Explicit induction principle and symplectic-orthogonal theta lifts

TL;DR

This paper develops an explicit induction principle for real dual pairs of type I and uses it to fully describe the local theta correspondence for pairs where p+q=4, simplifying calculations in this area.

Contribution

It introduces an explicit induction principle for dual pairs (O(p,q),Sp(2n,R)) with p+q even and applies it to explicitly determine the local theta correspondence when p+q=4.

Findings

Explicit induction principle formulated for dual pairs (O(p,q),Sp(2n,R)) with p+q even.

Complete description of local theta correspondence for p+q=4.

Elementary analysis on infinitesimal characters and K-types used in derivations.

Abstract

During the last two decades, great efforts have been devoted to the calculation of the local theta correspondence for reductive dual pairs. However, uniform formulas remain elusive for real dual pairs of type I. The purpose of this paper is twofold: to formulate an explicit version of induction principle for dual pairs $(O(p,q),Sp(2n,\mathbb{R}))$ with $p + q$ even, and to apply it to obtain a complete and explicit description of the local theta correspondence when $p + q = 4$. Our approach is very elementary by analysis on the infinitesimal characters and K-types under the correspondence.