A fusion construction of local L-factors

We propose a new conjectural way to calculate the local $L$-factor $L=L_\chi(\pi,\rho,s)$ where $\pi$ is a representation of a $p$-adic group $G$, $\rho$ is an algebraic representation of the dual group $G^{\vee}$ and $\chi$ is an algebraic character of $G$ satisfying a positivity condition. A method going back to Godement and Jacquet yields a description of $L$ using as an input a certain space ${\mathcal S}_\rho$ of functions on $G$ depending on $\rho$. A (partly conjectural) description of ${\mathcal S}_\rho$ involving trace of Frobenius functions associated to perverse sheaves on the loop space of a semigroup containing $G$ was developed %by Bouthier, Ngo and Sakellaridis, partly based on an earlier work of Braverman and Kazhdan. Here we propose a different, more general conjectural description of ${\mathcal S}_\rho$: it also refers to trace of Frobenius functions but instead of the loop space of a semi-group we work with the ramified global Grassmannian fibering over the configuration space of points on a global curve defined by Beilinson-Drinfeld and Gaitsgory (a relation between two approaches is discussed in the appendix). Our main result asserts validity of our conjectures where $\pi$ is generated by an Iwahori fixed vector: we show that in this case it is compatible with the standard formula for $L$ involving local Langlands correspondence which is known for such representations $\pi$. The proof is based on properties of the coherent realization of the affine Hecke category.