Formal degree and local theta correspondence; Quaternion case

Hirotala Kakuhama

Published 2020-12-08Version 1

In this paper, we determine a constant from the local analogue of Siegel-Weil formula when at least one unitary groups are anisotropic, and describe the behavior of the formal degree under the local theta correspondence of nearly equal rank for quaternion dual pairs over a local field of characteristic $0$. As an application, we prove the formal degree conjecture of Hiraga-Ichino-Ikeda and Gross-Reeder for the non-split inner forms of ${\rm Sp}_4$ and ${\rm GSp}_4$. These results extend a work of Gan-Ichino to quaternion dual pairs.