On the cubic Shimura lift to $PGL(3)$: Hecke correspondences

TL;DR

This paper proves a new Fundamental Lemma relating Hecke algebras of $PGL_3$ and a cubic cover of $SL_3$, advancing the understanding of Shimura lifts and automorphic representations.

Contribution

It establishes an algebra isomorphism between Hecke algebras of $PGL_3$ and a cubic cover of $SL_3$, and demonstrates a matching of distributions crucial for a new global Shimura lift.

Findings

Established an algebra isomorphism of Hecke algebras.

Matched distributions on $PGL_3$ and its cubic cover.

Extended previous matching results for Hecke algebra units.

Abstract

In this paper we establish a new Fundamental Lemma for Hecke correspondences. Let $F$ be a local field containing the cube roots of unity. We exhibit an algebra isomorphism of the spherical Hecke algebra of $PGL_3(F)$ and the spherical Hecke algebra of anti-genuine functions on the cubic cover $G'$ of $SL_3(F)$. Then we show that there is a matching (up to a specific transfer factor) of distributions on the two groups for all functions that correspond under this isomorphism. On $PGL_3(F)$ the distributions are relative distributions attached to a period involving the minimal representation on $SO_8$, while on $G'$ they are metaplectic Kuznetsov distributions. This Fundamental Lemma is a key step towards establishing a relative trace formula that would give a new global Shimura lift from genuine automorphic representations on the triple cover of $SL_3$ to automorphic representations on $PGL_3$, and also characterize the image of the lift by means of a period. It extends the matching for the unit elements of the Hecke algebras established by the authors in prior work.