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

Abstract

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.

20 pages

References in corpus (1)

• Cocenters and representations of affine $0$-Hecke algebras

Cited by in corpus (1)

• Jordan Decompositions of cocenters of reductive $p$-adic groups