The question at Oberwolfach, 2013
In May 2013 Friedrich Knop and Yiannis Sakellaridis ran the first workshop to bring experts on spherical varieties and on automorphic forms together [OWR13]. Sakellaridis's talk there, Satake transform and the L-function of a spherical variety (pp. 1477–1481), poses the question these pages are about and lists four ways the naive answer fails.
The workshopWhy were these two groups of people in one room?
The organisers' introduction gives the motivation in two sentences. Automorphic L-functions are hard to study directly. A common way in is “to take large enough – spherical – subgroups \(H\subset G\) and to consider period integrals”, which “are often equal to L-functions or special values of L-functions” [OWR13, pp. 1445–1446]. The structure theory of spherical varieties (Luna, Vust, Brion, Knop) and the dual group \(\GX\) of Gaitsgory–Nadler were expected to organize these phenomena. The talks that supply the automorphic motivation:
| speaker, page | content | here |
|---|---|---|
| Gan, p. 1451 | Plancherel decomposition of \(L^2(G(F))\), from Fourier analysis to the Langlands program | Satake |
| Prasad, p. 1457 | L-groups, local and global L-functions, distinguished representations; a uniqueness guess for \(\Hom_H(\pi,\C)\) that the audience corrected with \(\GL_3\subset\SO_7\) | Euler factors |
| Offen, p. 1474 | Jacquet's relative trace formula, the tool for studying periods | — |
| Sakellaridis, p. 1477 | the Hecke module of \(K\)-invariant functions: this page | below |
| Brumley, p. 1483 | Hecke's formula for torus periods of Eisenstein series (class numbers) and Waldspurger's formula (integer points on spheres) | periods |
The problemWhat exactly was asked?
The talk fixes a split \(G\) over \(\cO\), \(K=G(\cO)\), and a homogeneous spherical \(G\)-scheme \(X\) over \(\cO\). Under assumptions, the \(K\)-orbits on \(X(F)\) are indexed by a monoid of cocharacters (K-orbits). Then:
Problem 1. Consider the space \(C_c^\infty(X)^K\) (or another “nice” space of \(K\)-invariant functions); it has a basis indexed by \(\Lambda_X^+\) (the characteristic functions of these cosets). Describe its structure as a module for the spherical Hecke algebra \(\mathcal H(G,K)\). [OWR13, p. 1477]
In the notation here: describe \(\CK\) as an \(\HK\)-module. Sakellaridis's \(\Lambda_X^+\) consists of antidominant cocharacters, ours of dominant ones; the two labellings differ by a sign (the cone).
The first case: SatakeWhich classical theorem is an answer to Problem 1?
Take \(X=U\backslash G\), the basic affine space, with \(U\) the unipotent radical of \(B\) and \(A=B/U\). The Iwasawa decomposition \(G(F)=U(F)\,A(F)\,K\) labels the \(K\)-orbits by \(\Lambda_A\), and \(\one_{U\backslash UK}\) plays the role of \(\phiz\).
[OWR13, p. 1478, Lemma 2 and Thm. 1] Proved (cited)
- \(C_c^\infty(U\backslash G)^K\simeq\C[\Lambda_A]\) as a module for the Hecke algebra of \(A\), via \(a\mapsto a\cdot\phiz\) (normalized by \(\delta^{1/2}\)).
- The action map \(\HK\to C_c^\infty(U\backslash G)^K\simeq\C[\Lambda_A]\), \(h\mapsto h\ast\phiz\), is injective with image \(\C[\Lambda_A]^W\). This is the Satake isomorphism \(\HK\simeq\C[\As]^W\).
So the Satake isomorphism is itself a statement about what a basic function generates. Here \(\phiz\) does not generate over \(\HK\): \(\HK\ast\phiz\) is the \(W\)-invariants, a proper subspace of \(\C[\Lambda_A]\), and the torus action of \(A\) is needed for the rest. This is no contradiction with the Theorem, which requires \(X\) affine, while \(U\backslash G\) is only quasi-affine. See the Satake page.
The hoped-for squareWhat should the answer look like?
Langlands read \(\C[\As]^W\) as \(\C[\Gv]^{\Gv}\), invariant functions on the dual group (Chevalley restriction). Gaitsgory and Nadler attached to \(X\) a subgroup \(\GX\subseteq\Gv\). A function-theoretic analogue of their work “would suggest a commutative diagram” [OWR13, p. 1479, (1)]:
\[\begin{CD} \HK @>{h\,\mapsto\, h\ast\phiz}>> \CK\\ @V{\wr}VV @VV{\wr}V\\ \C[\Gv]^{\Gv} @>{\text{restriction}}>> \C[\GX]^{\GX} \end{CD}\]The model is Knop's Harish-Chandra homomorphism for invariant differential operators on \(X\) [OWR13, p. 1479, (2)]:
\[\begin{CD} \mathfrak z(\mathfrak g) @>>> \mathcal D(X)^G\\ @V{\wr}VV @VV{\wr}V\\ \C[\check{\mathfrak g}]^{\Gv} @>>> \C[\check{\mathfrak g}_X]^{\GX} \end{CD}\]By Chevalley restriction for \(\GX\) as well, the bottom row of the first square is \(\Res:\C[\As]^W\to\C[\AXs]^{\WX}\). The square therefore asks for the Theorem without the shift, and (SG) is the statement that its top arrow is onto. Labelled heuristic because the talk says: “(1) is not always true”.
Four complicationsWhat goes wrong with the square, and what replaced it?
“Instead of formulating a full theorem, which is not yet available, we describe several of the issues arising” [OWR13, pp. 1479–1480]. In order:
| # | the talk says | here |
|---|---|---|
| (1) | The bottom arrow needs a shift, as in Knop's case. For quasi-affine \(X\) it is trivial iff \(\PX=B\). A map \(\SL_2\to\Gv\) commuting with \(\GX\) explains it “in terms of Arthur parameters”. | the shift is \(\dXh\): δ(X) |
| (2) | \(\GX\to\Gv\) may have a finite kernel, or not be definable compatibly with multiplicities; then \(\C[\GX]^{\GX}\) “only takes care of a subspace” of \(\CK\). | \(\Res\) not onto, \(E_6/F_4\): computing r; \(\WX\) versus its normalizer: W_X |
| (3) | \(\one_{X(\cO)}\) “is not always the correct basic function”. One must pass to affine embeddings; when they are singular the basic function mirrors intersection cohomology. This links to normalized Eisenstein series, to extending the Rankin–Selberg method [Sak12], and to Ngô's L-functions from monoids [Ngô]. | non-homogeneous \(X\): L-factors, 𝔸ⁿ, beyond SV |
| (4) | “In some cases (1) is true”, citing [Sak]: “typically those cases are homogeneous affine varieties whose \(k\)-points have a unique open \(B(k)\)-orbit.” | the Theorem: item by item; one open orbit is (A5) |
Item (4) refers to [Sak, Thm. 8.0.2], then “to appear”. That theorem keeps the square, with \(\C[\GX]^{\GX}\) replaced by \(\C[\dXh\AXs]^{\WX}\), which builds in items (1) and (2). Item (3) is the non-homogeneous direction, where the Theorem says nothing and the \(\Aff^n\) tab supplies one case.
Where this goes
Why the answer matters for L-functions: Periods and L-functions. Then the geometry, starting from spherical varieties; the dated events are on History.