Theta correspondence for \(p\)-adic dual pairs of type I (Q2043598)

Let \((G_n, H_m)\) be a \(p\)-adic dual pair, where \(G_n\) is a symplectic, orthogonal, metaplectic, or unitary group, and let \(\pi\) be an irreducible admissible representation of \(G_n\). The Weil representation \(\omega_{n,m}\) of \(G_n \times H_m\) contains a maximal quotient of the form \(\pi \otimes \Theta(\pi,m)\). The representation \(\Theta(\pi,m)\) is called the full theta lift of \(\pi\). If \(\Theta(\pi,m) \ne 0\), it possesses a unique irreducible quotient, denoted by \(\theta(\pi,m)\) and called the (small) theta lift of \(\pi\). The first occurrence index of \(\pi\) is the smallest \(m=m(\pi)\) for which \(\Theta(\pi,m) \ne 0\). In this paper, the authors find the first occurrence index \(m(\pi)\) and describe, in terms of their Langlands parameters, the small theta lifts \(\theta(\pi,m)\) on all levels. The problem was solved for tempered representations by \textit{H. Atobe} and \textit{W. T. Gan} [Invent. Math. 210, No. 2, 341--415 (2017; Zbl 1394.11044)]. In the paper under review, the authors consider the tempered part \(\tau\) of the standard module of \(\pi\). They use certain ladder-representations to measure the discrepancy between the occurrences of \(\pi\) and \(\tau\). The central technical part is an algorithm which starts with the standard representation of \(\pi\) and transforms it, in a finite number of steps, into another representation which also has \(\pi\) as the unique irreducible quotient. The algorithm is used to find the ladder mentioned above, and also to describe the lifts themselves.