Spherical level · 3 · the target

The Theorem, item by item

The Theorem in one box, one row per item with its value for \(\GL_2/T\), that example worked to the end, the structure of Sakellaridis's proof, and a dictionary to his conventions.

Sections · arrows: “is used by”

The statementWhat does Sakellaridis prove about \(\CK\)?

[Sak, Thm. 8.0.2 and Cor. 8.0.4(1)] Proved (cited), under (A1)–(A5)

Let \(F\) be a \(p\)-adic field (characteristic \(0\)), \(G\) split reductive over \(\cO\) with connected fibres and simply connected derived group, and \(X=G/H\) a homogeneous affine spherical \(G\)-scheme over \(\cO\) satisfying the hypotheses (A2)–(A5) of Hypotheses. Then there is a canonical isomorphism of \(\HK\)-modules

\[\Phi:\ \CK\xrightarrow{\ \sim\ }\C\big[\dXh\AXs\big]^{\WX},\qquad\Phi(\phiz)=1,\qquad\Phi(f\ast\varphi)=\Res(\Sat f)\cdot\Phi(\varphi),\]

where \(\Res:\C[\As]^W\to\C[\dXh\AXs]^{\WX}\) is restriction of functions. Consequently

\[\HK\ast\phiz=\CK\iff\Res\ \text{is surjective}.\]

(the consequence) Proved here

Given \(\Phi\), \(\Phi(\HK\ast\phiz)=\im\Res\); so (SG) holds iff \(\Res\) is onto.

Proof

\(\Phi(f\ast\phiz)=\Res(\Sat f)\cdot1\), and \(\Sat\) is onto \(\C[\As]^W\). So \(\Phi(\HK\ast\phiz)=\im\Res\), and \(\Phi\) is bijective. The direction “\(\Res\) onto \(\Rightarrow\) (SG)” is Cor. 8.0.4(1) of [Sak].

The itemsWhat is each symbol, and what is it for \(\GL_2/T\)?

Table 1. The Theorem dissected; the last column is the running example \(X=\GL_2/T\).
item	what it is	for \(\GL_2/T\)
(i) \(F\), \(K\)	\(p\)-adic field; \(K=G(\cO)\), hyperspecial (setting; why char 0)	\(K=\GL_2(\cO)\)
(ii) \(X\)	homogeneous affine spherical, with a model over \(\cO\) (geometry)	pairs of distinct lines
(iii) \(\CK\)	\(K\)-invariant Schwartz functions; basis \(\one_{Kx_\lambda}\), \(\lambda\in\LXp\) (orbits)	functions of \(d\ge0\)
(iv) \(\phiz\)	\(\one_{X(\cO)}\) (functions)	\(\one_{\{d=0\}}\)
(v) \(\HK\), \(\Sat\)	spherical Hecke algebra \(\simeq\C[\As]^W\) (Satake)	\(T_1\mapsto q^{1/2}(x_1+x_2)\)
(vi) \(\As\), \(W\)	unramified characters of \(T\); Weyl group (A*)	\((\C^\times)^2\), \(S_2\)
(vii) \(\AXs\)	unramified characters of \(\AX\), inside \(\As\) (A_X*)	\(\{(y,y^{-1})\}\)
(viii) \(\dXh\)	\(q^{-\rho_L}\), \(L=\LevX\) (δ(X))	\(1\) (\(\LevX=T\))
(ix) \(\WX\)	little Weyl group, acting on the translate (W_X)	\(y\mapsto y^{-1}\)
(x) \(\C[\dXh\AXs]^{\WX}\)	polynomial functions on the translate, \(\WX\)-invariant	\(\C[y+y^{-1}]\)
(xi) \(\Res\)	restriction from \(\As\) to the translate (computing r)	\(x_1+x_2\mapsto y+y^{-1}\), \(x_1x_2\mapsto1\)
(xii) \(\Phi(\phiz)=1\)	normalization; fixes \(\Phi\) given the module structure	\(\one_{\{0\}}\mapsto1\)
(xiii) compatibility	\(f\in\HK\) acts on the right side as multiplication by \(\Res(\Sat f)\)	\(T_1\) acts as \(q^{1/2}(y+y^{-1})\)

What each hypothesis buys, and what breaks without it, is on Hypotheses.

Worked example: \(\GL_2/T\)What is \(\Phi\) on each orbit indicator?

(\(\Phi\) for \(\GL_2/T\)) Proved here

Put \(w:=q^{1/2}(y+y^{-1})\). The map \(\Phi(\one_{\{d\}}):=P_d(w)\), with

\[P_0=1,\quad P_1=w-2,\quad P_2=wP_1-(q-1),\quad P_{d+1}=wP_d-qP_{d-1}\ (d\ge2),\]

is an isomorphism \(\CK\to\C[y+y^{-1}]\) of \(\HK\)-modules with \(\Phi(\phiz)=1\); it is the isomorphism of the Theorem. So (SG) holds.

Proof

\(\HK=\C[\Sat^{-1}(q^{1/2}(x_1+x_2)),Z^{\pm1}]=\C[T_1,Z^{\pm1}]\), and \(Z\) acts trivially on \(\CK\) and \(\Res(x_1x_2)=1\). So it suffices that \(\Phi(T_1\varphi)=w\,\Phi(\varphi)\).
Apply \(\Phi\) to the three formulas for \(T_1\one_{\{d\}}\) of the triangular example: they are the defining recursion of \(P_d\).
\(P_d\) has degree \(d\) in \(w\) with leading coefficient \(1\), so the \(P_d\) form a basis of \(\C[w]=\C[y+y^{-1}]\): \(\Phi\) is bijective.
\(\Res(x_1+x_2)=y+y^{-1}\), so \(\Res(\Sat T_1)=w\): the module structures match.
Here \(\Res\) is onto, so an \(\HK\)-linear map to \(\C[y+y^{-1}]\) is determined by the image of \(\phiz\): \(\Phi\) is the Theorem's isomorphism.

\[P_3=w^3-2w^2-(2q-1)w+2q,\qquad P_d=q^{d/2}\big(y^d+y^{-d}\big)+(\text{lower powers of }y).\]

The extreme weights \(y^{\pm d}\) of \(P_d\) are the labels \(\pm d\): this is the triangularity in Sakellaridis's proof (next section). The lab on functions computes \(\Phi(f)\) this way.

The structure of Sakellaridis's proofWhere does \(\Phi\) come from in general?

Spherical functions. For \(\chi\in\dXh\AXs\) in general position, an explicit \(\HK\)-eigenfunction \(\Omega_\chi\in C^\infty(X(F))^K\) is computed [Sak, Thm. 1.2.1, 4.2.2, 5.2.1], and under Statements 6.3.1 and 7.1.5 its values are rewritten with a quotient of L-values [Sak, Thm. 7.2.1].
The polynomials. \(P_\lambda(\chi):=\Omega_\chi(x_\lambda)/L_X^{1/2}(\chi)\) [Sak, (8.1)]; \(P_0=1\).
Equivariance. \(\one_{Kx_\lambda}\mapsto P_\lambda\) is \(\HK\)-equivariant for the Satake isomorphism, because pairing with eigenfunctions intertwines convolution with multiplication by Hecke eigenvalues.
Triangularity. \(\lambda\) is the lowest weight of \(P_\lambda\) for an order defined by a strictly convex cone \(\mathcal T\) with \(\mathcal T\cap\Vc=\{0\}\); so the \(P_\lambda\) are a basis of \(\C[\dXh\AXs]^{\WX}\) [Sak, proof of Thm. 8.0.2, p. 64].

The project's run sakellaridis-char-p, audited, found two gaps in the printed argument: the identity behind triangularity (7.5) on p. 62 is off by a factor \(2\), and the nonzero leading coefficients of the \(P_\lambda\) are used without proof. Both are repaired there (positivity of open-orbit integrals), so the Theorem stands as stated (Characteristic 0?).

What surjectivity of \(\Res\) buysBesides (SG), why would anyone want \(\Res\) onto?

[Sak, Cor. 8.0.4] Proved (cited)

If \(\Res\) is surjective: (1) \(\CK\) is a principal \(\HK\)-module generated by \(\phiz\); (2) every Hecke eigenspace in \(C^\infty(X(F))^K\) is at most one-dimensional, and \(\dim\Hom_G(\pi,C^\infty(X(F)))\le1\) for every irreducible unramified \(\pi\); (3) every nonzero \(H\)-invariant functional on such \(\pi\) is nonzero on \(\pi^K\).

Without surjectivity, generic multiplicity one can hold while multiplicity one fails at special points, exactly where \(\iota\) glues points (W_X). Sakellaridis's Example 8.0.1, \(X=\SL_2^3\backslash\Sp_4\times\Sp_4\), is such a case [Sak, p. 64]; \(E_6/F_4\) is another (computing r).

Dictionary to Sakellaridis's conventions

Table 2. These pages against [Sak].
here	[Sak]
\(F\), \(\cO\), \(\vpi\)	\(k\), \(\mathfrak o\), \(\varpi\); \(k\) assumed unramified over \(\Q_p\) “to avoid complicated formulas” (§1.1)
\(X=G/H\), left action, \(K\backslash X(F)\)	\(X=H\backslash G\), right action, \(X/K\); exchanged by \(x\mapsto x^{-1}\)
\(T\), \(I\)	\(A\), \(J\)
\(\HK\), \(\CK\)	\(\mathcal H(G,K)\), \(C_c^\infty(X)^K\) (smooth measures, identified with functions by an invariant measure, which exists as \(H\) is reductive)
\(x_\lambda\), \(\lambda\in\LXp\)	\(x_{\check\lambda}\), \(X\)-anti-dominant \(\check\lambda\) (§2.1)
\(\Res\)	the map (8.2)
(A1)–(A5)	§1.1, §1.7, Axioms 2.2.1, 2.4.1, 2.4.2, §7 opening, Statements 6.3.1, 7.1.5, §8 opening