On the cubic Shimura lift to : The fundamental lemma

  • Solomon Friedberg

    Boston College, Chestnut Hill, USA
  • Omer Offen

    Brandeis University, Waltham, USA
On the cubic Shimura lift to $\operatorname{PGL}(3)$: The fundamental lemma cover
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Abstract

The classical Shimura correspondence lifts automorphic representations on the double cover of to automorphic representations on . Here we take key steps towards establishing a relative trace formula that would give a new global Shimura lift, from the triple cover of to , and also characterize the image of the lift. The characterization would be through the non-vanishing of a certain global period involving a function in the space of the automorphic minimal representation for split , consistent with a conjecture of Bump, Friedberg and Ginzburg (2001). In this paper, we first analyze a global distribution on 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 . We then match the corresponding local orbital integrals for the unit elements of the spherical Hecke algebras; that is, we establish the fundamental lemma.

Cite this article

Solomon Friedberg, Omer Offen, On the cubic Shimura lift to : The fundamental lemma. J. Eur. Math. Soc. (2024), published online first

DOI 10.4171/JEMS/1503