  This paper deals with the geometric local theta correspondence at the Iwahori
level for dual reductive pairs of type II over a non Archimedean field $F$ of
characteristic $p\neq 2$ in the framework of the geometric Langlands program.
First we construct and study the geometric version of the invariants of the
Weil representation of the Iwahori-Hecke algebras. In the particular case of
$(GL_1, GL_m)$ we give a complete geometric description of the corresponding
category. The second part of the paper deals with geometric local Langlands
functoriality at the Iwahori level in a general setting. Given two reductive
connected groups $G$, $H$ over $F$ and a morphism $\check{G}\times
\mathrm{SL}_2\to\check{H}$ of Langlands dual groups, we construct a bimodule
over the affine extended Hecke algebras of $H$ and $G$ that should realize the
geometric local Arthur-Langlands functoriality at the Iwahori level. Then, we
propose a conjecture describing the geometric local theta correspondence at the
Iwahori level constructed in the first part in terms of this bimodule and we
prove our conjecture for pairs $(GL_1, GL_m)$.
