Formal degrees and the local theta correspondence: the quaternionic case

TL;DR

This paper investigates the behavior of formal degrees in the local theta correspondence for quaternionic dual pairs, determining key constants and proving a conjecture for certain non-split inner forms of symplectic groups.

Contribution

It determines a crucial constant in the local Siegel-Weil formula and proves the formal degree conjecture for specific non-split inner forms of symplectic groups.

Findings

Determined a constant in the local Siegel-Weil formula.

Described the behavior of formal degrees under local theta correspondence.

Proved the formal degree conjecture for non-split inner forms of Sp_4 and GSp_4.

Abstract

In this paper, we determine a constant occurring in a local analogue of the Siegel-Weil formula, and describe the behavior of the formal degrees under the local theta correspondence for quaternionic dual pairs of almost equal rank over a non-Archimedean local field of characteristic $0$. As an application, we prove the formal degree conjecture of Hiraga-Ichino-Ikeda for the non-split inner forms of ${\rm Sp}_4$ and ${\rm GSp}_4$.