History

Why the Theorem looks the way it does

Three lines meet in Sakellaridis's Theorem: harmonic analysis on \(p\)-adic groups, which gave \(\HK\simeq\C[\As]^W\); integral representations of L-functions, which made basic functions of vector spaces and matrices into L-factors; and the structure theory of spherical varieties, which supplied \(\AX\), \(\Vc\) and \(\WX\).

Timeline

1950 · Tate's thesis

Obstacle: Hecke's L-functions needed a uniform local theory.
Advance: Zeta integrals \(\int_{F^\times}\Phi(x)\chi(x)|x|^s\,d^\times x\) of Schwartz functions on \(F=\Aff^1(F)\); for \(\Phi=\one_{\cO}\), the integral is \(L(s,\chi)\).
Left open: Which other spaces carry such a “basic function”.
Example: \(\Aff^1\) under \(\Gm\): the case \(n=1\) of the 𝔸ⁿ tab.
Developed in: 𝔸ⁿ: the open-orbit model

1963–1971 · Satake, Macdonald

Obstacle: Describe the algebra of bi-\(K\)-invariant functions on a \(p\)-adic group, and its eigenfunctions.
Advance: Satake: \(\HK\simeq\C[\As]^W\). Macdonald: an explicit formula for the spherical functions.
Left open: The same for functions on \(G/H\) instead of \(G\).
Sources: [Satake 1963], [Macdonald 1971], [Cartier]
Developed in: Satake

1965–1980 · Iwahori, Borel, Casselman–Shalika

Obstacle: Spherical vectors are hard to compute directly.
Advance: Iwahori–Matsumoto: the Iwahori subgroup and the presentation of \(\HI\). Borel: representations generated by \(I\)-fixed vectors are \(\HI\)-modules. Casselman, Casselman–Shalika: compute \(I\)-invariant vectors first, then average to \(K\).
Left open: The method for general \(G/H\).
Sources: [Iwahori–Matsumoto 1965], [Borel 1976], [Casselman 1980]
Developed in: Iwahori level; Sakellaridis's method [Sak, §1.2]

1972 · Godement–Jacquet

Obstacle: Standard L-functions of \(\GL_n\).
Advance: Zeta integrals on \(\Mat_n(F)\); the basic function \(\one_{\Mat_n(\cO)}\) has Satake transform the standard L-factor.
Left open: A general theory of which spaces give which L-functions.
Sources: [Godement–Jacquet]
Developed in: 𝔸ⁿ: contrast with \(\Mat_n\), Iwahori status

1983–1996 · Luna–Vust, Brion, Knop

Obstacle: Classify embeddings of homogeneous spaces with an open Borel orbit.
Advance: Luna–Vust: coloured cones. Brion–Luna–Vust: \([\LevX,\LevX]\subseteq H\). Knop: the valuation cone \(\Vc\) is a Weyl chamber of the little Weyl group \(\WX\), and \(W\) acts on Borel orbits.
Left open: What these invariants mean for harmonic analysis.
Pictures: Cones of valuations in \(\LX\otimes\Q\).
Sources: [Luna–Vust], [BLV], [Kn91]
Developed in: Geometry tab

2006–2012 · Gaitsgory–Nadler, Sakellaridis

Obstacle: Which unramified representations have \(H\)-invariant functionals, and is there a dual group governing them?
Advance: Gaitsgory–Nadler: a dual group \(\GX\) from the geometry of the loop space. Sakellaridis (2008): \(X\)-distinguished unramified representations are subquotients of \(I_{\PX}(\chi)\), \(\chi\in\AXs\). Sakellaridis (2012): basic functions of affine, possibly non-homogeneous \(X\), and the generalized Cartan decomposition.
Left open: The structure of \(\CK\) itself.
Sources: [GN10], [Sak08], [Sak12]
Developed in: δ(X), orbits

2009–2013 · the Theorem

Obstacle: Compute the unramified Hecke module of a spherical variety.
Advance: Sakellaridis (arXiv 2009, Amer. J. Math. 2013): spherical functions with L-values, and \(\CK\simeq\C[\dXh\AXs]^{\WX}\) with \(\phiz\mapsto1\); Example 8.0.1, where \(\phiz\) does not generate.
Left open: Statement 7.1.5 in general; characteristic \(p\); non-homogeneous \(X\).
Sources: [Sak]
Developed in: the Theorem

May 2013 · Oberwolfach mini-workshop

Obstacle: Experts on spherical varieties and on periods of automorphic forms had not met as one group.
Advance: Knop and Sakellaridis's mini-workshop. Sakellaridis's talk states the problem “describe \(C_c^\infty(X)^K\) as an \(\mathcal H(G,K)\)-module”, reads the Satake isomorphism as its case \(U\backslash G\), proposes a square of algebras through \(\GX\), and lists four complications: the shift, multiplicities, the right basic function, and the cases where the square holds.
Left open: Item 3 of the talk: the basic function of a singular affine embedding.
Example: \(U\backslash G\), where \(\phiz\) generates only \(\C[\Lambda_A]^W\).
Sources: [OWR13]
Developed in: Oberwolfach 2013, Periods and L-functions

2012–2017 · Sakellaridis–Venkatesh

Obstacle: A general framework for periods of automorphic forms.
Advance: The invariants of every homogeneous quasi-affine spherical \(X\), wavefront varieties, boundary degenerations, the dual group with its \(\SL_2\), and conjectures on the spectrum of \(X\).
Sources: [SV]
Developed in: Sakellaridis–Venkatesh

2017–2024 · Sakellaridis–Wang, Ben-Zvi–Sakellaridis–Venkatesh, Shtotland

Advance: [SW]: IC functions of singular affine \(X\) over \(\Fqt\) as L-values. [BZSV]: relative Langlands duality for hyperspherical varieties, with \(\Aff^n\) as an example. Shtotland: \(\HI\)-modules of symmetric spaces and distinction of the Steinberg representation.
Left open: A Hecke-module statement for non-homogeneous \(X\); the theorem in characteristic \(p\).
Sources: [SW], [BZSV], [Shtotland 2025]
Developed in: 𝔸ⁿ status, Iwahori: symmetric varieties

2026 · this project

Advance: The spherical- and Iwahori-generation workspaces; the satake-two-levels run decides \(E_6/F_4\) (\(\Res\) misses one dimension) and proves (SG) for the \(\GL\) families and spheres over every \(F\); the sakellaridis-char-p run transfers the Theorem to \(\Fqt\) for symmetric \(X\) with (T1)–(T5).
Developed in: computing r, characteristic