The local L-factor conjecture for the d4e2-Schwartz space

Let $G$ be a reductive group over the local field $d53d$, let $d4b3$ be a representation of the dual group, and let ${\mathcal{S}}_{\rho}(G(\sF))$ be the associated space of test functions. Write $\chi$ for the relevant character and let $\pi'$ denote the twist of an irreducible representation $\pi$ by the sign character.

Main conjecture. The following assertions hold: ${\mathcal{H}}(G(\sF))\subset{d4d2}_{\rho}(G(\sF))$; for every matrix coefficient $c(g)$ of an irreducible representation $\pi$, the integral

$$\int_G \phi(g)c(g)|\chi(g)|^s\,dg$$

for $\phi\in{d4d2}_{\rho}(G(\sF))$ converges for $\operatorname{Re}(s)\gg0$ and is rational in $q^s$; the resulting rational functions form a fractional ideal whose generator is $L(\pi',\rho,s)$; and for every suitable open compact subgroup $K$, the $K\times K$-invariant part of ${d4d2}_{\rho}(G(\sF))$ is generated by the functions $f_{\mathcal{F},\rho}$ attached to $K\times K$-equivariant irreducible perverse sheaves.

These properties are intended to characterize the local $L$-function through the proposed Schwartz space and to give a geometric generation statement. The paper proves the first three assertions for the spherical and Iwahori versions, while the full statement, including the generation assertion, remains conjectural.