Theta dichotomy for the genuine unramified principal series of \(\widetilde{\text{Sp}}_2(F)\)
Let \(F\) denote a local nonarchimedean field with odd residual characteristic and let \(Mp(n,F)\) denote a metaplectic group of rank \(n\) over \(F\), i.e., a non--trivial central extension of the symplectic group \(Sp(n,F)\). There exist two quadratic spaces of dimension \(2n+1\) over \(F\) having the same discriminant. The discriminant of the pair is commonly denoted by \((-1)^{n} \kappa\) for some \(\kappa \in F^{\times}\). We then label these spaces as \(V_{\kappa}^{+}\) and \(V_{\kappa}^{-}\) while the corresponding orthogonal groups are denoted by \(O(V_{\kappa}^{+})\) and \(O(V_{\kappa}^{-})\). Let \(\psi\) be a fixed non--trivial additive character of \(F\). The theta dichotomy conjecture, proved in [\textit{C. Zorn}, Am. J. Math. 133, No. 5, 1313--1364 (2011; Zbl 1228.22019)] states that for a genuine admissible representation \(\pi\) of \(Mp(n,F)\) there exists exactly one space in \(\{ V_{\kappa}^{+}, V_{\kappa}^{-} \}\) such that the theta lift \(\theta_{\psi}(\pi, V_{\kappa}^{ \pm })\) is non--zero. Thus, we may define the dichotomy sign \(\delta(\pi, \psi, V_{\kappa}) \in \{ \pm 1 \}\) by \(\theta_{\psi}(\pi, V_{\kappa}^{ \delta(\pi, \psi, V_{\kappa}) }) \neq 0\). A purpose of this paper is to explicitly determine \(\delta(\pi, \psi, V_{\kappa})\) for representations \(\pi\) which appear as irreducible subquotients of the genuine principal series of the group \(Mp(2,F)\).
scientific article; zbMATH DE number 6005059
scientific article