On the structure of theta lifts of discrete series for dual pairs \((\text{Sp}(n), \text{O}(V))\)
type: Item
instance of: scholarly article
title: On the structure of theta lifts of discrete series for dual pairs \((\text{Sp}(n), \text{O}(V))\)
zbMATH Open document ID: 1153.22020
author: Goran Muić
published in: Israel Journal of Mathematics
publication date: 2008-09-03T00:00:00Z
review text: 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 t…
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 5320434
zbMATH Keywords: \(p\)-adic groups
zbMATH Keywords: representations
zbMATH Keywords: discrete series
zbMATH Keywords: theta lifts
zbMATH Keywords: Howe correspondence
reviewed by: Jeffrey Adler
MaRDI profile type: Publication
full work available at URL: s11856-008-0022-5
OpenAlex ID: W2034656045
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DOI: 10.1007/S11856-008-0022-5
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description: scientific article; zbMATH DE number 5320434