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 $G=\mathrm{P}\mathrm{G}\mathrm{L}(2)$ and $\mathrm{S}\mathrm{L}(3)$.