Transfer factors for Jacquet-Mao's metaplectic fundamental lemma

TL;DR

This paper extends the proof of Jacquet-Mao's metaplectic fundamental lemma to more general representatives using Shalika germs, broadening the lemma's applicability in orbital integral identities.

Contribution

It generalizes the previous fundamental lemma proof from diagonal to more general representatives using Shalika germs for Kloosterman integrals.

Findings

Extended the fundamental lemma to broader classes of representatives.

Utilized Shalika germs to facilitate the proof.

Confirmed the identity of orbital integrals in the generalized setting.

Abstract

In an earlier paper we proved Jacquet-Mao's metaplectic fundamental lemma which is the identity between two orbital integrals (one is defined on the space of symmetric matrices and another one is defined on the $2$-fold cover of the general linear group) corrected by a transfer factor. But we restricted to the case where the relevant representative is a diagonal matrix. Now, we show that we can extend this result for the more general relevant representative. Our proof is based on the concept of Shalika germs for certain Kloosterman integrals.