Theta dichotomy and doubling epsilon factors for Sp͠ n ( F )

Abstract

In this work, we aim to prove the theta dichotomy conjecture for dual pairs of type $(\widetilde{{\rm Sp}}(W),{\rm O}(V))$ with $\dim(V)$ odd. The crux of the proof relies on the analysis of the reducibility of a certain degenerate principal series representation of the group $\widetilde{{\rm Sp}}(W')$ having double the rank of $\widetilde{{\rm Sp}}(W)$. This analysis is carried out by way of a normalized standard intertwining operator which is also used to define an epsilon factor coming from the doubling integral construction of Piatetski-Shapiro and Rallis. In this paper, we also illustrate the connection between the doubling epsilon factor and the theta dichotomy conjecture.