Matching of Hecke operators for Exceptional dual pair correspondences

TL;DR

This paper establishes a functorial correspondence between spherical representations of dual pairs involving a G_2-type group within a split E_n group over a p-adic field, via matching Hecke operators on the minimal representation.

Contribution

It proves the matching of spherical Hecke algebras for dual pairs in split E_n groups, demonstrating functoriality for spherical representations involving G_2.

Findings

Matching of spherical Hecke algebras for the dual pair

Functoriality of the representation correspondence

Implications for Langlands program

Abstract

Let $\mathbf{G}$ be a split algebrac group of type $E_n$ defined over a $p$-adic field. This group contains a dual pair $G \times G'$ where one of the groups is of type $G_2$. The minimal representation of $\mathbf{G}$, when restricted to the dual pair, gives a correspondence of representations of the two groups in the dual pair. We prove a matching of spherical Hecke algebras of $G$ and $G'$, when acting on the minimal representation. This implies that the correspondence is functorial, in the sense of Arthur and Langlands, for spherical representations.