Hecke action on tamely ramified Eisenstein series over $\mathbb{P}^1$

We study the space of automorphic functions for the rational function field $\mathbb{F}_q(t)$ tamely ramified at three places. Eisenstein series are functions induced from the maximal torus. The space of Eisenstein series generates a trimodule for the affine Hecke algebra. We conjecture a generators and relations description of this module and prove the conjecture when $G=\mathrm{PGL}(2)$ and $\mathrm{SL}(3)$.