Semi-Lie arithmetic fundamental lemma for the full spherical Hecke algebra

TL;DR

This paper extends the arithmetic fundamental lemma to the semi-Lie setting for the full spherical Hecke algebra, providing explicit formulas and proving the first non-trivial case, advancing the understanding of the Gan-Gross-Prasad conjectures.

Contribution

It formulates a new conjecture for the semi-Lie version of the arithmetic fundamental lemma and proves the first non-trivial case, expanding the scope of previous conjectures.

Findings

Explicit formulas for weighted orbital integrals in specific cases.

Proof of the first non-trivial case of the semi-Lie arithmetic fundamental lemma.

Extension of the conjecture to the full spherical Hecke algebra.

Abstract

As an analog to the Jacquet-Rallis fundamental lemma that appears in the relative trace formula approach to the Gan-Gross-Prasad conjectures, the arithmetic fundamental lemma was proposed by Wei Zhang and used in an approach to the arithmetic Gan-Gross-Prasad conjectures. The Jacquet-Rallis fundamental lemma was recently generalized by Spencer Leslie to a statement holding for the full spherical Hecke algebra. In the same spirit, Li, Rapoport, and Zhang have recently formulated a conjectural generalization of the arithmetic fundamental lemma to the full spherical Hecke algebra. This paper formulates another analogous conjecture for the semi-Lie version of the arithmetic fundamental lemma proposed by Yifeng Liu. Then this paper produces explicit formulas for particular cases of the weighted orbital integrals in the two conjectures mentioned above, and proves the first non-trivial case of the conjecture.