On the Cubic Shimura lift to $PGL(3)$: The Fundamental Lemma

TL;DR

This paper advances the understanding of a potential global Shimura lift from the triple cover of SL_3 to PGL_3 by analyzing orbital integrals and establishing the Fundamental Lemma, crucial for future trace formula applications.

Contribution

It provides the first analysis of orbital integrals related to the cubic Shimura lift and proves the Fundamental Lemma for the associated local orbital integrals.

Findings

Distribution on PGL_3 decomposes into orbital integrals.

Matching of local orbital integrals for spherical Hecke algebra elements.

Progress towards establishing a global Shimura lift via trace formula.

Abstract

The classical Shimura correspondence lifts automorphic representations on the double cover of $SL_2$ to automorphic representations on $PGL_2$. Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of $SL_3$ to $PGL_3$, and also characterize the image of the lift. The characterization would be through the nonvanishing of a certain global period involving a function in the space of the automorphic minimal representation $\Theta_{SO_8}$ for split $SO_8({\mathbb{A}})$, consistent with a 2001 conjecture of Bump, Friedberg and Ginzburg. In this paper, we first analyze a global distribution on $PGL_3({\mathbb{A}})$ involving this period and show that it is a sum of factorizable orbital integrals. The same is true for the Kuznetsov distribution attached to the triple cover of $SL_3({\mathbb{A}})$. We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the Fundamental Lemma.