Verdict · 2026-10-17

Satake generation at two levels

partial

Every variety of Table 1 except $E_6/F_4$ gets the same answer at both levels, and none has the reverse split (functions generate, sheaves do not). For $E_6/F_4$, $\ell$-adic sheaves over $\overline{\mathbb F}_q$, $p\ne2$, generate, given Chen–Yi's orbit geometry and the decomposition theorem; but over $\mathbb F_q(\!(t)\!)$ and over $p$-adic fields, $p$ odd, $C_K$ needs two generators, so $\varphi_0$ does not generate it.

Terms · every symbol used below

Symmetric space. $X=G/G^\theta$ for an involution $\theta$ of $G$; $S\subseteq T$, a maximal $\theta$-split torus in a $\theta$-stable maximal torus; orbit labels $\mu\in\Lambda_S^+$; $\varphi(\lambda)=\lambda-\theta(\lambda)$. A dominant lift of $\mu$ is a $\lambda\in\Lambda_T^+$ with $\varphi(\lambda)=\mu$.
Sheaves, over $k=\mathbb C$ (the problem) or $k=\overline{\mathbb F}_q$ with $\ell$-adic coefficients: $\mathcal O_\mu\subseteq LX$ the $L^+G$-orbit of label $\mu$; $\mathrm{IC}_\mu$ its intersection complex with constant coefficients; the Satake objects $V_\lambda\ast\mathrm{IC}_0$; $\mathcal C_K^{\mathrm{Sat}}\subseteq\mathcal C_K$ the thick subcategory they generate. Sheaves generate means $\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K$.
Functions, over a non-archimedean local field $F$ with residue field $\mathbb F_q$: $K=G(\mathcal O)$, $\mathcal H_K$ the spherical Hecke algebra, $C_K=C_c^\infty(X(F))^K$, $\varphi_0=\mathbf 1_{X(\mathcal O)}$, $\mathbf 1_{O_\mu}$ the indicator of $O_\mu=K\cdot x_\mu$. Functions generate, (SG), means $\mathcal H_K\ast\varphi_0=C_K$. Every $F$: every non-archimedean local field, of either characteristic.
(R) $G,\theta,X$ are defined over $\mathbb F_q$, $q$ odd, $T\supseteq S$ is split, and $G$ has a $\theta$-stable Borel pair $(B_0,T_0)$ over $\mathbb F_q$ with $T_0$ split. (C) $G^\theta$ and $Z_{G^\theta}(S)$ are connected.
Cited inputs over $\overline{\mathbb F}_q$, $p\ne2$ (full statements). D0 Chen–Yi's orbit geometry: $LX=\bigsqcup_{\Lambda_S^+}\mathcal O_\mu$, $\overline{\mathcal O_\mu}=\bigsqcup_{\nu\le\mu}\mathcal O_\nu$, $\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\overline{\mathcal O_{\varphi(\lambda)}}$, connected stabilizers under (C). D1 proper base change, the Grothendieck–Lefschetz trace formula, a perverse t-structure with simple objects the $\mathrm{IC}_\nu$. D2 each $V_\lambda\ast\mathrm{IC}_0$ is pure and a finite sum of shifted $\mathrm{IC}_\nu$. D3 Lang's theorem for $L^+G$ and its stabilizers. D5 "$\mathrm{IC}_\mu$ is a summand of $V_\lambda\ast\mathrm{IC}_0$" has the same answer over $\overline{\mathbb F}_q$ and over $\mathbb C$, for $p$ large (BBD §6.1).
Sakellaridis's theorem (Theorem 8.0.2), under his hypotheses (A1)–(A5) (list): $C_K\simeq\mathbb C[\Lambda_S]^{W_X}$ as $\mathcal H_K$-modules, $\varphi_0\mapsto1$, where $\mathcal H_K\simeq\mathbb C[A^*]^W$ (Satake, $h\mapsto\hat h$) acts through the restriction $r(f)(y)=f(\delta(X)^{1/2}y)$ from $A^*=\operatorname{Hom}(\Lambda_T,\mathbb C^\times)$ to $A_X^*=\operatorname{Hom}(\Lambda_S,\mathbb C^\times)$; $W_X$ is the little Weyl group. So (SG) holds iff $r$ is onto.
$E_6/F_4$. $G$ split simply connected $E_6$ (Bourbaki labels: chain 1–3–4–5–6, node 2 on node 4), $\theta|_T=-w_L$ with $w_L$ the longest element of the $W(D_4)$ of nodes 2–5, $G^\theta=F_4$, and the model $\mathcal M_\theta=(G/G^\theta)_{\mathbb Z}\otimes\mathcal O$, $2\in\mathcal O^\times$. Labels $(a,b)=a\omega_1^\vee+b\omega_6^\vee$, $\Lambda_S^+=\{a,b\ge0,\ a\equiv b\bmod 3\}$; $W_X=S_3$; $\delta(X)^{1/2}=q^{\rho_L}$, $\rho_L$ the half-sum of the positive roots of that $D_4$; $\sigma$ the diagram symmetry $1\leftrightarrow6$, $(a,b)\mapsto(b,a)$.

1 · The answer, variety by variety

$$\text{sheaves:}\ \ \mathcal C_K^{\mathrm{Sat}}\overset?=\mathcal C_K,\qquad\qquad\text{functions:}\ \ \mathcal H_K\ast\varphi_0\overset?=C_K .$$

Table 2. Generation comparison
$X$	sheaves	functions	same?	where
$(H\times H)/H$, $\mathrm{GL}_2/\mathrm{Sp}_2$, $\mathrm{GL}_2/T$	generate	generate	yes	pages
$\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}$, $n\ge2$	generate	generate, every $F$	yes	pages, V
$\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$	generate	generate, every $F$, pair model	yes	pages, V
$\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}$, $\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}$	generate over $\overline{\mathbb F}_q$, given D0–D3	generate, every $F$	yes	IV
$\mathrm{SL}_2/\mathrm{SO}_2$	fail	fail (split form $\mathrm{SL}_2/T$, every $F$)	yes	III
$\mathrm{GL}_n/\mathrm O_n$, $\mathrm{PGL}_2/\mathrm{PO}_2$	fail	fail	yes	pages
$E_6/F_4$	generate over $\overline{\mathbb F}_q$, $p\ne2$, given D0–D2	fail over $\mathbb F_q(\!(t)\!)$ and $p$-adic $F$, $p$ odd	no	VI–VIII

"Pages": the spherical-generation pages, whose sheaf cells reach $\mathbb C$ through the comparison D5. Over $\mathbb F_q(\!(t)\!)$, under (R), (C) and D0–D3, the reverse split cannot occur at all: Proposition II.

2 · Key propositions

Colour = proof status. Proved every step proved here or a cited published result. Conditional proved assuming a named statement that is not yet proved. Not proved open. Whether a level “holds” or “fails” is in the words, not the colour.

Independent propositions: Sakellaridis’s theorem (VI), with the characteristic-p proof · Sheaf generation and rational functions (IX).

I Summand criterion
- Proved a formal argument; the hypotheses are part of the statement.
Role in the verdict. The criterion: the sheaf question becomes “is every $\mathrm{IC}_\mu$ a summand of some Satake object?”
Let $G^\theta$ be connected. Assume H1–H3 (finitely many orbits below each orbit, recollement along orbits, constant coefficients on orbits), a finite-length perverse t-structure whose simple objects are the $\mathrm{IC}_\nu$, and that every $V_\lambda\ast\mathrm{IC}_0$ is a finite sum of shifted $\mathrm{IC}_\nu$. With $\mathcal M=\{\mu:\mathrm{IC}_\mu[s]\ \text{is a direct summand of some}\ V_\lambda\ast\mathrm{IC}_0\}$: $$\mathcal C_K^{\mathrm{Sat}}=\operatorname{Thick}\{\mathrm{IC}_\nu:\nu\in\mathcal M\},\qquad\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K\iff\mathcal M=\Lambda_S^+.$$
Over $\overline{\mathbb F}_q$ the hypotheses follow from (C) and D0–D2.
Statement, proof and proof graph →
II Functions $\Rightarrow$ sheaves over $\mathbb F_q(\!(t)\!)$
- Proved over $\overline{\mathbb F}_q$, given D0–D3.
Role in the verdict. The bridge: over $\mathbb F_q(\!(t)\!)$, functions generating forces sheaves to generate.
Let $X=G/G^\theta$ over $\mathbb F_q$, $q$ odd, satisfy (R) and (C), and assume D0–D3. If (SG) holds over $\mathbb F_q(\!(t)\!)$, then every $\mathrm{IC}_\mu$ is a shifted summand of some $V_\lambda\ast\mathrm{IC}_0$, and $$\mathcal C^{\mathrm{Sat}}_{K,\overline{\mathbb F}_q}=\mathcal C_{K,\overline{\mathbb F}_q}.$$
Frobenius traces of the Satake objects span $\mathcal H_K\ast\varphi_0$, so (SG) forces every $\mathrm{IC}_\mu$ to occur in one of them; then I applies. The converse fails: IX. Uses I.
Statement, proof and proof graph →
III $\mathrm{SL}_2/\mathrm{SO}_2$: both levels fail
- Proved sheaves over $\mathbb C$; functions for the split form $\mathrm{SL}_2/T$, over every $F$.
Role in the verdict. A row where both levels fail, for one reason: the central $-I$ of $\mathrm{SL}_2$.
Let $G=\mathrm{SL}_2$, $X=\mathrm{SL}_2/\mathrm{SO}_2$, $k=\mathbb C$. The central $-I\in L^+G$ acts trivially on $LX$, so it acts on every object, splitting $\mathcal C_K=\mathcal C_K^+\oplus\mathcal C_K^-$ by its eigenvalue $\pm1$. Satake objects lie in $\mathcal C_K^+$, while for $m\ge1$ the sign local system on $\mathcal O_m$, extended by zero, is a nonzero object of $\mathcal C_K^-$; so $\mathcal C_K^{\mathrm{Sat}}\ne\mathcal C_K$. For $X=\mathrm{SL}_2/T$ over every $F$, with $T_1$ the $\mathrm{GL}_2$ Hecke operator of the tree and $d$ the distance from the base vertex to the apartment of $T$: $$\mathcal H_K\ast\varphi_0=\mathbb C[T_1^2]\,\varphi_0\not\ni\mathbf 1_{\{d=1\}}.$$
The sheaf failure uses only $2\ne0$ in $k$; the function failure holds for every $q$.
Statement, proof and proof graph →
IV Symplectic family: both levels generate without dominant lifts
- Proved functions over every $F$, without (A1)–(A5).
- Proved sheaves over $\overline{\mathbb F}_q$, $p\ne2$, given D0–D3 (through II).
Role in the verdict. Rows where both levels generate although some labels have no dominant lift: so I's criterion, not dominant lifts, is the test.
Let $X=\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}$ or $\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}$, $m,n\ge1$. Then $\mathcal H_K\ast\varphi_0=C_K$ over every $F$, and $\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K$ over $\overline{\mathbb F}_q$, although for $m\ge2$, $n\ge3$ some labels have no dominant lift. For $\mathrm{GL}_4/\mathrm{Sp}_4$, up to an overall shift: $$V_{(1,1,0,0)}\ast\mathrm{IC}_0\simeq\mathrm{IC}_{(2,2,0,0)}\oplus\mathrm{IC}_{(1,1,1,1)}\otimes\big(k[2]\oplus k\oplus k[-2]\big),$$ matching $T_{(1,1,0,0)}\ast\varphi_0=\mathbf 1_{O_{(2,2,0,0)}}+(1+q)(1+q^2)\,\mathbf 1_{O_{(1,1,1,1)}}$.
Functions: a triangular system of lattice counts. Sheaves: II applied to it. Uses I, II.
Statement, proof and proof graph →
V Function-level generation without (A1)–(A5)
- Proved over every $F$, without (A1)–(A5); the last part given D0 and D3.
Role in the verdict. Rows where functions generate, proved directly, without Sakellaridis's hypotheses.
Over every $F$, $\mathcal H_K\ast\varphi_0=C_K$ for $\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}$, $n\ge2$, in the quadric model $\{\sum_{i\le n}x_ix_{2n+1-i}=1\}\subset\mathcal O^{2n}$, and for $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_{n-p})$ in the pair model (idempotents of rank $n-p$), which is the involution model when $2\in\mathcal O^\times$. Moreover, if $q$ is odd, $T\supseteq S$ is split over $\mathbb F_q$, $G^\theta$ and $Z_{G^\theta}(S)$ are connected and every label has a dominant lift, then given D0 and D3, (SG) holds over every $\mathbb F_{q^r}(\!(t)\!)$.
Hensel's lemma, Witt's theorem and lattice normal forms give a triangular system.
Statement, proof and proof graph →
VII $E_6/F_4$: $C_K$ needs two generators
- Proved the algebra, for every $q>1$, with no hypothesis.
- Proved the module $C_K$, over $\mathbb F_q(\!(t)\!)$ and $p$-adic $F$, $p$ odd, with VI.
Role in the verdict. The computation for $E_6/F_4$: $r$ is not onto, by one dimension.
Let $S=\mathbb C[\Lambda_S]^{W_X}$ and $y_1\in A_X^*$ with $y_1(\varphi\alpha_1^\vee)=q^{-5}$, $y_1(\varphi\alpha_6^\vee)=q$. For every real $q>1$: $\operatorname{im}r=\{g\in S:g(y_1)=g(\sigma y_1)\}$ and $S=\operatorname{im}r\oplus\mathbb C\,M_{3\omega_1^\vee}$ ($M$ the $W_X$-orbit sum), and $S^\times=\mathbb C^\times$, so $S$ needs exactly two generators over $\mathbb C[A^*]^W$. With VI, for $\mathcal M_\theta$, $p$ odd: no single function generates $C_K$, $$C_K=\mathcal H_K\ast\varphi_0\oplus\mathbb C\,\mathbf 1_{O_{3\omega_1^\vee}},$$ and $\mathcal H_K$ acts on the quotient by $h\mapsto\hat h(q^{\rho-\omega_4})$.
A single generator $s$ of $S$ would satisfy $1=r(a)s$, so $s$ would be a constant and $r$ onto. Uses VI.
Statement, proof and proof graph →
VIII $E_6/F_4$: sheaves generate, functions do not
- Proved sheaves generate over $\overline{\mathbb F}_q$, $p\ne2$, given D0–D2.
- Proved functions fail over $\mathbb F_q(\!(t)\!)$ and $p$-adic $F$, $p$ odd (VI, VII).
Role in the verdict. The answer for $E_6/F_4$, the one row of Table 1 where the levels differ.
For $k$ odd let $\mu_k=3k\omega_1^\vee$ and $\lambda_k=\tfrac{3k-1}2\omega_1^\vee+\omega_6^\vee$. Over $\overline{\mathbb F}_q$, $p\ne2$, given D0–D2, $\mathrm{IC}_{\mu_k}$ is a direct summand of $V_{\lambda_k}\ast\mathrm{IC}_0$, and likewise for $3k\omega_6^\vee$; every other label has a dominant lift, so by I $$\mathcal C_K^{\mathrm{Sat}}=\mathcal C_K .$$ For $\mathcal M_\theta$ over $\mathbb F_q(\!(t)\!)$ or a $p$-adic $F$, $p$ odd: $q^{\rho_L}y_1$ and $q^{\rho_L}\sigma y_1$ are $W(E_6)$-conjugate (both to $q^{\rho-\omega_4}$) while $y_1,\sigma y_1$ are not $W_X$-conjugate, so $r$ is not onto and (SG) fails.
The labels $3k\omega_1^\vee$, $k$ odd, are exactly where dominant lifts fail and where $r$ misses $M_{3\omega_1^\vee}$. Uses I, II, VI, VII.
Statement, proof and proof graph →

3 · What remains open

D0 in characteristic $p$. Every sheaf result over $\overline{\mathbb F}_q$ uses D0, and Chen–Yi prove its orbit parametrization (their Prop. 17(i)) by citing Gaitsgory–Nadler and Nadler, which work over $\mathbb C$. The function results do not use it: for $E_6/F_4$, $K\backslash X(\mathbb F_q(\!(t)\!))\simeq\Lambda_S^+$ is proved directly (VI).
Over $\mathbb C$. The sheaf results over $\overline{\mathbb F}_q$ reach $\mathbb C$, the field of the problem, only through D5.
Residue characteristic 2. Excluded at both levels; for $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$ the pair model and the involution model differ there.

The verdict as written at the end of the run

1 · Answer, row by row

$$\text{sheaves:}\quad\mathcal C^{\mathrm{Sat}}=\operatorname{Thick}\{V_\lambda\ast\mathrm{IC}_0\}\overset?=\mathcal C,\qquad\qquad\text{functions:}\quad\text{(SG)}\ \ \mathcal H_K\cdot\mathbf 1_{X(\mathcal O)}\overset?=C_c^\infty(X(F))^K .$$

Table 3. Historical verdict
$X$	sheaves	functions	same answer?	proved in
$(H\times H)/H$	equal (Prop. II)	holds, every $F$	yes	pages
$\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}$, $n\ge2$	equal (Prop. III)	holds, every $F$ (new for $n\ge3$)	yes	Theorem 5.3, Proposition 1.3
$\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$	equal (Prop. III)	holds, every $F$, in the pair model, which is the pages' $J$-model for $q$ odd (new; was under (A))	yes	Theorem 5.4
$\mathrm{GL}_2/\mathrm{Sp}_2$, $\mathrm{GL}_2/T$	equal	holds	yes	pages
$\mathrm{GL}_4/\mathrm{Sp}_4$	equal over $\overline{\mathbb F}_q$ (new; H5 fails)	holds, every $F$ (new)	yes	Proposition 3.3, Theorem 3.9
$\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}$ ($m\ge3$), $\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}$ ($n\ge3$)	equal over $\overline{\mathbb F}_q$ (new; H5 fails)	holds, every $F$ (new)	yes	Theorem 5.7, Audit 7.23
$E_6/F_4$	equal over $\overline{\mathbb F}_q$ (new; H5 fails at $3k\omega_1^\vee$, $3k\omega_6^\vee$)	fails under (A), $p$-adic $F$ (new)	no: categorical holds, functions fail	Theorem 6.6, Lemma 8.4, Theorem 8.8, Summary 8.10
$\mathrm{SL}_2/\mathrm{SO}_2$, $G=\mathrm{SL}_2$	not equal (new, over $\mathbb C$)	fails, every $q$ (new)	yes	Theorem 2.4, Computation 2.8
$\mathrm{GL}_n/\mathrm O_n$, $\mathrm{PGL}_2/\mathrm{PO}_2$	not equal	fails	yes	pages

"Every $F$" means every non-archimedean local field, of either characteristic, with no use of Sakellaridis's hypotheses (A1)–(A5). Apart from the group case, proved over $\mathbb C$ directly by Proposition II, every "equal" sheaf cell rests on $\ell$-adic inputs over $\overline{\mathbb F}_q$, $p\ne2$, and reaches the Betti category of the pages through D5; for the rows decided by Proposition III this is the pages' own open item 1. The cells marked "over $\overline{\mathbb F}_q$" are new: the symplectic rows are proved under D0–D3 of §6 ($\mathrm{GL}_4/\mathrm{Sp}_4$ also directly under D0–D2), and $E_6/F_4$ under D0–D2. The "not equal" cells are proved in the Betti category directly.

Reverse direction excluded. By Theorem 3, "functions hold, sheaves fail" cannot happen over $\mathbb F_q(\!(t)\!)$, $q$ odd, for any split symmetric space with $G^\theta$ and $Z_{G^\theta}(S)$ connected, under D0–D3. For every $F$ the table excludes it row by row: wherever (SG) holds, $\mathcal C^{\mathrm{Sat}}=\mathcal C$ holds too.

2 · The one divergence: E6/F4

Functions fail (under (A)). With $u=y(\varphi\alpha_1^\vee)$, $v=y(\varphi\alpha_6^\vee)$ on $A_X^*$, $\delta(X)^{1/2}=q^{\rho_L}$ and $W_X=S_3$: $$y_1=(q^{-5},q),\quad y_2=(q^{-5},q^4)\ \text{ not }S_3\text{-conjugate};\qquad q^{\rho_L}y_1\leftrightarrow\rho-9\omega_1-3\omega_6\ \sim_{W(E_6)}\ \rho-\omega_4\ \sim_{W(E_6)}\ \rho-9\omega_1\leftrightarrow q^{\rho_L}y_2 .$$ Every element of $\operatorname{im}r$ takes equal values at $y_1,y_2$, but the orbit sum of $3\omega_1^\vee$ does not: $$M_{3\omega_1^\vee}(y_1)=q^{-9}+q^{3}+q^{6}\ \ne\ q^{-6}+q^{-3}+q^{9}=M_{3\omega_1^\vee}(y_2)\quad\Longrightarrow\quad r\ \text{not onto}\ \Longrightarrow\ \text{(SG) fails, every }q .$$ The missing label $3\omega_1^\vee$ is the first label without a dominant lift, the same label at which H5 fails. Within the family $H_3(K_d)$ with restricted root system $A_2$ of multiplicity $d$, the map $r$ is onto for $d=4$ ($\mathrm{SL}_6/\mathrm{Sp}_6$, Prop. F and Theorem 4) and not onto for $d=8$. Computation 6.4, Theorem 6.6; checked independently in Audit 7.9 and Audit 9.3.

Sheaves hold. For $k$ odd put $\mu_k=3k\omega_1^\vee$, $\mu_k'=\mu_k+\varphi(\alpha_6^\vee)$ and $\lambda_k=\tfrac{3k-1}{2}\omega_1^\vee+\omega_6^\vee$. Then $$\varphi(\lambda_k)=\mu_k',\qquad\{\nu:\mu_k\le\nu\le\mu_k'\}=\{\mu_k,\mu_k'\},\qquad\operatorname{codim}=\langle\rho,\mu_k'-\mu_k\rangle=8,\qquad\dim\operatorname{Gr}^{\lambda_k}=\delta(\mu_k') .$$ Both convolution fibres lie in the open cell $\operatorname{Gr}^{\lambda_k}$: $\mu_k$ has no dominant lift, and no dominant $\lambda''<\lambda_k$ has $\varphi(\lambda'')=\mu_k'$. So the stalks of $V_{\lambda_k}\ast\mathrm{IC}_0$ are fibre cohomology shifted by $\delta(\mu_k')$, and the singularities of $\operatorname{Gr}^{\le\lambda_k}$ never enter: $$F_k'=m^{-1}(x_{\mu_k'})=\{t^{\lambda_k}\},\qquad F_k=m^{-1}(x_{\mu_k})\supseteq\mathrm{Spin}_9\cdot t^{\nu}L^+G\ \text{of dimension}\ \ge 7 .$$ The first fibre is a point because the black torus fixes $x_{\mu_k'}$, no root of $E_6$ vanishes on it, and the fibre has a single torus-fixed point (Objection 7.28). The second contains a 7-dimensional orbit of $Z_{G^\theta}(\mu_k)^\circ\simeq\mathrm{Spin}_9$ through each of its 8 torus-fixed points. The top class of $F_k$ sits in a degree no IC of the larger orbit reaches, because $2\dim F_k-8\ge 6\ge 0$: $$\mathrm{IC}_{3k\omega_1^\vee}\ \text{is a direct summand of}\ V_{\lambda_k}\ast\mathrm{IC}_0,\qquad E_{\mu_k,\,2\dim F_k-8}\ne 0 .$$ The mirror labels $3k\omega_6^\vee$ follow from the same computation with nodes 1 and 6 exchanged, covered labels by Proposition III, and Proposition 7 gives $\mathcal C^{\mathrm{Sat}}=\mathcal C$. Inputs: the decomposition theorem and finite-type model already used by Proposition III, over $\overline{\mathbb F}_q$; Betti via D5. Lemma 8.2 to Summary 8.10; audit Audit 7.26 to Summary 7.32. Heuristic refinement: $F_k\cong Q^7$ and the multiplicity space is $k^7$ in degrees $-6,\dots,6$, whose trace is $c_8(q)-1$ from Computation 6.5 (Conjecture 8.11).

Why the levels can split here and nowhere else. Theorem 3 says (SG) over $\mathbb F_q(\!(t)\!)$ forces every $\mathrm{IC}_\mu$ to be a summand of a Satake object, so every row of the multiplicity matrix $\big(m_{\nu\lambda}(q^{1/2})\big)$ (rows indexed by orbit labels $\nu$, columns by Satake labels $\lambda$) is nonzero. The converse does not follow from the dictionary: (SG) needs the columns, evaluated at the given $q$, to span, and a nonzero row only detects $\mathrm{IC}_\mu$. On the $p$-adic side, under (A1)–(A5), a Weyl-group coincidence of points of $\delta(X)^{1/2}A_X^*$ kills the span of the Hecke translates of $\varphi_0$ without removing any orbit label: at $E_6/F_4$ the glued points move with $q$; in Sakellaridis's Example 8.0.1, where $\delta(X)$ is trivial, they do not. Whether the columns of the dictionary matrix span for $E_6/F_4$ over $\mathbb F_q(\!(t)\!)$ is open (§6). Proposition 4.7.

3 · Theorems proved in this run

Theorem 1 ($\mathrm{SL}_2/\mathrm{SO}_2$: both levels fail). Let $G=\mathrm{SL}_2$, $X=\mathrm{SL}_2/\mathrm{SO}_2$, $k=\mathbb C$. The central $-I\in L^+G$ acts trivially on $LX$ and on $\operatorname{Gr}$ and defines a natural involution $\varepsilon$ of $\mathrm{id}_{\mathcal C}$: $$\mathcal C=\mathcal C^+\oplus\mathcal C^-,\qquad\mathcal C^{\mathrm{Sat}}=\operatorname{Thick}\{\mathrm{IC}(\overline{\mathcal O_{2n}})\}\subsetneq\mathcal C^+\simeq\operatorname{Shv}_{\mathrm{cons}}(L^+\mathrm{PGL}_2\backslash LX)\subsetneq\mathcal C,\qquad 0\ne a_{m!}\mathcal L^-_m\in\mathcal C^-.$$ Here $\mathcal L^-_m$ is the sign local system of $\pi_0\operatorname{Stab}(x_m)=\{\pm I\}$, $m\ge1$, and $V_{n\alpha^\vee}\ast\mathrm{IC}_0\simeq\mathrm{IC}(\overline{\mathcal O_{2n}})$ because the convolution map is finite and birational. At function level, for $X=\mathrm{SL}_2/T$ over every $F$, $\mathcal H_K\varphi_0=\mathbb C[T_1^2]\varphi_0\not\ni\mathbf 1_{\{d=1\}}$. For $q$ odd the square classes give a second obstruction: the Frobenius trace of $\varepsilon$. Steps: Computation 2.1, Lemma 2.2, Lemma 2.3, Theorem 2.4, Lemma 2.6, Proposition 2.7, Computation 2.8, Proposition 1.2. Two further independent proofs of the sheaf failure: Theorem 4.11, Proposition 1.7. Audit: Audit 7.1, Audit 7.7.

Theorem 2 ($\mathrm{GL}_4/\mathrm{Sp}_4$: both hold, although H5 fails). For $\lambda=(1,1,0,0)$, the convolution map is an isomorphism over $\mathcal O_{(2,2,0,0)}$ and has fibre $LG(2,4)\cong Q^3$ over the closed orbit $\mathcal O_{(1,1,1,1)}$ of codimension $4$. The decomposition theorem, the support condition and Verdier duality give $$V_{(1,1,0,0)}\ast\mathrm{IC}_0\simeq\mathrm{IC}_{(2,2,0,0)}\ \oplus\ \mathrm{IC}_{(1,1,1,1)}\otimes\big(k[2]\oplus k\oplus k[-2]\big),$$ and the scalar shift reaches every uncovered $(a,a,a,a)$. At function level, over every $F$, $$T_{(1,1,0,0)}\ast\varphi_0=\mathbf 1_{\mathcal O_{(2,2,0,0)}}+(1+q)(1+q^2)\,\mathbf 1_{\mathcal O_{(1,1,1,1)}},\qquad T_{(2,0,0,0)}\ast\varphi_0=q(q+1)\,\mathbf 1_{\mathcal O_{(2,2,0,0)}} .$$ Computation 3.2, Proposition 3.3, Theorem 3.6, Theorem 3.9; same counts in Theorem 4.9, Proposition 1.4, Computation 5.5; recomputed for $q=2,3$ in Audit 7.6.

Theorem 3 (functions $\Rightarrow$ sheaves over $\mathbb F_q(\!(t)\!)$). Let $X=G/G^\theta$ be defined over $\mathbb F_q$, $q$ odd, $T$ split, with $G^\theta$ and $Z_{G^\theta}(S)$ connected. Under D0–D3, $$\mathcal H_K\cdot\varphi_0=C_K\ \text{ over }\mathbb F_q(\!(t)\!)\quad\Longrightarrow\quad\mathcal C^{\mathrm{Sat}}_{\overline{\mathbb F}_q}=\mathcal C_{\overline{\mathbb F}_q}.$$

Lang's theorem gives $K\backslash X(F)=\Lambda_S^+$.
$\operatorname{tr}(V_\lambda\ast\mathrm{IC}_0)=\operatorname{tr}(V_\lambda)\ast_K\varphi_0$, so $\mathcal H_K\varphi_0$ is spanned by the traces of Satake objects.
Each Satake object is $\bigoplus\mathrm{IC}_\nu[-i]\otimes E_{\nu,i}$; its trace has coefficient $\sum_i(-1)^i\operatorname{tr}(\mathrm{Fr}\mid E_{\mu,i})$ on $\operatorname{tr}\mathrm{IC}_\mu$, in a unitriangular basis.
(SG) makes some coefficient at $\mu$ nonzero, so $E_{\mu,\bullet}\ne0$ and $\mathrm{IC}_\mu$ is a summand.
Proposition 7 concludes.

Three independent write-ups: Theorem 4.6, Proposition 3.10, Theorem 1.6; audit Audit 7.14.

Theorem 4 (function level, no (A), every $F$). (SG) holds for $\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}$ ($n\ge2$), $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$, $\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}$ and $\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}$, over every non-archimedean local field. Spheres: $K$-orbits are the pole-order shells, by Hensel and smoothness of even-rank orthogonal groups; Corollary C applies. $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$: a lattice normal form and a componentwise support bound. Symplectic families: for $b=(b_1\ge\dots\ge b_m\ge0)$, put $\lambda^*(b)=(b_1,\dots,b_m,0,\dots,0)$ and $N(\lambda,b')=(T_\lambda\ast\varphi_0)(x_{b'})$. Then $$N(\lambda^*(b),b)\ge1,\qquad N(\lambda^*(b),b')\ne0\ \Longrightarrow\ b'\ge_{\mathrm{dom}}b,$$ the second from $0\to\Lambda''/\Lambda_0\to\Lambda_0^\#/\Lambda_0\to\operatorname{Hom}(\Lambda''/\Lambda_0,F/\mathcal O)\to0$ and the characterization of partial sums of module types by $k$-generated submodules. Descending induction on dominance inside the finite sets $\{b:\ b_m\ge2c,\ \textstyle\sum b_i=d\}$, with central translates, gives every orbit indicator. Lemma 5.2 to Theorem 5.7; audit Audit 7.20 to Audit 7.23. For an arbitrary split symmetric space satisfying Proposition III's two conditions, (SG) over $\mathbb F_q(\!(t)\!)$, $q$ odd, follows under D0 and D3 from Lang, the support bound and Corollary C: Lemma 4.8.

Corollary 5. Theorems 3 and 4 give $\mathcal C^{\mathrm{Sat}}=\mathcal C$ over $\overline{\mathbb F}_q$ for $\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}$ and $\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}$ for every $m,n$, although H5 fails for $m\ge2$, $n\ge3$. Audit 7.23

Theorem 6 (Sakellaridis's restriction map). With $\delta(X)^{1/2}=q^{\rho_L}$, calibrated on Proposition F: for the spheres $r(m_{\varepsilon_1+\varepsilon_2})=c_n(q)(u+u^{-1})+\mathrm{const}$ with $c_n(q)>0$, so $r$ is onto; for $E_6/F_4$, $r$ is not onto (§2). Setup 6.1, Proposition 6.3, Theorem 6.6.

Proposition 7 (summand criterion). Assume H1–H3 and that every Satake object is a finite sum of shifted $\mathrm{IC}_\nu$ with constant coefficients. Then $$\mathcal C^{\mathrm{Sat}}=\operatorname{Thick}\{\mathrm{IC}_\nu:\nu\in\mathcal M\},\qquad\mathcal M=\{\mu:\mathrm{IC}_\mu\text{ is a retract of some }V_\lambda\ast\mathrm{IC}_0\}\supseteq\varphi(\Lambda_T^+),\qquad\mathcal C^{\mathrm{Sat}}=\mathcal C\iff\mathcal M=\Lambda_S^+ .$$ H5 is sufficient, not necessary. A stratum of maximal defect $2\dim(\text{fibre})-\operatorname{codim}$ always contributes its IC as a summand. Proposition 3.4, Lemma 3.5, Lemma 4.5, Lemma 1.5.

4 · Three fields, two bridges

$$\mathcal C_{\mathbb C}\ \overset{\text{D5}}{\longleftrightarrow}\ \mathcal C_{\overline{\mathbb F}_q}\ \xrightarrow{\ \operatorname{tr}\ }\ C_K\big(\mathbb F_q(\!(t)\!)\big)\ \overset{?}{\longleftrightarrow}\ C_K(p\text{-adic }F)\ \overset{\text{Thm E under (A)}}{\longleftrightarrow}\ \operatorname{im}r .$$

The sheaf statement is defined over $\mathbb C(\!(t)\!)$; Theorem E needs $\operatorname{char}F=0$; the trace of Frobenius relates sheaves over $\overline{\mathbb F}_q$ to functions over $\mathbb F_q(\!(t)\!)$. Theorem 4 holds over every $F$, so the positive function cells sit on both sides of the "?" bridge. The $E_6/F_4$ function failure sits only on the $p$-adic side. Objection 1.8, Objection 4.12, Audit 7.16.

5 · Corrections from the audit

The remark in Proposition 1.2 that $\mathrm{IC}_0$ is a summand of $V_{(1,0)}\ast\mathrm{IC}_0$ is false; that object is simple. A1's conclusions do not use it. Objection 7.15
Lemma 3.1 cites Chen–Yi Prop. 64, which assumes $LX$ connected; the scalar shift reaches component 0 only from components whose Pfaffian valuation is a multiple of $m$, so the formula is instead proved in every component by the finite-type stabilizer count (Correction 10.2). Objection 7.11
Theorem 5.7 invokes Proposition B with reversed order, whose lower sets are infinite; the proof is a direct induction on the finite sets $S_{d,c}$. Audit 7.23
Stale cells: Summary 1.9 lists $E_6/F_4$ as open at function level, $\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)$ as open over $\mathbb F_q(\!(t)\!)$, and the $\mathrm{SL}_2/\mathrm{SO}_2$ sheaf failure as $\ell$-adic only; Dead end 5.9 says nothing suggests a failure at $E_6/F_4$, and Summary 5.10 says $\mathrm{SL}_2/T$ fails only for $q$ odd. Each was written before A6's result or A2's count reached that agent. Objection 7.17, Objection 7.24
Lemma 8.5 bounds the fibre over the covering orbit by a dimension count in a congruence quotient; that count is wrong as written, since in the group case every orbit has the same quotient dimension. The replacement is a torus-fixed-point argument showing the fibre is the single point $t^{\lambda_k}$. Objection 7.28
Lemma 8.9 transports the result by the diagram automorphism, which must first be normalized to preserve $G^\theta$; rerunning the computation with nodes 1 and 6 exchanged avoids this. Objection 7.31

6 · Obligations

D0 Chen–Yi's orbit geometry over $\overline{\mathbb F}_q$, $p\ne2$: Prop. 17, Thm 18, the support bound, the stabilizer lemma.
D1 proper base change and Grothendieck–Lefschetz on convolution fibres; the Frobenius-compatible perverse t-structure.
D2 decomposition of $V_\lambda\ast\mathrm{IC}_0$ into shifted $\mathrm{IC}_\nu$ on finite-type equivariant models (Sun, Thm 1.2; Chen–Yi Prop. 70).
D3 Lang's theorem for $L^+G$ and the stabilizers.
D5 comparison $\overline{\mathbb F}_q\leftrightarrow\mathbb C$ of "$\mathrm{IC}_\mu$ is a summand of $V_\lambda\ast\mathrm{IC}_0$" (BBD §6.1, large $p$); the pages' open item 1. Theorem 1 does not need it.
(A) Sakellaridis's (A1)–(A5), assumed only in the $E_6/F_4$ function cell. (A5) is verified there (Lemma 6.9); (A2), (A3) at the given place and (A4) beyond "$L_\gamma$ classical" are not.
$E_6/F_4$ on one field. The divergence compares sheaves over $\overline{\mathbb F}_q$ (Betti via D5) with functions over a $p$-adic field under (A). Putting both on $\mathbb F_q(\!(t)\!)$ needs Sakellaridis's Theorem 8.0.2 in characteristic $p$, or the $E_6(\mathcal O)$-orbits on the norm-one Albert hypersurface with a Hecke count showing the dependence at $3\omega_1^\vee$.

7 · Routes ruled out

A weaker H5 (support in the closure and nonzero restriction) at an uncovered label: no Satake object satisfies it, and nonvanishing alone does not generate the standard object (the cone of $u$ on $B\mathbb G_m$). Proposition 3.4
Sheaves $\Rightarrow$ functions in general: inside the dictionary over $\mathbb F_q(\!(t)\!)$ it needs the columns of the multiplicity matrix to span at the given $q$, which nonzero rows do not provide; on the $p$-adic side, under (A1)–(A5), the corresponding condition is the surjectivity of $r$, and $E_6/F_4$ shows that it can fail while $\mathcal C^{\mathrm{Sat}}=\mathcal C$. Proposition 4.7
Feeding "(SG) under (A)" into Theorem 3: Theorem E is $p$-adic only. Objection 4.12
Proposition B with reverse dominance on all symplectic labels at once: lower sets are infinite. Audit 7.23
$E_6/F_4$ at function level by lattice counts: needs the $E_6(\mathcal O)$-orbits on the norm-one Albert hypersurface, and A6 predicts the triangular system fails at $3\omega_1^\vee$. Dead end 5.9, Objection 7.24

Problem

Question. Among the symmetric spaces $X=G/G^\theta$ of the spherical-generation pages (Table 1), is there one where sheaves generate but functions do not, or the reverse?

$$\text{sheaves:}\quad \mathcal C_K^{\mathrm{Sat}}:=\operatorname{Thick}_{\mathcal C_K}\{V_\lambda\ast\mathrm{IC}_0:\lambda\in\Lambda_T^+\}\overset{?}{=}\mathcal C_K:=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash LX),$$ $$\text{functions:}\quad \text{(SG)}\qquad \mathcal H_K\ast\varphi_0\overset{?}{=}C_K:=C_c^\infty(X(F))^K,\qquad \varphi_0=\mathbf 1_{X(\mathcal O)}.$$

A divergence is a variety $X$ for which exactly one of the two equalities holds.

Setting and conventions

$k=\mathbb C$; $G$ connected reductive, $\theta$ an involution with $G^\theta$ connected, $X=G/G^\theta$, $x_0=eG^\theta$; $S$ a maximal $\theta$-split torus, $T\supseteq S$ a $\theta$-stable maximal torus; $\Lambda_T=X_*(T)$, $\Lambda_S=X_*(S)$, dominant cones $\Lambda_T^+$, $\Lambda_S^+=\Lambda_S\cap\Lambda_T^+$.
$LX=X(k(\!(t)\!))$, $L^+G=G(k[\![t]\!])$; orbits $LX=\bigsqcup_{\mu\in\Lambda_S^+}\mathcal O_\mu$, $\mathcal O_\mu=L^+G\cdot x_\mu$, $\tau(x_\mu)=t^\mu$ for the Cartan embedding $\tau(gG^\theta)=g\theta(g)^{-1}$ (Chen–Yi, Prop. 17); leading-term map $\varphi(\lambda)=\lambda-\theta(\lambda)$, $t^\lambda x_0\in\mathcal O_{\varphi(\lambda)}$; a dominant lift of $\mu\in\Lambda_S^+$ is a $\lambda\in\Lambda_T^+$ with $\varphi(\lambda)=\mu$.
$\mathcal C_K$: bounded $L^+G$-equivariant constructible complexes on $LX$, each supported on a finite union of orbit closures; $\mathrm{IC}_0$ the constant sheaf on $\mathcal O_0=L^+X$; Hecke category $\mathcal H=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash LG/L^+G)$ acting by $\mathcal A\ast\mathcal F=m_!(\mathcal A\,\widetilde\boxtimes\,\mathcal F)$; $V_\lambda$ the Satake sheaf of $\overline{\operatorname{Gr}^\lambda}$; the Satake objects are the $V_\lambda\ast\mathrm{IC}_0$, and $\mathrm{IC}_\mu$ is the intersection complex of $\overline{\mathcal O_\mu}$ with constant coefficients. Thick = closed under isomorphisms, shifts, cofibers and retracts.
Function level: $F$ a non-archimedean local field with residue field $\mathbb F_q$, $K=G(\mathcal O)$, $\mathcal H_K=C_c(K\backslash G(F)/K)$, $T_\lambda=\mathbf 1_{Kt^\lambda K}$, $\operatorname{vol}K=1$; $\mathbf 1_{O_\mu}$ is the indicator of the orbit $O_\mu=K\cdot x_\mu\subseteq X(F)$.

Sources

References

symmetric.html · Proposition II, the group case; all definitions
summary.html · Table 1: every variety at both levels
criterion.html · Proposition III
generation.html · Proposition I, H1–H5
classical.html · the function level: Lemma A … Proposition J
families/ · explicit families (symplectic, split forms, E6/F4, spin, GL(p,q), group)
Chen–Yi, arXiv:2310.20006v4 · orbits, dimension theory, Example 88, Lemma 90
Sakellaridis, Spherical functions on spherical varieties · Theorem 8.0.2, Corollary 8.0.4, Example 8.0.1

Attempt tree

One row per agent, time running left to right; coloured bands mark the phases of the search. The whole tree is in view; zoom (− / +, or a pinch) stretches time only, so rows and agents stay put. Drag to pan; ⤢ opens full screen. Hover a step for its name; click it to open it.

Animate Speed

Style Rows Time zoom100%

516 M tokens ≈ $373 during the run, root session and every agent. This conversation, start to 05:19 UTC: 802 M tokens ≈ $490.

rowstime →

proved
refuted
blocked
open
heuristic
superseded
error
relies on an error
correction
final verdict
builds on
across rows
audit
error carried forward
corrected path
★ key step

\(X\)	sheaves	functions	same?	where
\((H\times H)/H\), \(\mathrm{GL}_2/\mathrm{Sp}_2\), \(\mathrm{GL}_2/T\)	generate	generate	yes	pages
\(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), \(n\ge2\)	generate	generate, every \(F\)	yes	pages, V
\(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\)	generate	generate, every \(F\), pair model	yes	pages, V
\(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\)	generate over \(\overline{\mathbb F}_q\), given D0–D3	generate, every \(F\)	yes	IV
\(\mathrm{SL}_2/\mathrm{SO}_2\)	fail	fail (split form \(\mathrm{SL}_2/T\), every \(F\))	yes	III
\(\mathrm{GL}_n/\mathrm O_n\), \(\mathrm{PGL}_2/\mathrm{PO}_2\)	fail	fail	yes	pages
\(E_6/F_4\)	generate over \(\overline{\mathbb F}_q\), \(p\ne2\), given D0–D2	fail over \(\mathbb F_q(\!(t)\!)\) and \(p\)-adic \(F\), \(p\) odd	no	VI–VIII

\(X\)	sheaves	functions	same answer?	proved in
\((H\times H)/H\)	equal (Prop. II)	holds, every \(F\)	yes	pages
\(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), \(n\ge2\)	equal (Prop. III)	holds, every \(F\) (new for \(n\ge3\))	yes	Theorem 5.3, Proposition 1.3
\(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\)	equal (Prop. III)	holds, every \(F\), in the pair model, which is the pages' \(J\)-model for \(q\) odd (new; was under (A))	yes	Theorem 5.4
\(\mathrm{GL}_2/\mathrm{Sp}_2\), \(\mathrm{GL}_2/T\)	equal	holds	yes	pages
\(\mathrm{GL}_4/\mathrm{Sp}_4\)	equal over \(\overline{\mathbb F}_q\) (new; H5 fails)	holds, every \(F\) (new)	yes	Proposition 3.3, Theorem 3.9
\(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) (\(m\ge3\)), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) (\(n\ge3\))	equal over \(\overline{\mathbb F}_q\) (new; H5 fails)	holds, every \(F\) (new)	yes	Theorem 5.7, Audit 7.23
\(E_6/F_4\)	equal over \(\overline{\mathbb F}_q\) (new; H5 fails at \(3k\omega_1^\vee\), \(3k\omega_6^\vee\))	fails under (A), \(p\)-adic \(F\) (new)	no: categorical holds, functions fail	Theorem 6.6, Lemma 8.4, Theorem 8.8, Summary 8.10
\(\mathrm{SL}_2/\mathrm{SO}_2\), \(G=\mathrm{SL}_2\)	not equal (new, over \(\mathbb C\))	fails, every \(q\) (new)	yes	Theorem 2.4, Computation 2.8
\(\mathrm{GL}_n/\mathrm O_n\), \(\mathrm{PGL}_2/\mathrm{PO}_2\)	not equal	fails	yes	pages

Table 1. Generation comparison
	tokens	US $
This conversation: the search, the audit, the pages and propositions, and the VI characteristic-p run01:44–05:19 UTC	802 M	$490
Setup pages16:17–19:42 UTC	37 M	$19
Shareable animation17:16–17:22 UTC	1 M	$1
Proposition IIb: figures and the linear-algebra lemma19:48–20:05 UTC	14 M	$7
Audit pass: the repairs by Agents X1–X4, their re-checks and Propositions VI–VII (with the audit skill)19:35–02:12 UTC	275 M	$113
Proposition IX: converse search, verdict and trace formalism19:56–02:12 UTC	199 M	$106
Audit pass on the VI characteristic-p run; the simplified VIb page and the VI corrections05:21–07:33 UTC	268 M	$107
Removing the detail switch; Proposition IX restated; the hub and verdict reviewed and simplified05:34–15:56 UTC	38 M	$20
C_K for the categorical statements, in the Iwahori run's conversation18:27–18:32 UTC	9 M	$3
Total	1,642 M	$866