Conjectural properties of the Schwartz space on an affine spherical variety

Let $X$ be an affine spherical variety over $k$ with trivial arithmetic multiplicity. Let $\mathcal S(X(\mathbb A_k))$ be a Schwartz space in the sense described in the source, let $\Phi_v^0$ denote its basic functions, let $\Lambda_X^+$ be the parameter space in the generalized Cartan decomposition, and let $E(\Phi,\omega,g)$ denote the $X$-Eisenstein series. Conjectural properties of the Schwartz space. There exists a Schwartz space $\mathcal S(X(\mathbb A_k))$ such that:

• The basic functions $\Phi_v^0$ factor through the map

$$\{G(\mathfrak o_v)\text{-orbits on }X_v^+\}\to\Lambda_X^+$$

from the generalized Cartan decomposition, and, as functions on $\Lambda_X^+$, equal the functions obtained by the function-sheaf correspondence from the basic sheaf of Gaitsgory and Nadler.

• For every $\Phi\in\mathcal S(X(\mathbb A_k))$, the $X$-Eisenstein series $E(\Phi,\omega,g)$, originally defined for sufficiently $X$-positive characters, admits a meromorphic continuation everywhere.

These properties are presented as very speculative assumptions intended to provide a framework unifying methods of integral representations of $L$-functions. The source does not state a resolution of these conjectures, so their status remains open.