The Sakellaridis–Venkatesh invariants

• Field: “the group will always be split over the base field \(k\), and the base field will be of characteristic zero” [SV, §1.1, p. 1]; for the harmonic analysis, “a locally compact \(p\)-adic field \(k\) in characteristic zero” [SV, §1.7, p. 11].
• Variety: “We will assume throughout that \(X\) is homogeneous and quasi-affine” [SV, §2, p. 18]; quasi-affine is “not a serious” restriction.
• The structure theory of Brion, Luna, Pauer and Vust is used over “an arbitrary field in characteristic zero, as long as the group \(G\) is split” [SV, Rem. 2.3.2].

So the invariants on the Geometry pages are defined for every homogeneous quasi-affine spherical \(X\) in characteristic \(0\), a wider class than the Theorem's homogeneous affine \(X\). Page numbers are those of arXiv:1203.0039v4.

Symmetric varieties are wavefront; \(U\backslash G\) is not (its cone is everything), nor is \(\GL_n\backslash\SO_{2n+1}\) [SV, §1.9]. All the running examples are wavefront. For \(\GL_2/T\) and the group the cone is the image of the chamber by the torus page; for \(\GL_2/\SL_2\) and \(\Aff^n\smallsetminus\{0\}\) the cone is the whole line, and so is the image of \(\{\lambda_1\le\dots\le\lambda_n\}\), since \(\lambda_1+\lambda_2\), resp. \(\lambda_n\), takes every integer value on it. \(U\backslash G\) differs because there \(\LT\to\LX\) is an isomorphism, so the image of the chamber is a chamber. Wavefront is a standing hypothesis of [SV] from §5 on; Sakellaridis's Theorem does not assume it.

Under further axioms (GN2)–(GN5) there is a distinguished morphism \(\GX\times\SL_2\to\Gv\) [SV, Thm. 2.2.3]. Its torus part is the map \(\AXs\to\As\) of A* and A_X*, and its \(\SL_2\) part produces the translate \(\dXh\) (δ(X)). Examples from [SV]: \(T\backslash\SL_2\) has \(\GX=\SL_2\) [SV, Ex. 2.2.5]; \(\Sp_{2n}\backslash\GL_{2n}\) has \(\GX=\GL_n\), embedded by \(\chi\mapsto\diag(\chi_1,\chi_1,\dots)\) [SV, Ex. 14.2.2; Ex. 1.3.2].

[SV, Thm. 5.3.3] states the generalized Cartan decomposition under smooth integral models, \(K\) hyperspecial and \(x_0\in\Xo(\cO)\), and attributes it to [Sak12] following Gaitsgory–Nadler, with the symmetric case due to Delorme–Sécherre. To each face \(\Theta\) of \(\Vc\), [SV, §2.4] attaches a boundary degeneration \(X_\Theta\), the open \(G\)-orbit in a normal bundle of a toroidal compactification; it models \(X\) near infinity in the directions of that face, and is the basis of their asymptotics. Neither is needed for the Theorem's statement; both are in its proof's background.

The invariants are attached to the open \(G\)-orbit. An embedding, such as an affine \(X\) containing \(\Xb\), adds a coloured cone (Luna–Vust theory, [SV, Thm. 2.3.1, §2.3.7]); for affine \(X\), \(F[X]=\bigoplus_{\lambda}V_\lambda\) over a saturated submonoid of \(\Xc(X)\) [SV, §15.3]. Their harmonic analysis does not treat non-homogeneous \(X\): a search of [SV] (2026-10-19) found no discussion of \(\Aff^n\) or \(\Mat_n\) as spherical varieties.

No. [SV] import the unramified computations from Sakellaridis's paper: the unramified Plancherel formula [Sak, Thm. 9.0.1] in [SV, §16.3] and §17.3, the generalized Cartan decomposition [Sak, Axiom 2.4.1] and the L-value [Sak, Def. 7.2.3] in §17.3, the volume of \(X(\cO)\) [Sak, Thm. 9.0.3] in §17.6. There is no theorem describing \(\CK\) as an \(\HK\)-module, and the phrase “basic function” does not occur. The unramified spectrum is stated in the form of the Proposition on δ(X) (subquotients of \(I_{\PX}(\chi)\), \(\chi\in\AXs\)), under conditions that include “\(X\) is homogeneous affine … or Whittaker-induced” [SV, §16.3].

None of these states a Hecke-module theorem like the Theorem for \(\Aff^n\). For \(\Aff^n\) it can be proved directly, which the 𝔸ⁿ tab does. Checked 2026-10-19 against arXiv:1203.0039v4, 0905.4245v3, 2009.03943v3, 2409.04677v1, 1410.2312v2.