V_{\lambda_\mu}\ast\mathrm{IC}_0)=m^{(n)}_{\mu\lambda_\mu}\mathrm{tr}_n\mathrm{IC}_\mu+\sum_{\nu\prec\mu}(\ast)\mathrm{tr}_n\mathrm{IC}_\nu\) with nonzero leading coefficient, and induction on the finite lower sets of \(\prec\) puts every \(\mathrm{tr}_n\mathrm{IC}_\mu\) in the span (Proposition B of the classical page, in the IC basis).

7. *(g).* Combine (e) with [[A1-6]](S). Mechanisms (a)–(c) are excluded; [[A1-2]](i) shows \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) is the non-vanishing of rows of \(M\), which constrains neither the rank of the \(U\)-reduced system nor the root-of-unity parts of the \(A_{\mu,i}(\lambda)\). For the known non-covered families: \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) satisfy (SG) (Theorem 4 of the run); \(E_6/F_4\) is excluded from this question and open over \(\mathbb F_q(\!(t)\!)\). \(\square\)

**Sources cited:** trace-b.html (IIb step 1); criterion.html (Proposition III, steps 2 and 4); classical.html (Lemma A, Proposition B, Corollary C); zhu-satake-v2.pdf, (5.6.2); verdict.html (Theorem 4, Corollary 5)


## A2-12 · Remark: why the trace misses thick generation, and what (R) removes

**Kind/status as reported:** objection / proved

**Builds on:** A2-3, A2-6, A2-8, A2-9, A2-10, A2-11

### Statement

1. **(Proved) Two statements.** (SG)\(_n\) holds iff the traces \(\operatorname{tr}_n(V\ast\mathrm{IC}_0,\beta\ast\mathrm{can})\), over Weil objects \((V,\beta)\) of \(\mathrm{Sat}_{G_k}\), span \(C_{K_n}\otimes\overline{\mathbb Q}_\ell\) (IIa, [[A2-8]]). \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) holds iff every \(\mathrm{IC}_\nu\) is a retract of a shift of some \(V_\lambda\ast\mathrm{IC}_0\), \(\lambda\in\Lambda_T^+\) (Proposition I). The two differ by three Frobenius effects:
   - (G1) retracts of \(V_\lambda\ast\mathrm{IC}_0\) with \(\sigma\lambda\ne\lambda\), whose Weil structures come from no Satake object and which are invisible at \(n=1\) ([[A2-3]], [[A2-8]]);
   - (G2) for \(\sigma\lambda=\lambda\), cancellation in \(\operatorname{tr}A_{\nu,i}(\lambda)\), or linear dependence among the rational columns ([[A2-9]] has dependence with every row nonzero);
   - (G3) Frobenius on orbit labels, which removes orbits from \(C_K\) and never adds obligations ([[A2-3]](3), [[A2-10]]).
2. **(Proved) Forcing criterion.** Assume D0, D3, connected stabilizers, and a rational point on every Frobenius-stable orbit. If \(\varphi\) is injective on \((\Lambda_T^+)^\sigma\) (modulo elements acting trivially on \(X\)) and \(\varphi((\Lambda_T^+)^\sigma)\ne(\Lambda_S^+)^{\mathrm{Fr}}\), then (SG) fails. If moreover \(\mathcal M=\Lambda_S^+\) over \(k\), the pair is a counterexample to IId. Unitary twists are of this kind: \(\ker\varphi=X_*(T)^\theta=\{(a,-a)\}\) meets \(X_*(T)^\sigma=\{(a,a)\}\) only in \(0\).
3. **(Proved) Not an invariant of \((G_k,X_k)\).** (SG) over \(\mathbb F_q(\!(t)\!)\) is not a function of \((G_k,X_k)\) ([[A2-10]]).
4. **(Proved, using the run's Lemma 4.8) Under (R) and (C).**
   - \(\sigma=1\) on \(\Lambda_T\) and every \(\mathcal O_\mu\) has the rational point \(x_\mu\), so (G1) and (G3) are void.
   - \(\varphi\) is never injective on \(\Lambda_T^+\) itself in rank \(\ge1\). If it is injective modulo cocharacters acting trivially on \(X\), then (C) forces every label to lift, so (SG) holds. Either way the criterion of item 2 never produces a failure.
   - If every \(\mu\in\Lambda_S^+\) lies in \(\varphi(\Lambda_T^+)\), (SG) holds over \(\mathbb F_q(\!(t)\!)\) (verdict, Theorem 4, last sentence, Lemma 4.8 of the run).

   Hence a counterexample within Proposition II's setting needs an uncovered label \(\mu\notin\varphi(\Lambda_T^+)\) with \(\mathrm{IC}_\mu\) a summand of some \(V_\lambda\ast\mathrm{IC}_0\) but \(\mathbf 1_{\mathcal O_\mu(\mathbb F_q)}\notin\operatorname{span}\{T_\lambda\ast\varphi_0\}\). That is a cancellation question among Hecke operators sharing a leading orbit, which Frobenius twisting cannot produce.
5. **(Heuristic) Categorical trace.** Once \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\), \(\mathrm{Tr}(\mathrm{Fr},\mathcal C_k^{\mathrm{Sat}})=\mathrm{Tr}(\mathrm{Fr},\mathcal C_k)\) tautologically, and it contains every class \([\mathrm{IC}_\nu,\mathrm{can}]\). (SG) concerns instead the image of \(\mathrm{Tr}(\mathrm{Fr},\mathrm{Sat}_{G_k})\) under the action functor \(a(V)=V\ast\mathrm{IC}_0\), which is not full. For example \(\operatorname{Hom}_{\mathrm{Sat}}(V_{(c,0)},\mathrm{Fr}^*V_{(c,0)})=0\), while \(\operatorname{Hom}_{\mathcal C_k}(\mathrm{IC}_c,\mathrm{Fr}^*\mathrm{IC}_c)\ni\mathrm{can}\).

### Argument

**1. Two statements.** The first equivalence is IIa together with [[A2-8]](2)–(3). The second is Proposition I. (G1) is [[A2-8]](3)–(4): the summand \(\mathrm{IC}_c=V_{(c,0)}\ast\mathrm{IC}_0\) carries Weil structures, but only induced Weil objects of \(\mathrm{Sat}\) contribute to \(\mathcal H_K\ast\varphi_0\), and those see \(V_{(c,0)}\oplus V_{(0,c)}\) with an off-diagonal Frobenius. For (G3), a non-stable orbit has no rational points, so it does not index a \(K\)-orbit.

**2. Forcing criterion.**
- *Support.* \(\operatorname{tr}V_\lambda=\pm T_\lambda+\sum_{\mu<\lambda}a_{\lambda\mu}T_\mu\) for rational \(\lambda\), and \(\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\overline{\mathcal O_{\varphi(\lambda)}}\) (D0). Induction on \(\lambda\), using that \(\varphi\) is order-preserving, gives \(\operatorname{supp}(T_\lambda\ast\varphi_0)\subseteq\bigcup_{\nu\le\varphi(\lambda)}O_\nu\).
- *Top value.* \(t^\lambda x_0\in\mathcal O_{\varphi(\lambda)}(\mathbb F_q)\) gives \((T_\lambda\ast\varphi_0)(t^\lambda x_0)\ge1\) (Lemma A(2)).
- *Conclusion.* [[A2-2]](b) with \(\psi=\varphi|_{(\Lambda_T^+)^\sigma}\) shows that the labels outside \(\varphi((\Lambda_T^+)^\sigma)\) are not generated.
- *Unitary twist.* \(\theta(a,b)=(-b,-a)\) and \(\sigma(a,b)=(b,a)\) give \(\varphi(a,a)=(2a,2a)\), injective, with image \(2\Lambda_H^+\ne\Lambda_H^+\).

**4. Under (R) and (C).**
- (R) makes \(T\) split, so \(\sigma=1\), and \(S\) split, so \(t^\mu\in S(F)\) and every orbit is Frobenius-stable with a rational point.
- *Injectivity on \(\Lambda_T^+\).* A nonzero \(\kappa\in\ker\varphi=X_*(T)^\theta\) gives \(\varphi(N\xi+\kappa)=\varphi(N\xi)\), with both arguments dominant for \(\xi\) regular dominant and \(N\gg0\). So injectivity on \(\Lambda_T^+\) forces \(X_*(T)^\theta=0\), hence \(T=S\), \(Z_G(S)=T\) and \(Z_{G^\theta}(S)=T[2]\cong\mu_2^{\operatorname{rk}G}\), which is disconnected for \(p\) odd and rank \(\ge1\). Under (C), then, \(\varphi\) is never injective on \(\Lambda_T^+\).
- *Injectivity modulo trivially acting cocharacters.* Since \(T_\lambda\) and \(T_{\lambda+\kappa}\) act identically on \(C_K\) when \(t^\kappa\) acts trivially on \(X\), the relevant condition is injectivity modulo such \(\kappa\). By the same \(N\xi+\kappa\) argument, it forces every \(\kappa\in X_*(T)^\theta\) to act trivially on \(X\) after passing to a multiple, so \(Z':=(T^\theta)^\circ\) acts trivially on \(X\). Divide \(G\) by any simple factor that acts trivially on \(X\); such a factor only rescales Hecke operators by their degrees. The kernel of the action is then a normal subgroup whose identity component is a central torus, so \(Z'\) is central in \(G\). Every root is nonzero on \(S\), since a root vanishing on \(S\) would vanish on \(SZ'=T\); hence \(Z_G(S)=T\) and \(T=S\cdot Z'\).
   - Consider \(f:T\to S\), \(t\mapsto t\theta(t)^{-1}\). It is surjective (\(f(s)=s^2\) on \(S\)) with kernel \(T^\theta=Z_{G^\theta}(S)\), and \(X_*(f)=\varphi\). The cokernel of \(X_*(f)\) has order \(|\pi_0(\ker f)|\), so under (C) \(\varphi:\Lambda_T\to\Lambda_S\) is onto.
   - A preimage \(\lambda=\tfrac12\mu+\zeta\), \(\zeta\in X_*(Z')_{\mathbb Q}\), of a dominant \(\mu\) is dominant, because roots vanish on the central \(Z'\).
   - So every label lifts dominantly, and (SG) holds by Lemma 4.8 of the run. The forcing criterion yields nothing under (R) and (C), and a failure there must come from cancellations among Hecke operators sharing a leading orbit.
- The positive statement is the run's Lemma 4.8, quoted in the verdict: split symmetric spaces meeting Proposition III's two conditions (connected \(Z_{G^\theta}(S)\) and dominant lifts) satisfy (SG) over \(\mathbb F_q(\!(t)\!)\) under D0, D3.

**5. Categorical trace.** This item is heuristic; no statement of AGKRRV or Zhu on categorical traces is used or checked here. On the level of trace functions, the precise content is [[A2-8]](4). In \(\mathrm{HH}_0(\mathcal C,\mathrm{Fr}^*)\), a retract \(M\xrightarrow{i}N\xrightarrow{p}M\) satisfies \([M,\beta]=[N,\mathrm{Fr}^*(i)\circ\beta\circ p]\) by cyclicity. The map \(\mathrm{Fr}^*(i)\beta p\) is a morphism of \(\mathcal C_k\) that need not come from \(\mathrm{Sat}\). Thick closure therefore enlarges the trace by exactly such classes, while (SG) uses only the classes coming from \(\mathrm{Sat}\).

**Sources cited:** satake-two-levels/verdict.html, Theorem 4 (Lemma 4.8 of the run) and §7; trace-pages/trace.html, Limits; spherical-generation/classical.html, Corollary C, Lemma A(2); trace-pages/_definitions.html, Proposition I