Is characteristic 0 needed?
The Theorem is stated for \(p\)-adic fields. The function-field case \(F=\Fqt\) is where the geometric (sheaf) version of the same question lives, so it matters whether the Theorem survives there. Short answer: no published proof covers \(\Fqt\); nothing known forces characteristic \(0\); and for symmetric varieties with good integral models, \(E_6/F_4\) included, this project proves the characteristic-\(p\) statement by transfer from ramified \(p\)-adic fields.
Two different characteristicsWhich “\(p\)” is the question about?
\(F\) always has a residue field \(\Fq\) of characteristic \(p\gt0\). The question is about the characteristic of \(F\) itself:
\[\operatorname{char}F=0:\ F\supseteq\Q_p\ \text{finite};\qquad\operatorname{char}F=p:\ F\simeq\Fqt .\]A separate condition, “\(p\) odd”, is about the residue field and is needed for symmetric varieties in either case (quadratic forms, \(2\in\cO^\times\)). This page is about the first.
Where characteristic \(0\) entersWhich steps of the proof use it?
- The standing assumption. [Sak, §1.1]: “a local non-archimedean field \(k\) in characteristic zero”. [SV, §1.1, §1.7] likewise.
- The structure theory. Brion, Luna, Vust and Knop's results on spherical varieties (the open orbit, \(\LevX\), the cone, \(\WX\), Knop's action) are used over fields of characteristic \(0\) [SV, Rem. 2.3.2].
- Orbit counts. Finiteness of \(G(F)\)- and \(B(F)\)-orbits on \(X(F)\) comes from finiteness of Galois cohomology, true for \(p\)-adic \(F\) (orbits). Over \(\Fqt\) it can fail: \(F^\times/F^{\times p}\) is infinite, because the principal units \(1+t\Fq[\![t]\!]\) form a \(\Z_p\)-module of infinite rank.
- The earlier paper. [Sak08], whose results on morphisms \(\Sw(X)\to I(\chi)\) Sakellaridis uses, is stated for local fields of characteristic \(0\).
- Triangularity. The colour combinatorics behind (7.5) uses Lemma 7.1.4 of [Sak], a characteristic-\(0\) statement.
None of these is a counterexample; in examples each has a characteristic-\(p\) substitute (next section).
What is known over \(\Fqt\)Has anyone proved the Theorem in characteristic \(p\)?
Published. No paper checked here states Theorem 8.0.2 over \(\Fqt\). [SW] work over \(\Fqt\) only, but prove a different statement (the unramified Plancherel density of the IC function, under \(\GX=\Gv\)); [BZSV, §9.4] expect \(C_c^\infty(X_F/G_{\cO})\simeq\C[\GX]^{\GX}\) “in the situations to which [Sak13] is applicable”, as a heuristic.
(transfer to \(\Fqt\)) [run sakellaridis-char-p, Thm. 4.8, Prop. 4.9; satake-two-levels, Prop. VI] Proved in this project (audited run, unpublished)
Let \(p\) be odd, \(F=\Fqt\), and \(X=G/G^\theta\) a symmetric variety over the Witt vectors \(W(\Fq)\) satisfying five conditions (T1)–(T5): \(G^\theta\) split with connected fibres; one \(G\)-orbit of points over \(F\) and over the test fields; \(\tau(x)=x\theta(x)^{-1}\) detects integral points; the Cartan decomposition over the test fields \(F'_e=\Q_q(p^{1/e})\); and the Theorem over the \(F'_e\). Then the Theorem holds for \(X\) over \(F\). For \(E_6/F_4\), (T1)–(T5) hold.
The idea (close local fields)
- \(\cO'_e/\vpi'^m\simeq\Fq[t]/(t^m)\) for \(m\le e\), \(\cO'_e\) the integers of \(F'_e\) [Deligne 1984; Kazhdan 1986].
- Membership in \(K\), in \(Kt^\lambda K\) and in Iwasawa cells is read off valuations of matrix coefficients in fundamental representations, so each count \(\#K\backslash Kt^\lambda X(\cO)\), each structure constant \((T_\lambda\ast\one_{Kx_\mu})(x_\nu)\) and each Satake coefficient is decided modulo \(\vpi^D\), with \(D\) independent of the field.
- Over \(F'_e\), \(e\ge D\), the Theorem holds (characteristic \(0\), any ramification); the images \(P_\kappa\) of orbit indicators depend only on \(q\).
- So \(\one_{Kx_\kappa}\mapsto P_\kappa\) is \(\HK\)-linear over \(F\), checked one pair \((\lambda,\mu)\) at a time against a field \(F'_e\) with \(e\) large.
(SG) directly, over every \(F\). Without the module isomorphism, (SG) is proved in any characteristic for \(\GL_{2m}/\Sp_{2m}\), \(\SL_{2n}/\Sp_{2n}\) ([satake-two-levels, Prop. IV]), \(\GL_n/(\GL_p\times\GL_{n-p})\) and \(\Spin_{2n}/\Spin_{2n-1}\) ([satake-two-levels, Prop. V]), \(\GL_2/T\) (the tree) and \(\Aff^n\) (𝔸ⁿ); and it fails for \(E_6/F_4\) over \(\Fqt\) as over \(p\)-adic fields (computing r).
Repairs found on the way. The same run found two gaps in [Sak]'s printed proof (a factor \(2\) in the derivation of (7.5), p. 62; nonzero leading coefficients of \(P_\lambda\) not proved) and repaired both, so the characteristic-\(0\) Theorem stands as stated.
Verdict
| \(X\) | over \(\Fqt\) | status |
|---|---|---|
| general homogeneous affine | open | Open no published proof; no counterexample |
| symmetric, (T1)–(T5), \(p\) odd | holds | Proved in this project by transfer |
| \(E_6/F_4\), \(p\) odd | holds; (SG) fails | Proved in this project |
| \(\GL\) families, spheres, rank one, \(\Aff^n\) | (SG) holds | Proved in this project directly |
Heuristic The transfer argument uses only that \(X\), its Cartan decomposition and the Theorem are defined uniformly over the test fields; this suggests the statement holds over \(\Fqt\) whenever \(X\) has such a uniform model over \(W(\Fq)\). Turning that into a theorem needs, for each \(X\), the characteristic-\(p\) orbit geometry of step 3 of the list above.