Cocenter of $p$-adic groups, II: induction map

In this paper, we study some relation between the cocenter $\bar H(G)$ of the Hecke algebra $H(G)$ of a connected reductive group $G$ over an nonarchimedean local field and the cocenter $\bar H(M)$ of its Levi subgroups $M$. Given any Newton component of $\bar H(G)$, we construct the induction map $\bar i$ from the corresponding Newton component of $\bar H(M)$ to it. We show that this map is surjective. This leads to the Bernstein-Lusztig type presentation of the cocenter $\bar H(G)$, which generalizes the work \cite{HN2} on the affine Hecke algebras. We also show that the map $\bar i$ we constructed is adjoint to the Jacquet functor and in characteristic $0$, the map $\bar i$ is an isomorphism.