Langlands parameters for Moy-Prasad types

Suppose $G$ is a tamely ramified $p$-adic reductive group. We construct a partial local Langlands correspondence between the set of irreducible smooth representations of $G$ having depth $r$ and a certain set of $G^{\vee }$-conjugacy classes of continuous homomorphisms $\phi :I_F^r\to G^{\vee }$. Here $G^{\vee }$ is the dual group of $G$, and $I_F^r$ is the $r^{\text{th}}$ upper-numbering filtration subgroup of the inertia subgroup $I_F$.