On the structure of theta lifts of discrete series for dual pairs \((\text{Sp}(n), \text{O}(V))\)
Let \(F\) be a non-archimedean local field. Let \(\{ G_n\}_{n \in {\mathbb N}}\) be a tower of symplectic groups over \(F\), and \(\{ G'_n\}_{n \in {\mathbb N}}\) a tower of even-orthogonal groups over \(F\), or vice versa. Then for each pair \((n,m)\) of natural numbers, one has a dual reductive pair \((G_n,G'_m)\). Thus, from an irreducible representation \(\sigma\) of \(G_n\), one can construct a smooth representation \(\Theta(\sigma,m)\) of \(G'_m\). This representation, the maximal \(\sigma\)-isotypic subquotient of the oscillator representation \(\omega_{n,m}\), is called the \textit{full lift} of \(\sigma\) to \(G'_m\), and is known to have finite length. Let \(m(\sigma)\) denote the smallest number \(m\) such that \(\Theta(\sigma,m) \neq 0\). Suppose \(\sigma\) is discrete series. Then one of the main theorems of this paper determines a number \(m_{\text{temp}}(\sigma)\) such that whenever \(m(\sigma)\leq m \leq m_{\text{temp}}(\sigma)\), we have that all irreducible subquotients of \(\Theta(\sigma, m)\) are tempered, or are discrete series. Moreover, if \(m> m_{\text{temp}}\), then every irreducible subquotient of \(\Theta(\sigma,m)\) is a subquotient of a representation that is parabolically induced from data described in terms of \(\Theta(\sigma,m_{\text{temp}})\). If the residual characteristic of \(F\) is odd, then the Howe conjecture holds, meaning that \(\Theta(\sigma,m)\) either is zero or has a unique maximal proper subrepresentation, in which case we will denote the corresponding irreducible quotient by \(\sigma(m)\), and the previous theorem can be made more precise, generalizing a theorem on supercuspidal representations by [\textit{C. Mœglin, M.-F. Vignéras}, and \textit{J.-L. Waldspurger}, Howe correspondences on a \(p\)-adic field. (Correspondances de Howe sur un corps \(p\)-adique.) Lecture Notes in Mathematics, 1291 (Berlin) etc.: Springer-Verlag (1987; Zbl 0642.22002)]. Under the same assumption on the field \(F\), a further theorem shows that the full lift \(\Theta(\sigma,m)\) is irreducible when \(m(\sigma)\leq m \leq m_{\text{temp}}(\sigma)\), generalizing a corresponding result for supercuspidal representations. The proofs in this paper do not rely on the recent classification (due to Mœglin-Tadić) of the discrete series representations, which relies on a hypothesis that is yet to be verified in some cases.
scientific article; zbMATH DE number 5320434
scientific article