Spherical level · 5

Hypotheses (A1)–(A5)

Sakellaridis's standing assumptions are spread over §1.1, §1.7, §2, §7 and §8 of [Sak]. The spherical-generation workspace collected them as (A1)–(A5).

Sections · arrows: “is used by”

(A1) Field and groupWhy \(p\)-adic, split, simply connected?

Statement. \(F\) is a local non-archimedean field “in characteristic zero” (§1.1, p. 2), assumed unramified over \(\Q_p\) “to avoid complicated formulas”, the modifications being “trivial”; \(G\) is reductive, split, with connected fibres over \(\cO\) and simply connected derived group, with \(B\supseteq A\) fixed (§1.7, p. 6).

Used for. \(K\) hyperspecial, so the Satake isomorphism and the Iwahori subgroup are available; the simply connected derived group “causes no harm to generality” (pass to a cover). Characteristic \(0\) enters through the cited results of [Sak08] and the structure theory of spherical varieties; whether it is needed is the subject of Characteristic 0?.

Status. The unramified-over-\(\Q_p\) clause only fixes the additive character and the self-dual measure, which §1.7 already fixes with conductor \(\cO\); the project's run sakellaridis-char-p checked that the proof works for every finite extension of \(\Q_p\) (its Audit 4.7).

(A2) The integral modelWhich \(X(\cO)\), hence which \(\phiz\)?

Statement (§1.7, p. 7). \(X\) is a quasi-affine \(G\)-scheme over \(\cO\) with an open \(B_F\)-orbit \(\Xo_F\), such that \(\Xo:=X\smallsetminus\overline{X\smallsetminus\Xo_F}\) is smooth and surjective over \(\cO\) with special fibre homogeneous under \(B_{\Fq}\); \(X\) admits a \(G\)-eigenmeasure and \(\Xo\) a \(B\)-invariant measure.

Used for. A base point \(x_0\in\Xo(\cO)\) exists (§2.1), the basic function is \(\one_{X(\cO)}\) for this model, and the Mackey functionals are integrals against the measures.

Fails. The model matters: \(\GL_2\) acting on \(\Aff^2\) and on \(\Aff^2\smallsetminus\{0\}\) share their open orbit but have different \(X(\cO)\), and at the Iwahori level opposite answers (Iwahori examples).

(A3) Axioms 2.2.1, 2.4.1, 2.4.2Do orbits over \(\cO\) behave like orbits over \(\C\)?

Status. Sakellaridis: “always the case at almost every place if \(X\) and \(G\) are defined over a global field” (§1.7), proved in [Sak12]. At a given place they must be checked; for \(E_6/F_4\) the project's run proves (A3a) over every \(\cO\) with \(2\) invertible.

(A4) Spherical roots and coloursWhat makes the L-value formula, and hence triangularity, work?

Statement (§7 opening, p. 58). No spherical root of type \(N\); for every simple spherical root \(\gamma\) the Levi \(L_\gamma\) is classical; the simple spherical roots satisfy Statement 6.3.1; the set \(\Theta\) of virtual weighted colours satisfies Statement 7.1.5.

Used for. Theorem 7.2.1, the formula for \(\Omega_\chi(x_\lambda)\) with the L-value \(\beta\), from which the polynomials \(P_\lambda\) and their triangularity come (proof structure).

Status. Statement 7.1.5 is “certainly not true for every spherical variety. However, I conjecture that it is true if \(X=H\backslash G\) is (homogeneous and) affine” (p. 59). For his table of examples (p. 76) the groups are split with “convenient choices for the centers … so that the assumptions of Theorem 7.2.1 are satisfied”; rows 1, 2 and 12 are \(\GL_m\times\GL_n\backslash\GL_{m+n}\), \(\Sp_{2n}\backslash\GL_{2n}\) and \(H\backslash H\times H\). The spheres \(\Spin_{2n}/\Spin_{2n-1}\) and \(E_6/F_4\) are not in the table; \(E_6/F_4\) is checked in the project's run.

(A5) Homogeneous, affine, one open \(B(F)\)-orbitWhat if \(\Xo(F)\) splits into several \(B(F)\)-orbits?

Statement (§8 opening, p. 63). \(X\) is homogeneous and affine, and \(\Xo\) “has a unique \(B\)-orbit”, i.e. \(\Xo(F)\) is one \(B(F)\)-orbit. Then \(K\backslash X(F)\simeq\LXp\) and \(\AXs\subseteq\As\).

(connected stabilizers give (A5)'s last clause) Proved here

If the stabilizer \(B_x\) of a point of \(\Xo\) is connected, then \(\Xo(F)\) is a single \(B(F)\)-orbit.

Proof

\(\Xo(F)/B(F)\simeq\ker\big(H^1(F,B_x)\to H^1(F,B)\big)\). \(B_x\) is a connected solvable subgroup of the split \(B\): its maximal torus is split (a subtorus of a conjugate of \(T\)) and its unipotent radical is split, so Hilbert 90 and \(H^1(F,\mathbb G_a)=0\) give \(H^1(F,B_x)=1\).

(\(\SL_2/T\): (A5) and (SG) fail) Proved here

Let \(X=\SL_2/T\), \(q\) odd. The generic \(B\)-stabilizer is \(\{\pm1\}\), \(\Xo(F)\) has \(|F^\times/F^{\times2}|\) orbits of \(B(F)\), and \(\HK\ast\phiz\ne\CK\).

Proof
  1. Let \(u\in\cO^\times\) be a non-square, \(s:=\diag(u,1)\in\GL_2(\cO)\), and \(\sigma(gT):=sgs^{-1}T\) on \(X(F)=\SL_2(F)/T(F)\).
  2. \(s\) normalizes \(K\), \(T\) and every \(Kt^\lambda K\), and fixes \(x_0=eT\); so \(\sigma\) fixes \(\phiz\) and commutes with every Hecke operator. Every element of \(\HK\ast\phiz\) is \(\sigma\)-invariant.
  3. For \(\val x=-1\), the points \(y=\begin{pmatrix}1&x\\0&1\end{pmatrix}T\) and \(\sigma(y)=\begin{pmatrix}1&ux\\0&1\end{pmatrix}T\) lie in different \(K\)-orbits: \(k\begin{pmatrix}1&x\\0&1\end{pmatrix}\diag(a,a^{-1})=\begin{pmatrix}1&ux\\0&1\end{pmatrix}\) forces \(k=\begin{pmatrix}a^{-1}&xa^{-1}(ua^2-1)\\0&a\end{pmatrix}\), in \(K\) only if \(ua^2\equiv1\bmod\vpi\), impossible for a non-square \(u\).
  4. So \(\one_{Ky}\) is not \(\sigma\)-invariant, and \(\one_{Ky}\notin\HK\ast\phiz\).

For \(\GL_2/T\) the same \(\sigma\) is an element of \(K\) itself, which is why \(\GL_2/T\) generates and \(\SL_2/T\) does not. A second failure is \(\GL_n/\Orth_n\): \(X(F)\) has several \(G(F)\)-orbits (quadratic forms of different discriminant or Hasse invariant), and Hecke translates of \(\phiz\) stay in \(G(F)\cdot X(\cO)\).

Summary
Table 1. Each hypothesis, what it buys, and a variety where it fails.
hypothesisbuysfails for
(A1)Satake, Iwahori, [Sak08]\(\Fq(\!(t)\!)\) (open in general)
(A2)\(x_0\), \(\phiz\), measuresa model with bad reduction
(A3)orbits over \(\cO\)finitely many places, in general
(A4)L-value formula, triangularitytype \(N\) roots, e.g. \(N(T)\backslash\PGL_2\)
(A5)\(K\backslash X(F)\simeq\LXp\)\(\SL_2/T\), \(\GL_n/\Orth_n\)