Local theta correspondence and Galois periods

TL;DR

This paper investigates how Galois periods transform under the local theta correspondence for orthogonal and symplectic groups, providing explicit transfer maps and establishing key relations.

Contribution

It introduces explicit transfer maps and proves adjoint and relative character relations for Galois periods in the context of local theta correspondence.

Findings

Comparison of Galois period multiplicities under theta correspondence

Construction of explicit transfer maps for periods

Establishment of adjoint and relative character relations

Abstract

We study the behavior of Galois periods under the local theta correspondence for even orthogonal and symplectic groups. Specifically, we compare their multiplicities and construct explicit transfer maps. Furthermore, we establish both an adjoint relation and a relative character relation for these periods.