Explicit induction principle and symplectic-orthogonal theta lifting

Xiang Fan

Published 2014-07-10, updated 2015-07-28Version 3

First, an explicit version of induction principle is formulated to compute the local theta correspondence for $(O(p,q),Sp(2n,\mathbb{R}))$ with $p+q$ even: when $p+q\leqslant2n$, the Langlands parameters of the theta $(n+k)$-lift of a representation of $O(p,q)$ is read off from the parameters of its theta $n$-lift, if the $n$-lift is nonzero; similarly when $p+q\geqslant2n+2$, a nonzero theta $(p,q)$-lift of a representation of $Sp(2n,\mathbb{R})$ determines its theta $(p+k,q+k)$-lift explicitly. Secondly, after reducing computations by our explicit induction principle, a complete and explicit description of the local theta correspondence is obtained for all the dual pairs $(O(p,q),Sp(2n,\mathbb{R}))$ with $p+q=4$. Our strategy is to determine the theta lifts under consideration by their infinitesimal characters and lowest $K$-types.