1 · The two statements
$$\text{sheaves:}\qquad\mathcal C^{\mathrm{Sat}}:=\operatorname{Thick}_{\mathcal C}\{V_\lambda\ast\mathrm{IC}_0:\lambda\in\Lambda_T^+\}\overset{?}{=}\mathcal C:=\operatorname{Shv}_{\mathrm{cons}}(L^+G\backslash LX),\qquad\qquad\text{functions:}\qquad\text{(SG)}\quad\mathcal H_K\cdot\varphi_0\overset{?}{=}C_K:=\mathcal S(X)^K .$$Sheaves. Proposition I proves the equality under H1–H5. Proposition III reduces H1–H5, for \(X=G/G^\theta\) with \(G^\theta\) connected, to two conditions on the root datum: \(Z_{G^\theta}(S)\) connected, and the leading-term map \(\varphi:\Lambda_T^+\to\Lambda_S^+\) onto. When a condition fails, H1–H5 fail and the equality is undecided, except where a support argument decides it. Proposition IV treats \(\mathbb A^n\) one level at a time.
Functions. On the classical page: Proposition B is the triangular criterion, the function-level form of Proposition I; Theorem E is Sakellaridis's criterion, (SG) iff the restriction map \(r:\mathbb C[A^*]^W\to\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\) is onto, under his hypotheses (A1)–(A5), listed there and never verified here (“under (A)” below); Proposition F computes \(r\) for the two \(\mathrm{GL}\) families. No implication between the two levels is proved; they are shadows of each other through the dictionary.
2 · Every variety, both levels
| \(X\) | sheaves: H1–H5 → \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) | functions: (SG) | why |
|---|---|---|---|
| \((H\times H)/H\) | hold → equal (II, III) | holds (D) | \(Z_{G^\theta}(S)=T_H\) connected, \((a,-a)=\varphi(a,0)\); \(\varphi_0\) is the unit of \(\mathcal H_{K_H}\). |
| \(\mathrm{Spin}_{2n}/\mathrm{Spin}_{2n-1}\), \(n\ge2\) | hold → equal (III) | \(n=2\) holds; \(n=3\) under (A); \(n\ge4\) open | \(Z=\mathrm{Spin}_{2n-2}\), \(2m\varepsilon_1=\varphi(m\varepsilon_1+m\varepsilon_2)\). \(n=2\) is the group \(\mathrm{SL}_2\), \(n=3\) is \(\mathrm{SL}_4/\mathrm{Sp}_4\) (F); \(r\) not computed for \(n\ge4\), and the spheres are not in Sakellaridis's table. |
| \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\), \(p\le q\) | hold → equal (III) | holds under (A) (F); \(p=q=1\) unconditionally (J) | \(Z=\mathbb G_m^p\times\mathrm{GL}_{q-p}\), lift \((a_1,\dots,a_p,0,\dots,0)\). \(r(p_k)=\sum_i(y_i^k+y_i^{-k})+c_k\) generate the \(W(C_p)\)-invariants; for \(\mathrm{GL}_2/T\) the adjacency operator of the tree. |
| \(\mathrm{GL}_2/\mathrm{Sp}_2\simeq\mathbb G_m\) | hold → equal (III, H1 tab) | holds (J) | Rank one through \(\det\): \(K\backslash X(F)=F^\times/\mathcal O^\times\). |
| \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(m\ge2\) | H5 fails → unknown (III) | holds under (A) (F) | \(Z=\mathrm{SL}_2^m\) connected, but \((1,1,1,1,0,\dots,0)\) has no dominant lift. \(r(p_{\pm k})=(q^{k/2}+q^{-k/2})p_{\pm k}(y)\) generate the \(S_m\)-invariants. The function level points to equality beyond H5. |
| \(\mathrm{SL}_4/\mathrm{Sp}_4\cong\mathrm{Spin}_6/\mathrm{Spin}_5\) | hold → equal (III) | holds under (A) for \(\mathrm{GL}_4/\mathrm{Sp}_4\) (F) | Every label lifts in rank \(2\); the \(\mathrm{SL}\) case is the degree-\(0\) part of the \(\mathrm{GL}\) case. |
| \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\), \(n\ge3\) | H5 fails → unknown (III) | holds under (A) for \(\mathrm{GL}_{2n}/\mathrm{Sp}_{2n}\) (F) | \((1,1,1,1,0,\dots,0,-2,-2)\) has no dominant lift; Chen–Yi's semisimplicity criterion still applies. Functions: degree-\(0\) part of the \(\mathrm{GL}\) case. |
| \(E_6/F_4\) | H5 fails → unknown (III) | open | \(Z=\mathrm{Spin}_8\) connected; \(3\omega_1^\vee\) has no dominant lift. Not in Sakellaridis's table; \(r\) not computed. |
| \(\mathrm{SL}_2/\mathrm{SO}_2=\mathrm{SL}_2/T\) | both conditions fail → undecided here (III) | fails, \(q\) odd (J) | \(Z=\{\pm1\}\) disconnected and \(\alpha^\vee\) has no lift (\(\varphi\) doubles). Functions: the square classes \(\mathcal O^\times/\mathcal O^{\times2}\) split the \(K\)-orbits and the Hecke translates of \(\varphi_0\) cannot see them; (A5) fails since the generic \(B\)-stabilizer is \(\{\pm1\}\). Outside the paper's class. |
| \(\mathrm{GL}_n/\mathrm O_n\), \(\mathrm{PGL}_2/\mathrm{PO}_2\) | outside III → not equal (split forms) | fails | \(G^\theta\) disconnected. Supports: every Satake object lives on \(LG\cdot x_0\), a proper union of orbits (the paper's remark); at function level \(X(F)\) has several \(G(F)\)-orbits (classes of quadratic forms) and the translates of \(\varphi_0\) stay in \(G(F)X(\mathcal O)\). |
| \(\mathbb A^n\) under \(\mathrm{GL}_n\) | H1 fails; level form → equal (IV) | holds (G) | Infinitely many orbits in each closure; the central Satake objects \(\underline k_{t^m\mathcal O^n}\) suffice, as do the central translates \(\mathbf 1_{\varpi^m\mathcal O^n}\). |
| \(M_n\) under \(\mathrm{GL}_n\times\mathrm{GL}_n\) | not on these pages | \(n=2\) holds; \(n\ge3\) open here (H) | Nonsingular orbit indicators are translates because the Satake transform of \(\varphi_0\) is the standard \(L\)-factor; near the singular locus only \(n=2\) is settled. |
| \(\mathrm{SL}_2^3\backslash\mathrm{Sp}_4^2\) (Sakellaridis) | not symmetric; not on these pages | fails (E) | \(W_X\)-nonconjugate points of \(A_X^*\) are \(W\)-conjugate, so \(r\) is not onto. |
Reading across: wherever Proposition III's two conditions hold, the function level agrees (unconditionally for the group, \(\mathbb G_m\) and \(\mathrm{GL}_2/T\); under (A) otherwise). Where H5 fails by coverage (\(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\)), the function level still holds under (A), so the sheaf statement is open rather than false. Where the orbit set has a structure the Hecke translates cannot reach (square classes, several \(G(F)\)-orbits, Weyl conjugacy), both levels fail or are expected to.
3 · Cited theorems and their hypotheses
| result | used for | hypotheses there | here |
|---|---|---|---|
| Chen–Yi, arXiv:2310.20006v4: Proposition 17 (orbits \(\leftrightarrow\Lambda_S^+\), representatives \(x_\mu\)), Proposition 41 (dimension theory), Theorem 87 (semisimplicity criterion) | Orbit geometry of \(LX\) in II and III; H1, H2, H3 | \(X=G/G^\theta\) symmetric with \(G^\theta\) connected, over \(k=\mathbb C\); their sheaf theory on placid ind-schemes | \(G^\theta\) connected is verified on each family page; \(\mathrm{GL}_n/\mathrm O_n\) and \(\mathrm{PGL}_2/\mathrm{PO}_2\) are excluded. That their sheaf theory realizes \(\mathcal C\) is assumed (definitions). |
| Sakellaridis, Spherical functions on spherical varieties (2013), Theorem 8.0.2, Corollary 8.0.4, Example 8.0.1 | (SG) at function level \(\iff r\) onto (Theorem E); Proposition F | (A1)–(A5), listed on the classical page: \(\operatorname{char}F=0\), \(G\) split with simply connected derived group, his Axioms 2.2.1, 2.4.1, 2.4.2, no type-\(N\) spherical roots, Statements 6.3.1 and 7.1.5, \(X\) affine homogeneous with a single open \(B(F)\)-orbit | Assumed, never verified. He asserts the hypotheses of his Theorem 7.2.1 for the rows of his table (the \(\mathrm{GL}\) families and the group); (A3) he proves only at almost all places; (A5) follows from the paper's connected generic stabilizers (classical, 4) and fails for \(\mathrm{SL}_2/\mathrm{SO}_2\). |
| Sakellaridis–Venkatesh, Periods and harmonic analysis on spherical varieties, Astérisque 396 (2017), §2.1 | The invariants \(A_X\), \(\Lambda_X^+\), \(W_X\), the spherical roots, in Theorem E and Proposition F | \(X\) a spherical variety over an algebraically closed field of characteristic \(0\); nothing more for the definitions | Verified: every \(X\) here is spherical over \(\mathbb C\). Their theorems on the unramified spectrum (wavefront and further hypotheses) are not used. |
| Ben-Zvi–Sakellaridis–Venkatesh, Relative Langlands duality (2024), Conjecture 7.5.1 | Only as the origin of the question: its automorphic side is \(\mathcal C^{\mathrm{Sat}}\), and (SG) asks whether that is all of \(\mathcal C\) | \(T^*X\) polarized hyperspherical: \(X\) smooth affine spherical with connected generic \(B\)-stabilizers (§3.5.1, Proposition 3.7.4), characteristic \(0\) | Not used in any proof. The paper's standing Definition; \(\mathrm{SL}_2/\mathrm{SO}_2\) (generic stabilizer \(\{\pm1\}\)) and \(\mathrm{GL}_n/\mathrm O_n\) lie outside it. |
| Geometric Satake (Mirković–Vilonen; Zhu's notes) | The Satake sheaves \(V_\lambda\) and the Hecke action (definitions) | \(k=\mathbb C\), \(G\) connected reductive | Verified as standing setting. |
| Borel–Serre (1964); Kneser; Hilbert 90 | \(X(F)\) a finite union of open \(G(F)\)-orbits (Lemma A); \(X(F)=G(F)/H(F)\) in Propositions F and J | \(\operatorname{char}F=0\) for the finiteness; \(H=\mathrm{Sp}_{2n}\) or a split torus for the vanishing of \(H^1\) | Assumed where used (\(\operatorname{char}F=0\) is standing on the classical page). |
| Tamagawa (1963); Godement–Jacquet, LNM 260 | The Satake transform of \(\mathbf 1_{M_n(\mathcal O)}\) (Proposition H) | none beyond \(G=\mathrm{GL}_n\) over a local field | Recomputed in the proof of H. |
4 · Open
- Sheaves. \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) for \(\mathrm{GL}_{2m}/\mathrm{Sp}_{2m}\) (\(m\ge2\)), \(\mathrm{SL}_{2n}/\mathrm{Sp}_{2n}\) (\(n\ge3\)) and \(E_6/F_4\), where H5 fails by coverage: it would follow if the intersection complex of each uncovered orbit closure were a retract of some \(V_\lambda\ast\mathrm{IC}_0\). For \(\mathrm{SL}_2/\mathrm{SO}_2\) a support argument in the spirit of the square-class obstruction may decide it.
- Functions. (A1)–(A5) for any family, checked rather than assumed; \(r\) for the spheres with \(n\ge4\) and for \(E_6/F_4\); \(M_n\) for \(n\ge3\).
- Between the levels. Any implication between (SG) for sheaves and (SG) for functions; over \(F=\mathbb F_q(\!(t)\!)\) the function level is the trace of Frobenius of the sheaf level, and the \(\mathcal D\)-module setting has no Frobenius.